CONTRIBUTIONS TO STATISTICAL THEORY
OF LIFE TESTING AND RELIABILITY

Thesis for the Dogma of Pk. D.
MICHIGAN STATE UNIVERSITY
Safya Deva Dubey
E960

0-169

Date

This is to certify that the

thesis entitled
CONTRIBUTIONS TO STATISTICAL THEORY

OF LIFE TESTING AND RELIABILITY

presented by

Satya Deva Dubey

has been accepted towards fulfillment
of the requirements for

 

Ph. 0. degree in Statistics

 

Major professor

‘./

May 19, 1960

 

 

LIBRARY

Michigan State
University

 

CONTRIBUTIONS TO STATISTICAL THEORY OF LIFE TESTING
Am RELIABILITY -

BY

SATYA DEVA OUOEY

A THESIS

Sub-itted to the School of Graduate Studies of
Hichigan State University of Agriculture and
Applied Science in partial fulfiliaant of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Depath of Statistics

I960

Satya Deva Dubey
Candidate for the degree of
Doctor of Philosophy

Final exenination, Hey ll, I960, 2:00 P.H., Physics-Mathematics
Building

Dissertation: Contributions to Statistical Theory of Life Test-
lng'and Reliability

Outline of Studies

lIaJor subjects: Probability, ltathanatlcal and Applied Sta-
tlstics
Itinor subjects: Hethenatics .

Biographical Itees
Born, February l0, I930, Sahara Bazld, India
Undergraduate Studies: Patna University, ISM-5i

Graduate Studies: indian Statistical Institute, Calcutta
”SI-53; Carnegie Institute of Technology, Pittsburgh, Pa.,
Fall, l956; Michigan State University, l957-60.

Experience: Technical Assistant, Indian institute of Technology,
Kharagpur, India, l953-56: Graduate Assistant, Carna-
gie institute of Technology, Fall l956; Tanporary
Instructor (Feb. to June l957), Full-tine Research
tiorlter (Si-er l957) and Teaching and Special Gradu-

, ate Research Assistant 0957-60) Michigan State Uni-
versity, Editorial Collaborator of the Journal of
suggest Statistical Association (Dec. i957 s March
l .

Founder “or of Indian Society of Theoretical and Applied ite-
chanlcs, I955: ‘full member of Society of Sign Xi, nae-bar of
Institute of Mtheutlcal Statistics.

 

 

ACKMULEOGEHEIITS

I wish to express ay deep gratitude to Professor
Kenneth J. Arnold for his expert supervision,
valuable criticisms and active interest throughout
this investigation. I an also grateful to Profes-
sor Leo hat: for his encouragement at an early
stage of this work and to Professor Fritz Herzog
of the Departunt of Hathenatlcs for helping ne

in the proof of Theoren' 2 of Chapter ll. Hy
sincere thanks are due to hrs. Jane Joyaux for

the excellent typing of the nanuscript. Finally,
I an indebted to the Office of National Science

Foundation for financial support.

 

D E O I C A T E O

T 0

My Beloved Parents

 

CONTRIBUTIONS TO STATISTICAL THEORY OF LIFE TESTING

AND RELIABILITY

BY
SATYA DEVA DUBEY

AN ABSTRACT

Submitted to the School of Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics
i960

' 7./ .‘fl, "I :'/:’,”. //' "
Approved ”/1"! “I “(J/M {/1711' x/

 

54

55

91
92

11b

Line

7

3 (from
below)

16 (nth
from
below)

Errata Sheet

Printed

location
6‘
Student's t test

ti and t3

Involing

-1 1
“393765)
1 .1 3
are)

with

Read

scale

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Student's t

t1 and tr

Invoking

531893-59

[i- r<i> ]3

which

 

C‘-

SATYA OEVA OUBEY . ABSTRACT

Several problems in testing hypotheses about and in estimating para-
meters of Heibull distributions, particularly the exponential, are con-
sidered. Some attention is given to the possibility that simple assump-
tions about the intensity function will lead to classes of distributions
of wide applicability in describing distributions of length of life.

In the case of the exponential failure law with known location para-
meter, the minim-variance-slngle-observation-unbiased estimator of the
location parameter is Investigated. It is found that if the rth ob-
servation in order of increasing time is the single observation on nhich

this estimate is based and if we write r e "S" , then

llm Snug-30.8 .

n ---> (D

Several tests of parameters are developed for the case in which no
observations beyond the rth in order of magnitude are used. than the
scale parameter is know, the likelihood ratio test that the location
parameter is a given value is unifome most powerful against all altern-
atives. then the scale parameter is unknown, the likelihood ratio test
that the location parameter is a given value is uniformly most powerful
unbiased. For the latter situation a simple test function based on the
first and rth observation Is prOposed. This test function Is shown
to be unbiased and for the left-sided alternatives the power of the like-

lihood ratio test and of the simple test function is show: to be identical.

 

SATYA OEVA OUBEY ABSTRACT

For a simple hypothesis on the location and the scale parameters the
test function derived by means of the layman-Pearson lemma is shown to
be uniformly most powerful against alternatives confined to the south-
west quadrant. A uniformly most powerful unbiased test for the scale
parameter is derived for the case in which the location parameter is
unknown and for the case in which it is known a similar test function
of first r observations is suggested. The power functions for-these
tests are expressed in terms of the Incomplete B‘s Function. Two
slaple test functions for testing the hypothesis on the scale parameter
than the location parameter is know and unknown respectively are pro?
posed and their power functions are derived.

Some results are extended to two sample problems. For the likeli-
hood ratio test based on the first rI (s n') and r2 (5 n2) ob-
servations to test the hypothesis on the equality of two location para-
meters assuing the same but unknown scale parameter, the power function
is derived and it is shout that the test is unbiased.

The percentile estimators for the parameters of the exponential laws
are derived for various situations. The choices of the cumulative prob-
abilities are made so that we have minimum variance unbiased percentile
estimators for the estimators. The asymptotic results are given for the
sampling distributions, the means, the variances and the coverlance of

the unbiased percentile estimators.

 

SATYA OEVA OUBEY ABSTRACT

The moment-recurrence formulas for the Heibull laws are established
and the moment estimators of Heibull parameters are derived through them.
The percentile and the modified percentile estimator for these parameters
are derived explicitly and by using the reliability and the intensity
functions other estimators are obtained.

Starting from the intensity function, a large nunber of potentially
useful failure laws are generated and the estimation of the parameters
is considered. Finally the applications of some of these failure laws

are pointed out.

 

IV.

V.

VI.

CONTENTS

INTRODUCTION

SONE RESULTS RELEVANT TO EXPONENTIAL FAILURE LAN

SONE TESTS ON PARAMETERS OF EXPONENTIAL FAILURE LAN

PERCENTILE ESTIMATORS FOR PARAMETERS OF EXPONENTIAL
FAILURE LAN

NEIBULL FAILURE LAWS

INTENSITY FUNCTION: GENERATOR OF FAILURE LABS

BIBLIOGRAPHY

3O

73

II2

I20

 

I. INTRODUCTION

9%
Epstein [23] has derived simle estimators of the parameters of
exponential distributions those probability density functions (p.d.f.)

are
( J:
9 \
JC<X-9):,/—Lfi 2 X703 670
2 ‘ 6 (l)
i o , otherwise
\
and , V76
. J“ _,-R G i” .5 F ./ ‘a/ LN”? Cr 7 O
‘f‘, . f“ .'f. 'a '- \ /, I
j\"') "i “7/"; (2)

i0 , otherwise
k

ahen sales are censored. There he is led to investigate the proper-
ties of an unbiased estimator for O which involves only the r(§ n) th
observation of the sample of size n drawn from (I). Denoting the

rth observation by x" n , his unbiased estimator of O is given by
3

. :- ‘ more P\ ‘1 , " '-
rBTL fihv'r. ‘7 I‘ 3‘ r'

F ) ‘ k 'i.‘ i “y
z.--“— "I\- \.+ I
‘ /

W 4

t.
i

From [23] we find that in [No] he has shove: that Or n is of very high
P

efficiency (>96 percent) if :k :7: ,[ [>90 percent if ég...
- 1‘“ J1 - [Y‘I

ehen cowared with the best estimator 8r n more
1

LL!

 

9k lhlmbers in parentheses refer to the bibliography.

 

‘r;
A. _ 23cm "I" (“f-hfifihm

. ——_—.—_— ‘s-v— __—-e .

 

. (*3.
7Z7“ )‘v

(The reference [Ill] was not available to this author.)

Here we have considered the problem of finding the most efficient
single-observation estimator of O . The results concerning this have
been investigated in Chapter II. those smary we give below.

In the second chapter we show that the smallest sanle observation
in case of the l-paramater exponential failure. law provides a worse as-
timator (in the sense of minimum variance unbiased estimator) among
single-observation estimators for average life than any one of the
(n-l) remaining swle observations in a sale of size n . it
fol lows from [23] that the largest samle observation, up to the eagle
size five, provides the best estimator for average life in the same
sense. llare we show that a single-observation unbiased estimator for
average life based on the rig n) th statistic were Q1 1-“an with
[513" gm 90 :2: 0. <3
possesses minim:- variance. It is about 66 percent efficient In com-
parison with the minimum variance unbiased estimator based on all ob-
servations in the sample. The saqle median has only '08 percent effi-

 

ciency. The smallest sample observation ls—%9~ (n, sawle size) percent
1
efficient and the largest sanla observation has fiélin (*0) asmtotlc

efficiency. Since the life test data are naturally ordered we have 9,;
£29223 random variables to work with. in this connection we have fond

 

-3-

the product mount correlation coefficient to order (n e 2) 4 between

any ith and jth (i <1) ordered sanle observations from exponen-

tial population. This correlation is

‘J tJC- :— 'L(m‘!+‘i_ I-— I J. “H
{< H 3‘) Jézfln—LH) B Q(’Y\+Z) (”“84“ 'A'H‘ j
It is asymptotically equal to j“ TUTTI)“. .
)rC’fi-L‘l i)‘

Epstein and Sobel have derived the best test for the it: D e D

 

 

l
against A: 0 e 02 (< 0') based on the first r(§ n) observations

out of a saamle of n draws from (i) in [i]. In the first part of
Chapter III . we have considered tests for the hypotheses:

i) ll: 0-00 against A: Odbo, C knot-m

ll) it: 0-00 against A: 0400, 6 unknown.

ill) ll: 6-60 against A: Cdfio, O luioam,

and
iv) ll: 6-60 against A: 6460, 0 unknown.

Paulson [Ill and Lelvaann [2] have considered these hypotheses under the
asswtion that all n senile observations are available. llare we have
extended the results of Paulson and Lehman for all the cases. The ex-
tension consists in the fact that our tests are based on only the first
r(5 n) ordered observations from a sawle of size n . Furthermore

we have also considered v) ll: 6‘ a 62 against A: C. d 62 and

 

assuming the same but mimosa: scale parameter. Paulson [it] has consid-
ered this hypothesis under the assuqtlon that all the sample ob-
servations are available. Epstein and Teen [7] have derived the reduced
likelihood ratio test when samples are ansored from the right. tiara
following Paulson [II] we have derived the power function of this test
and have shoe: that the test is unbiased. For the tests concerning the
hypotheses i), ii), iii) and iv) we have derived the power functions
and have investigated sue of their properties. Following Lehmann [2]
we have obtained the miformly most powerful (W) unbiased test for the
hypothesis ll) when sample is censored from the right. in the case of
the hypothesis lil) we have shown that the likelihood ratio test is ll!i
against all alternatives and for the hypothesis iv) we have show, by
following Lehmaltn [2], that the corresponding likelihood ratio test is
ID. unbiased test. Furthermore we have pointed out that the best test

for the II: D - 0 against A: 0 e 02 (< 0') , considered by Ep-

l
stelnandSobel [l], lsll'for the li: 6-60 and 0-00 against
A: C<¢o and D<Do. Thistestisalsolltforthe ll: 0200

against A: O<Oo with known 6 .

Epstein and Sobel have considered a test based on the rth observa-
tion only to test ll: 0 e 0' against A: D :- 02 (< 0') in [i]. This
has led us to consider simple test functions for the hypotheses i), ii),
and iv) in the secoltd part of this chapter. Since the test function for
the hypothesis iii) is based on the first observation alone and is W
we have considered simple test functions for the remaining three cases.

liven guarantee time (location parueter) is lotown, the proposed slamle

 

-5-

test function to test the hypothesis about ,9 , the scale parameter, is
based on only the rth observation of the sample. linen guarantee time
Is nnnnlononaa, a corresponding simle test function ls based on the first
and the rth observations. For both cases the power functions have
been derived and have been reduced to the lncoqlete Data Function.
linen the scale parameter is unknown a siwle test function based on the
first and the rig n) th observations has been suggested to test hy-
potheses on guarantee time. its power function has been derived adnich
shows that the test is unbiased and for the left-sided alternatives its
power function and the power function of the corresponding likelihood
ratio test is identical. Two moment-recurrence formulas have been es-
tablished to comute hiyner moments of this siqle statistic.

In Chapter N. we have made an extensive study of the percentile
estimators for both the location and the scale parameters of the expo-
nential failure law in three different cases. Sue of the results of
this report are extensions of the results of Chapter II. lie have derived
A the sampling distributions of the percentile estimators; have derived
‘ their asymptotic distributions, have given expressions for their kth
moments and have made choices for cuulative probabilities such that the
corresponding percentiles in first two cases insure minimum variance
single-observation unbiased percentile estimators provided the sample
size is large. For the third case we have given the asymtotic form of
the covariance matrix.

Kao III] has derived the maximnnm likelihood estimators (m.l.e.) of
0 and m for the Z-parameter Uaibull law then a randm sale is

 

 

censored from the riynt. ills likelihood equations demand the use of
successive approximations. Duggan [l3] has obtained moment estimators
for all the three parameters of the liaibull law which requires use of a
table especially prepared for this purpose. Representing the p.d.f. of
the 3-parameter lieibull law by

WI

7: ’i 2134-3090261

9 ,
91m 6 x 0,90),

x-I _._L(+__(
, (gm-GA- 6 v ”J
int): w

L0 , otherwise,
we have presented in Chapter v. the results listed In the following
paragraph.

Fl rst we show that 6' (measure of skewness) and D2 (measure of
kurtosis) for the llaibull laws are functions of m , the shape para-
meter, only. Then we establish a lane which reveals the relationship
between the rth moment and the rth power of the first moment of the
liaibull law. This lama is used for deriving muent estimators for the
Helbull parameters. lie obtain percentile and modified percentile esti-
Imators for the lielbull parameters in the form of formulas. By means of
the reliability function and the intensity function we have derived
some other estimators for the Ueibull parameters as wall.

in the sixth chapter, a large nuaber of failure laws have been gen-
erated by various reasonable assmtlons about the form of the intensity
function. The applications of some failure laws, generated in this man-
ner, have been pointed out and the estimation of parameters of such
failure laws has been considered.

II. SONE RESULTS RELEVANT TO EXPONEHTIAL FAILURE LAN

in this chapter we shall derive some results of interest in life
testing problems where the random variable has the following exponen-
tial probability density function

——,1-(x-—G) p
S" be C’ / )X_7(nE(-c~9/>0)¢r
/

: ӎ ,
(X 0 C {0, 00)

0 , otherwise.
lie shall assnaae C to be known throughout this chapter and since we are
concerned with life test data in time units we shall write t instead
of X - C and reduce the above exponential probability law to the one-

parameter exponential probability density function (p.d.f.) ndnose form

is given by

Wit
(Jé‘fi 9 ,‘t70

-}(t)=<>[/

/'

0 , otherwise.

liow we proceed to prove' the following slqnle theorem.

Theorem i: For the above one-parameter exponential failure law, the max-
lmn- of samle observations provides a more efficient estimator of average

life than its minimum.

PM: Let (t', t2, ..., tn) be a sample of size n . It does not
matter it». ndnether they are ordered or not. Hanover, it is clear that
in the life testing situations our observations will always appear order-
ed. Let E emax(t', t2, ..., tn) and ”l eminitl, t2, ..., tn) .
lbw the p.d.f. of g is given by

-3 “E m—i
§<f):0{%€ E {3—6 9) , E20

0 , otherwise.

 

This gives
’1”: I
T“ /m-\ If I
K
E? “ ”“9 VOW) Z ' )W ' KTH
:0 ? (Ci/H)
Hence',

 

 

"°°' "‘""‘ (1‘1)=ZO(;’)(-u)x

 

 

Therefore
\ ( )m m m i ‘ i
- \—-x __ -l
I. :- Z ( (3’ >( I) x for I #0
=\
“QC - :2 CI“X) for 1’. if}

\ , "0'
. m 0" . I ‘

\ 3)-! Jpn ...1.
~ §§7W 1 *1 g— a

llancethe lane is proved.

’Yl , m .\
Returning to E: :1 9 {—12 ,we see that §[ZJE~] is
K=I K—I
an unbiased estimator of O . its variance is found to be

2.
z_l‘::l VINE) . An expression for Var (E?) is given in
K=l ’

the fol lowing leu.

Lane 2: If tr is the rth ordered sawie observation out of a random

 

sale of size n(l f r 5 n) drawn frona the above one-parameter expo-
nonetial law then the

\thfit 9 22*: 7‘2 '

Proof: The p.d.f. of tr is given by

 

 

_ _. I _, . t t
—(’n—}z+t)—-7L , « ”‘- 91"
mi, 9 __ ‘6‘ 0
flittiaiihwznn—afféfi (’ ”g ) Q7

0 , otherwise.

For the sake of convenience we drop subscript r from t . The charac-

teristic function of the above p.d.f. is given by

no): Mm -= 1%me (1f

. t ._,
.. m] SOC —(/V!—?L'¢“+V'§ @i-Efaéfo
(WWW-9 0 f
To integrate out the above integral, wring: ?, then
00
Lil? MI -(rn—}L‘H"C‘l)yr d} 91.}
Be :: ___..._-.-_--,....._.... g—e l—c J v .
y _} Cit—DICTI-fi)! O C j (1*
Let“; : 2 than
I
0i ”I! ’n— )L—C‘L fi.‘
R 2W @371) z 0- 2) o
0
Since the real parts of (n-rel-iu) and r are positive, the above in-

tegral is a late function whose argn-ants are n-Nl-iu and r respec-
tively ([22], p. 2l2). Thus

$1M): Egcu}:~ji.mm_ BC’l’l-hil-CLL)‘ it)

-32-
where any branch can be chosen.
The rth cnuulant is given by kk: (4)

10% m / .... __
duh. ‘qu'O

M _ OLZWMIOL) __ ~ .
leT W“— [120 iéi-Mflz Vmght9 limb

2 ...,...— .
TI'POTOI’O, VM tn :2. 9 \‘_‘ :Q‘h+g)z
J-

Q.E.O.

4M9}; (L) :Z’Lmlh-t

 

I

Tiling/3 E”

incidentally K :1 (- U

 

O

I

 

:_ E % wilt/Iv Err/vow Eli/L :: 9 [mt-)1?

UTE

, the maximnmn of sale observations is t" ndnen sawle observations

m
\ + Z
are ordered, therefore, Veg-LE = V032, gm 2

8
m F‘ T

incidentally, EE 2: {ft/71:9 Z—r“ which cheeks with the direct

i“ i

derivation of the expectation of the maxim. of sample observations.

,... new 975?- é 92'

The p.d.f. of 0L, the minlmnm of sale observations is given by

 

liance {501.1% OMOL Vanni:

The unbiased estinator of 0 based on the nlninun saqile observation is
n02- idiose variance is 02 .

M

£1

Thus we see that Ver<f >5 Verb?) idiich l—adiately proves

'

the theoran.
The following lane is useful in deriving the p.d.f. of ”L fro.

the joint p.d.f. of E and "Z .

i (

Lane}:

 

 

m\ a}: __i
9/)(“9 3"“ M ”WM

_ :rr-o m i .
2.9.21: Hewite 0—1)!“ : Z (x; >908)”: 0060.
5r
\ m . '. xy‘) .
’ \‘ 3’ 2’ “m 3’s
1" \-1 MAX: J... . (TV’V g1 (7sz ,° (‘91-'"3
”JOQ ) m+i "dz/:0 0)) j 0 &;(cf/ fl

hence the lei-la is proved.

-31.-

”...,"...ffiwm >+Y+Om my ..

l
Suler's constant, we see that the unbiased estinetor of 0 based on the
naxinin of senile observations has asynptotlc variance zero. liowavu
the variance of the unbiased estiiaator of O besed on the nlnini-

2 , independent of sawle size n . Thus its

2

senile observation is O
asywtotic variance is also 0
It is shown not: that in fact the unbiased estintor of O , based
on the nininiae sample observation provides the worst estinator of O in
the sense of nininun variance unbiased estimator in a class of single-

observetion estinators.

Proof: For the r(i 5 r 5 n) th ordered sane observation we have
established that

i

Etch: "“7“? Cg VOW—t“); —ch: ,(m—h+g)7—'

a”. r-l gives :_Et‘:E’YZ_-:%end Va, L|ZV:;L :: 9.1 ;

 

m
m
fizrh gives Efm: EE: 92:;- end Vaktmzvahgz Biff;

and for l < r < n we have the expressions for Etr‘ and liar t"

given above.

 

 

 

r;
Y‘ 1
2 2 /‘ 73:}«2. / t
lion Lam/MU: 9 7 e E1 k 1# 2 : V0-2 fit“ — ,
' /,;’l \ é‘ *
I' \ \ '\_ / Y‘z— +3
I: Z: m‘h+d. j $11 h 6
6—!

for r > i ,
hence it is proved that the minimum sample observation provides the

morst estimator of O in a class of single-observation estimators.

 

Since ’ ”W
T— l

_ \..‘ *‘T
\«w( E ‘\ ‘— 92 4 3’ ~ £ 52

m l —_ m 2

\ Z—aw 2M

‘ 9r . w

and

 

 

2. .9... . I / ‘
VORKTTL \ :1 9 ”ii ’ Tfl§ei

 

W —‘-_-
\ ”...,—i. <7' 1
\ 1.. ’n-h-hgf W—“fiw' ,
\\&:\ \\ }:\ (I

it is not clear at this stage anther the maximum sample observation pro-

vides the best estimator of 9 among estimators based on a single ob-
servation. luarical mutation ([23]; Table ii) shuts that the maxi-
m:- sqle observation provides the best single-observation estimator

of 0 so long as the swiesize does not exceed five. beyond five the

-l6-

mimi- of sample observations no longer provides the best single-obser-
vation estimator of O . lie would like to know for which rth ordered
saqle observation we obtain the best estimator of O .

in other words we want to find an integer r , say ra (depending
on n) such that F(n, r") is minimi- of flu, r) for fixed at

 

 

 

 

 

 

do." 7 h. { ”Tl !
’7‘“ 2 K1
E ,p J‘H‘ ..\2 ..
-._ K“ ' a / KW“
x‘p‘ I \— J ‘ r 4— --.. "‘ f __-. a
i“ (W- MH) K=M—h+\ K

it shall answer this question for large n in this chapter. it is clear
' 7:

that 0 < f(n, r) 5' . Let gm:.71. Iowwe shall shew that the limit
’71

inferior of Sn is larger than zero. First we prove the following lama.

 

Lem-all: F(n, "2"; .

‘I

f: Let us define a random variable X with the following probabil-

ity distribution.

 

 

 

 

'7 (”Eh”); r
P[X:m-h+3’j: r"-i~-~-T~:.- §0)L {2 l3 2’. ”-71.
(“n—Mn?-
k 0 , otherwise.

this gives % '

. ”(N—n+3, 1 9.;
EX 1 * J; «Ind E X r: 7.. -

..L... ' _
Zffi'h‘hg)?‘ '=l (CA-fi+é'jz—

iiow Var X 2 O iqiles Ha, r) 2%- . Q.£.D.

A very simle upper estimate of fin, r) is computed as

 

 

 

m
J... 32.
° -~+\ 1K?- W2— L
Hwy“ 4' J *m
)“"‘ m L\t)zvl.“ ._‘:L—
n/z—I— (-.-.- “W
2\ k
”NA“

“2
From F(n, r) 2%- and min F(n, r) (min m , wehave

2 ' z «
J- 5 Nu, r n) < min " mich gives :1- < min n
r" r “..-,...” r r(n-r+i)
Therefore 5 >max ' n "I (l 4- 33¢”)
" - r n 27

£- (l+%>(2+%)2 or :27(2+%)2 accordingas

n-Z mod3, lmod3 and mod} respectively. llence

n- inf 8:3" 57- : 0.11.8 .
n
i 2
A better upper estimate of E ‘2- sim I’Y “9;.“—
k
n-r-tl

fol lows free the result of blom ([25], p. 80 equations 7.3." s 7.3.i3).
Further using the fact that 'E is a strictly monotone nonincreaslng

function, an upper estimate of F(n, r) is computed as

 

 

« - ‘ ‘ ‘ZT}~ Umé‘f
(3')" )L‘Tr 66w- fifr-tJj‘L m w?” j C! ..rr-fl‘? (0’)”)
a. i. . I .1: '
\diich gives lim inf l 2 - 0.329 . ' v
6“ (Va!
liow we proceed to prove the following Theorem.
Theorem 2: Let ’Wvlm FM. )1\‘_-_-F/’Y}?z and g) : 32’“ nhere
when") ‘3’" m m
’YN
‘2'” ‘l
K2.

 

\ ’f‘x-hH
F< r“! h) :

m ‘X'Z
/7“*H

. \\ ”fj'i“ /
O
The» I) 5mgo<<l for san- o<< \ and sufficiently large n; in fact
ii) % tends to the mm 80 as It tends to plus infinity

where go is the positive root of the ematim,prn(t-g>+1g: 0;

”0. What)

iii) converges to unity on the open 8 interval (0, l)

 

as It tends to plus infinity were “(3(6):

 

( l- -§)PmQ- -3)
in other words, f(n, r) is asymtoticaily equal to - n“ g) , and
in particular F(n, r n) is asywtotically equal to ; ”Mg-,0) ; and

mFWh)
WW)

as n tends to plus infinity, provided a is any constant satisfying

/

   

        

uniforoly in §(04g:z%éo()m unity

O<0<l .

-19-

i) first we show that for large n , F(n, n) > F(n, n-l) .

 

 

Proof:
T\ ”h
r- J ...,
Let if; 1: m OWOL TIT Tl? T t?)
‘ T r.‘ n
'3. A "J
Then FT’MTYJ); 1m]. ) p(m)’7‘)—\>::; m\2
L’m (Tap)

WW" WW‘J: ---~ [W J (got/:1

Wit/fly?“

+— (J ~29?
x. /" .

11“.“ Tr“ ““11? —T-TT-——-—‘:~TTTT‘ /0 TjT/LT MgLTYT 1&va
+ 4 , a
5m Cw

Fromnowonweassuae that lsrsn-l . Put 5.5T.

Let H -- ST 0<T<I .
T T) O~9Jfl TS) T

n.” V“) : "T\3)1)3<l__gj

iiere the denominator, (l- 8'): final-a) , is < O for 0 <gT<l,
and the nuaerator, Try-(l- 5) «I» 2 ST , is > O for 0 < 5; go there
go is the positive root of the equation, 25+ [MU- g) . O, and it

is <0 for So <g<l . [The function 9(8) .283 Emu-5') ,

O < S< i has the following properties.

‘ —>*me
ii

-20-

z) m- 9(5).“ m n- «Sh-a. and
890 89\

iii) max g(g)-g':1,-( )>O . Therefore, 50, apositive root
of 9(8) lies between T5 and l . it is obvious that there is no real

root of 9(8) greater than one]. This means that “8) decreases

 

"0. 4a) to “80) as 8 increases from 0 to 50 0M "8) (1)

increases from M80) to to as Sincreases from SO to l.

—7

[Son .797 , which gives "80) . L516].

An upper estimate of i-‘(n, r) is found as

’W
9L1 \ _L
FTTTT 72.) J‘hj: -—- TT:.-T.TTT
' ””9“ l {.Jl,//'Q+)_,____\
J n 93%- “WW

 

 

iiow we prove the following lens.

Lane 5: if p is constant,0<a<l,and 0<u<l then

/ ITJYTE- UL
TTTT TT 9 DTHTQIJ

"7' .. __...._:JT3(‘-9J (1-9990- 99
‘T’TTT‘T <\- X 99sz <\—- 99)

 

increases with u .

 

-21-

Thedenominator, (i-u)(l-m) £n2('-fil) , is >0 .

=-9<I-9><9+—'.-92+~~>+99999949999199
=Tfi(\5‘[31)+ (.9- 9) + +(;&;<9~9“)9’“+~

+ >O~Q q.£.o.

Hence, by virtue of the lea-a 5,

9(9).) 4 1992 “at???" )— 99919911 (2)

Hit—1m"

He may note that

Tm (m) 1T9 W

A |
E—m—(mw) "‘ )W 9+1” \+_I_")‘ {11‘7—‘>|% Tl->+oo

l ,
Similarly we find a lower estimate of F(n, r).
fi+l
x l ’
F< .1: h+i~ m—h“- '4‘"
’Y\ e"I. 7——— g: __.__.____,.,._
\L '2.
T 99.99 9 9.9.1199
3 :r i \ ”99h
99—h /

 

iTKfl __(‘)
?z. [/Y\2- {“Eéi;\ -~ .1}:L_Qélh.__t._...i.._32.u"9 “
:(mflhm 7L) 9 m ~ m 2+? \+_,__§—r

V
llowchoou a and a suchtlut go<a<a<l and “(fl-3N3) -

lb consider the following cases.

“ul:o<gfa<l.

Tho following lower «tin-to of F(n, r) actually holds for any "mg.

Since L'- S‘__,_ -- l deems» with g:

 

 

 

\- oL
¥(mlh)7 m H... +£3.—o< (3)
the" l...- ‘_(:( -->l as n-—§+O
A}: "s 'l‘ HR.- O<

Caull: a<c<l.

 

 

.3.
“((8) \ “(\“rr)_ $100 I _1_
C ~—----~——~——~ — """"
‘(mifl7 m Hui-n Pr}; 0—577/ m “"3; 7—
For largo n , l i 7 _L wf/W (m) a: , thoroforo,
7. \JNWS'R 3 (In?)

Thus we so. that for large n ,

F/m h) 7mJ£ KN ‘J for $6893] . (u)

-..o-u-”

Consequently, F(n, r) > F(n,V r') than 9 - 32>“ and
5' ..filetg‘o , B] and n large . ihere is at least one such value

of r' than n is large. llance F(n, r) does not asst-a its rainin-
( .

inn 7:»- 5 >0 . That is, Ha, r) assues its niniwun, F(n, r”) ,
r
M 83;" s
Q.:.o.
r “z
m u. have here 5 _—_. -.fLJ OJ and
in FY“

. F n
§(_F\'J)([—€f;‘) ’ FM >7 K is}; W; 5.”)ror O<§ in“ where

e’ .
P (1’).“ 6:; (70) approachn zero as n tends to plus infinity.

“” .3 m -‘ 0
~: :F F F 4" 2F l"! r F...“ "7 ‘\ “1*-
l/Slncs “WWW. W5)? F’F‘FU 1 ._. Vii) 2 1" 3” ~."‘*‘/’( ‘; ‘
\ ., bi <‘_.LI:..»J—: -1.-- -__ -_ ‘ b“; ‘ ._ / ..-- 7’ r l
x ' " Y" *1} {F‘ ~--\ m} I /
1"] K. ..
and, (J
l , a" I —\ _‘
FF“ fr 7uii)..—!. 1:04.. KLQF‘\+__L (‘4‘ ...i w“.‘
)‘2 ( J, .7 -~ F. [p '7‘ fl".
’ I \T (Y! ‘*;7‘*§< n ( (...J \ F 01.))
f" 0 Z: 3 :ECX‘ ).
/
5’“ K“

Ha proceed to show that :3” tends to do as n -->+ a . Let E> 0
hagiven. Mkwechoosag in (90 -€,} +6) such that 4m
and “(30) «(S )<nin “(if E) , “go +6)! - N; 19,

“Yo (That this is possible follows froe properties ant-erated
under (l) ). llowuchoosa ll so large that for n>li,

(“VIM )< “#(SJEX‘ em)- “’1‘

(That this is possible follows from properties onwarated under (I).
my" (>0) -->o, €;'(>o) ---->0 and

"0513253. {we}. ) For lF-lee, liglzflgotél

hence
“la/F30 . /F
«vi;- J-Fl—QQQ “if? (I—eF 4 rim/‘2) m

 

3.1.1.1)(14U 1599.)“ +5" m); PM Xi)”

Y\
fiance fr“ (5). (7) and (8) it imadiately follows that for lg: SI >5,
F(n, r) > F(n, r 2:)

Thus we have F(n, r) > f’(n, r :) > F(n, r n') This iqlies that
S" ”'33 lust "0 in (80 -€, 5.04%) . That is, Is" -5014
for n > ll

Q.E.D.

('1
iii) Fro. (2) we have for 0 <0< I

Q
1.
FMQLM LF I F £}\([\J\ \
J x“. NHL-1‘ .5.
This gives
1711(9)“
ll“; ---— ~

"119$

Fro- (3),wehave for 0<b<a<l ,

1.1111,.7/7’1 ...,.-W . ”1:35.... A

”1* 1+4: Arm—x
This gives \
!. mf (7‘)?) > 1

~ r /
Hence for every (2) in O<g§a , 14W“ 2:11 11)__

Since we can choose a as close to unity as we wish, it lie-edlately

f H h (“FOB/Qt} i i l (O l)
o ows t at converges to un ty on open nterva , as
W (E

n -—> + on . Further-ore, fro. the definition of asmtotic equivalence
([28],p p. l0) it follows that No, r) is asynptotically equal to

-.(§) and in particular F(n, r u) is asymptotically equal to

m8}.

Q.E.D.

iv) To prove the state-out iv) true we show that for every E > 0 ,

there exists an “depending on 6 and a) such that

...,-..n.312_\ é
‘ 1(11j 1 \‘ /7\F1m§/

for nail end ell 1gin (0, a] . That is, LEW» “11‘7“”!
("n +11)

0 K“ .
for ell Q satisfying 0 <3 5 a < l . From 2) we get, for,
. ,7.

‘ n .
(“H F1 (1; 11,1 1‘ 11,11») 1 ’1 fl
0 < S; M t ’ g. < , ~""*"'“““'-‘ “1211“" e 17 )

\115) t 111131}:

8‘
p

tends to zero as n tends to plus infinity. That is, for every E) O ,
a an We) such that Iéé'l < E for nail

Fro-3),weget for O< 53a,

31E('W7‘:K27
W 5)

7”: win.“

where a; (> 0) tends to zero es n tends to plus Infinity. Thet ls,

forevery €>0,

3 "(6,0) suchthetfor nzl, |5$|<€

Nouteke N-lergestof "(5) and IKE, a) . Thenfor nzl,

m 32).
w 8)

every 5 In (0,

the existence of

\

. // T

b-
0:]; end , ”A...

e single I ,

for

W m MF(:U)L)4 é‘

--}7—&glves

W? '“

...Lfimé/zo finch-sens
m m

, for every 8 In (0, a] . Hence
(1
for every 5 in (0, a] . By virtue of

the unlfore convergence ls established.
0.6.0.

m: W 7)

Ueclele thet ~- X" ~~ does not converge unlforuly to unity on the

(o, I). ‘VK S)
{hr( A ,
2592:: Suppose thetw- $5 converges unlforely to unity on the (0,
Then, for any sequence :ngwlth O < f“ < I , It uouId be true thet
mevwv
WUT‘.’ «- . n.3, 71—1

Iowchoose fn-L. 'l‘hen\q‘j’_'.. ..... :'_T-i--/«-;,

I).

This is e centrediction.

 

Q.E.D.

it seees interesting to point out, beyond the result of Theoree 2,

the following fects. if we epproxieete

m “A
\f— \ A '* " 2
A '7 j :1 . 3.x" (1»... ' f H. I Le by
A )n 3‘ {I )A;
. 5
<: A . A .4. , A
/ ..._-.- _' 1 “~31 .- , end
...,. A" ”V , 3 J“ i /L i
.' fr" ‘ l ’ f
m . rm f k
A A ......... :LAQ. -.
i l.» " L/
"n- M! “A" 7%

respectively then we can write
1-- e ..3

F / it); ' :4: 21.“. .

m (A. A . A
Let 0 - :- . Then this epproxieetion for F(n, r) reduces to nil-2-
The ulnl; of nail gives the sea 5‘0 which hes been shone! in

Theorem 2 to be the limit of ¢ n . This result can else be derived by

means of the Euler's Sunstion Fomle.

Following (Mil, [26] end [2719 we derive the expressions for the
expectetion end the verience of the rth ordered semle observetion fro-

one-perueter exponentiei lew up to order (n + 2)" . They ere:
f," .. ..

.-. f ‘5‘. 9,
En: a L m-’-“—*~‘«-~‘ "2’

r‘r‘.~ 7% ~\ ‘5

end,
'3 553..

\/w, 1 II" 5 t . ...,. ...m-‘v-e ‘ moo - O

1‘

As n increeses with fi fixed, the esmtotic distribution of tr

‘A

’ ‘. \, ‘ “ 1 H
h.’ W hen, OET'" i\:\ , OM verience 9- ,. ‘ 4H \‘\ respective-

.‘f‘;' [(3 '\ fl—y "v/I
ly. This follows free the results in [2b]. lience the esyeptotlc veri-
ence of the unbiesed estieetor, besed on the tr th stetistic, for 0

works out to be

llow we went to detereine r so thet this verlence is einieue. Treeting
the expression for the verlence es e function of r , teklng its first
derivetive with respect to r end setting it equel to zero, we get,
efter siqiificetion,
C. . : A»:
(v. {\ \ - ‘."‘,]"‘§:'A:‘T‘ {I} “l- X h A
writing FIT - x , we reduce the ebove equetion to z‘nO-x) + 2x - 0 ,

O<x<i . if x

o is the solution of the ebove equetion then

r“ .'. (n+i) xo provides einleue verience for lerge n . lie get

xo 4 0.797 which gives esyieptotlc mini” verlence e LEI-fi 02 . This
result elso follows free the feet thet F(n, r") is esywtoticeiiy
equei to %O( 50) idiere H go) #- l.$ii$ . lie know thet in the present
situetion the supie ween is the unique einiein verience unbiesed esti-

2
eetor for 0 [l9]. The verience of this estieetor is 9- . llence

n
the esmtotic efficiency of the estieetor besed on the . r" th stetistlc
is found to be ebout 66 percent. The subject eetter for the Cese l in
Chapter ill. is cleerly reieted to the present discussion. Free the re-
sults derived there, it follows thet the suple eedlen hes epproxleetely
li8 percent esyieptotic efficiency.

As e point of interest, the product-mot correletion coefficient
between two order stetistics, sey t' end tJ(l < j) in e rendoe sale

of size n drew from one-per-eter .exponentlei lew to order (n e 2)"'

is obtelned by eeens of the foreule given in [2's] end [26]. This works

out to be
or..- . -,-.- _..- _.... ' ‘ . ' \
e O 0 ‘ ,\ A s ‘ L.
+ -~ \ r m.- A .. r . I
i . .i ‘- 1' i : ‘ ‘ _ ' ‘_ - _’ .... .. *
e I ’“‘-o—.o—.~ _ . .L - ...,-.- Q. .~ 1 v“ Q - can. . In. . .5. ...- ~ r . - .-
b ‘ ‘ {sf ~‘ It . I . ’ ‘ \ l ’ ' ‘. “' 1‘? C ‘ ‘Q ‘ '4‘ ‘ i ’ A‘
i i ' . \ ;, ' . l l l. ‘13 \V‘ \ \1 l . l(.-,’ ‘V
x -: /i.,-..--Avs~, min.“ .
6 6 f ' V ' / ' (l ( ll
\ .
V
M..- —'-'n€~t-‘r'n=¢‘ . ['4
iv " ‘ ‘ w; i
i 4\ ..-. H A ‘l { i."-
. I ...g , 1‘ “ ~ -~ -RI .1

‘ .
5“

\/”‘-

I
- i
‘ _' I‘
'. i \ \ r . _4 -
. \ A g. ." \ -.- rt '- -' 1 i '
‘ . w- r ‘ 4 .I .
S r ’\ ‘- -' . A -. ~ ~ ,3 ' . Q ‘ ' ‘
I ,‘ \ . ‘ ' ‘ 'I’L ' h‘ - ' f
, - a N .. A : v _ ‘ v , ‘

o -' - ~ , .
v ‘ ~ ‘ \ O
l. f '0' 0
fl - -
\.

there p' - "1-f- , and pz «- 3-11- . its esymtotic expression

‘ r1”, \

.-

is cleerly

 

1 1:31.. or 2:){»A:f.1\ . This result elso follows
1 ’91 1‘1- 0‘) ./ YKM'LJ‘“) .

free Creee'r: (“0], p. 369).

-30-

ill. SM TESTS Oil PAINTER!» 0F WIN. FAILME LAN

Pert i.

if e rendoe verleble (r.v.) x hes the probebility density func-
t‘m (Pedefe)

Wit—'5 {ft

the likelihood retlo test of ll: 0 . 0

“fix

, X70 can! 970

, otherwise 3

' egeinst A: O - 02k 0')
given the first r(5 n) ordered observetlons free e rende— senle of

n yields, es shown by Epstein end Sobel [i], the criticel region
a.

21L 4(1‘187 )(h1g.EpsteinendSobel [l] showthet—z'fow-fmm

Ln
hes e chi-squere distribution with 2r degrees of freedm. The pere-

eeter O is the expected velue of X end is, in life testing, celled
everege life.

Let flx) be the probebility of ecceptlng the elternetive when the
observetion vector is x . liow following Freser [3] it is eesy to see
thet Eo {fin} s a for 0 Z 0' end further-ore the test does

notdependon 02 soionges Oz<0 lienceitetoncefoilowsthet

' .
the test function, flx) derived by eeens of lleyeen-Peerson lens is e

1—}
hi

A,

uni foreiy eost powerful (“0) test for the ii: 0 z 0' versus

A: O < 0' . Sieileriy we cen show thet the corresponding test function,
neeeiy,
<73.
/ . wt ' I"“\ \ "’ I” .-
’ O ”4 i ”i "I "2"}1 ‘ A. fr L»
(Mi): . . ‘ .

l, otherwise 9

for the ii: 0 . O egeinst A: 0 - 02 (> 0') is W test for the

l
codified ii: 05 0| egeinst A: O>O' , elthough in life testing
problen, u: o 2 o, seees in generei to be of precticel interest. How-
ever, there does not exist e I!" test for the ii: 0 - 0‘ egeinst
A: .0 9i 0' . But if we restrict ourselves to e ciess of unblesed tests
there does exist e ”if unblesed test for the ll: 0 . 0' egeinst
A: 0 d 0' .

Here we propose to consider testing stetistlcei hypotheses connected
with the two-pereeeter exponentiel lew idiose probebility density function

(p.d.f.) is given by

...- f \' r" .
'1' ‘ -.- 5} xx. W 'Jf' /‘ , \ f" 'l
‘/ (6 fl 9 :‘f‘ :7 Lb E foflffiji- O c (_ 9/“
j’ \ {\s '33:;
\. 1" ‘
i o ’ omw‘“,
'\

where O is know es the scele pereeeter end 6 the iocetion pereeeter.
G is identified es guerentee tine or einieue life in life testing situ-

etlons.

Since our life test dete refer to eeesureeent of tine it sees eppro-

priete to denote the observetion vector by t insteed of x . in the

'sequel we shell use ‘I’ end t in the sense of rende— end observetion
vector respectively.

low, in feilure enelysis we generete dete by destructive tests end
so free en eoonoeic point of view we consider e censoring scheee in
vhich we use only the first r(§ n) ordered observetlons:
65:. <t2<...<tr<m .

to test the ll: 0 - 0' egeinst A: 0- 02k 0') , essi-lng 6
to be Imam, sey 60 (not necesserily zero), the best a- level test
function besed on the first r out of n ordered observetlons cen of
course be derived directly froe the test function given ebove for G - 0 .
Obviously, this test function would provide W test for eodified
ii: 0 2 0‘ egeinst A: O < 0' end would possess eli the other proper-
ties doich have been pointed out regerdlng the hypothesis considered by
Epstein end Sobel.

men 6 ismimown, the best a - level test function for simple
H: 0 - 0' egeinst sinle A: O o 02 (< 0') is found to be

h.
V"““ f -
t ‘\ ‘I t "" . \ ‘- ‘ifi A N: . I 4* I ~ “

O J; I a: ' f “ V") "71‘ (‘"}.;1 t5: ('1 i "3' "a
l \ ,1 ‘ v ' .i‘ i ‘ . . - ‘ r M
Catt): .1. ’
‘ . ..

i if t; C:

2 1 3 I "\

Y." a ' l‘\ ‘- "
there Mr.“ ,7 ...f" :4: NJ“? {27L 1] hes chi-squere distribution

1 h. \ K, ‘ : .l .2 K. .V -’i

C" L 2 V“ '4! ‘ ’1 "J:

with 2r - 2 degrees of freedom For eodified ii: 0 e 00 egeinst
A: 0 d 00 end 6 unknown, the “if unblesed a - level test function

5

isfeuedto be

- 9,. \
$66)"; 0 £6 6’ é gaff!metflfflcz

\. ?

there cI end c2 ere detereined froe

‘L --
. \ ( ‘% ELVZ ea 4 3 ’
We..- . fif) . TV ' 7:? "-C’\ (3)
‘ ~ I
. .... ‘ ~11J“ . j 'l'-/
Q? I). ‘50 j C’ <,/
C I
in conjmction with
F
I“ ~2- ._\
r-\‘ . \ ‘
!— CL \ ‘H 1() 0‘ L l “:2. O
1 777% e'w } ‘ G i (2)
l C7 7:} "'10 ...j 9- ‘ 6,
e... V‘ —~ '3
«‘7- ’ a n
f‘ \ ‘ )1 "‘~ ," J, \f‘. \j
( ‘3 5‘"-‘"-‘"""”"‘1'». '. ‘ 9‘ s C “
f. \ r “4).“ :3 C’
—'+-( *1 1': J
J “G I
3’ 0 , otherwise.
\
The reietion (22 yields, A C ,3-
?» f" ‘35: { haw on
C, ’ ‘ ”C T x 7 L...

lhe nueericel solution for c' end c2 cen be cerrled out by successive

-31.-

epproxinetions or grephicelly. This test function is derived by follow-
ing the hint given in Leh-enn ([2), Problee l2 i), p. 202)

For the ii: 9 a 00 egeinst A: O i .0 essmlng 6 to be low,
sey 6° , en a - level test function is defined by

)1

‘3‘:
\ ) Rfm‘ / \ ,' , \‘if i ‘- ‘ \ ; in
O is, C \ .4 4 3-: t: 670) +~(\f~}2j-\_ ‘ ‘ UM.) \- Mi"

l , otherwise.

w

\t

ilotlng thet
j) {Jvfiu g" i ."\ ‘ ‘\ (r ~¥ . {a \ .
..r ‘\‘ f, ._ , ‘H . _ - A.“ _._- ..m .‘o‘ ...,, '
- .63. . i! .f I.“ K. .‘ .3 i a) "\ ( ' i K J’k ‘sb h (A 0”

l 4.1:‘"‘ \d

L...

hes chi-squere distribution with 2r degrees of freedom we obtein the
Pedefe 0f

3%
5"- . _
v." / .
m a 1‘ 4: (WW I -- Go) '
w ‘ a x. \ v 1, ,
x 6- ' ‘ h
Th“ '3 \L

CIR-44‘. ., '
gt“): Q )

“-.-,“ .... ,.‘ . . K 3 ' "' J 9
: ’91. J A " (: -"‘ J“ :11!“ C/ (3)
C»
N.

. 3 1 3
i ...-.- .2... \‘ -~ : ’j‘ \1 ft .’ !\ : O (A)
i n '. ' . . x , cl \.
' I “a ..u 1" \ (I ‘
L. C” :4 “If: .... Q: 30
The reietion (Ii) yields ,0
..., 2.
(1 fl- ..
C}, i @1- ?L 9 5’
Ci ~€, ._.. C'— 2. C«

i

The equetions (I) end (3) cen be expressed in terns of the lncomlete

cue Function es

0< —~ «.1 1324‘ . -~ ; -:: s
"“ :1- ) 90‘! ’~)(>; ,

end
.: \m‘ G '“ 1 i“ Ci, "i
C\ .. \m 11 f1. ,_.....: 4; ,.‘«. Lfifl 1’, 96 4

respectively, where {[p,‘ q] is the incoqslete cue Function those
veiues ere tebuleted in [5].

And fi'nelly the power functions corresponding to ll: 0 - 0° egeth

-35-

A: 09‘00 essuing 6 union-mend ll: 0-00 egeinst A: Odoo

essuing 6 lmown ere expressed es

. O ..‘ -1
i ‘ \‘3
P(B) \ “ T 2 ...l :‘L. E ..L- 0 I K a \ f
h“. l K i , 9 z. t "p l , t) 1
we» '--I ...-
end
" '3 "‘1 a 1 1
PKG) 1:. K v“ T ‘ k» ...2' i “I" ‘11 g '3“ .. .. 1
/ a u 3 e} l: 1 .. 1 / L j {I}
'. “va Mr I "'-
respectively.

Eeriier we heve seen thet the test function

3 “L .:/{f1j"_t Go ‘4‘): L\ (:w :"
2

l , otherwise

, a..-
,.. o/
i
4

.h. C

l

\
is I” for the ll: 0 2 00 egeinst A: O < 00, essmlng G to be
blown, sey co . He shell show ieter on thet this test function is

egeinllfl’for the li: 6-60 end 0-00 egeinst A: c<60 end
0 < .0 .
For testing the ll: 6 - 60 egeinst A: 6 d 60 end 0 km,
sey 00 the W test besed on the first r(5 n) ordered observetlons
out of e rendou sale of size n fro. the two-pereneter exponentlel
populetion is obteined by eeens of likelihood retlo. The likelihood

retio,

£3; Wt)
:W’W— ’

where Q - {so , Gog end—(Lu- {6, 9025 gives in the present situ-

etion

- “30(437690)
‘e- if t. > 60
C

)1:

This inpiiss that the a - level likelihood retio t... is given by

if “<60 .

M

vdiere A“ is deternined m- the integrei, Sam ax - a . “>9.

0
the p.d.f. of )\ under null hypothesis is e unifore distribution over

unit intervel idiich inedieteiy gives A“ . a . An equivelent a - level

test function is then given by
<c --'-0 {ma
l- 0 n O

' X.
i if :,<i:o or s'zco-noo a .

circgs

0
Mt) -

For r - n where n is the size of e rendoie seeple froe the seee
exponentlel iew, Peulson [1i] hes considered, eeong other things, ii: 6 - 0

egeinst A: G d 0’ . iiere we shell reforeulete Peulson's problees end

-33-

obteln eore generei results besides considering other probleiiis es well.

He ieey note thet whenever e W test exists it is unblesed since
its power cennot fell below thet of the test fit) :0 , end in eddition
iieynen end Peerson heve shouei thet if e 0" test exists, it is the lilie-
lihood retio test.

iiow we shell derive power function, "6) of the likelihood retio
test for ii: 6 - 60 versus A: 6 d 60 , essi-ing O - 00 linown end
show thet the likelihood retio test is unblesed end W.

Fro- the equlveient a - level test function given ebove, we write

the power function es

m1. {:3}: At + (swat

where
-9. t-«Q‘
[Igo‘e‘ 6o(\ ),t1?(>7
” faz‘): 0 otherwise

ii) :2: Go'- -le1~ 90 EYWN.

Now we consider three ceses.

Cese i: 6560

-39-

CL

f :31. IA -.
.5. (555955 /. a»),
idiich gives N60) Inca , a; expected.

Ceseli: cases. .

00 .11. /
W63): E(t.10(51..;--55(51w _ «1515.5

idilch gives "60) . (2 es expected.

CO“ "I: 6?. ,

936
CG .4-
== MM = 3:5)“. .
For 6330,
{go (61" {3H)é
‘6' 4] 9 which iepiies

59(67): (_ (I MM) %(é’ -06)//q

For Go<6<e ,

m.
{13(6‘ 60) 7 E idiich iwpiies

4.0-

llence N6) 20 for every 6£(- oo, :3) end so the likelihood retio
test is unblesed. . I

To show thet this test is W, we need to estebllsh thet the criti-
cel region provided by the ebove a - level equlvelent test function
gives nexinm power of the‘test for eech end every elternetive to
ll; 6 . co . To see this, we consider two sic-pie elternetives, neeely,
6 I 6' (> 60) end 6 - 62 (< 60) for the null hypothesis 6 - ‘0 .
For siqle hypothesis egeinst si¢le elternetive we find the a - level
best test by the epplicetion of the lieyeen-l’eerson lone. llere hey-en-

Peerson lens gives the following a - level best test function for the

II: 6 - Go egeinst A: 6 - 6' (> 60) end elso egeinst A: 6 - 62k 60)

o if ‘05‘15‘0‘;°o a
lit)-

! 1,.
i if “<60 or t'>60-;Oo a.

This test function is identicel with the equivalent a - level test
function besed on the likelihood retio test end further-ore it is inde-
pendent of 6' end 62 end hence of every elternetive for the hypothesis
6 - 6° ; so cleerly the likelihood retio test is W. lhls property is
elso the subject of Leheenn's problen l3.2 (i) of pege llo. [1].

4n-

For testing the ll: 6 - Go egeinst A: 6 d 60 then the scele

pan-wt» o is unluioen, the likelihood retio )\ besed on the first

r(5 n) ordered observetlons out of e rende- semle of size n is

given by

/\=- 2* /
(L;- 35% WW X51569)

L

which cen be written es

5* L «(tr-Go) Y

§\+

L. _L
Frau-[i 1 end [l9] it follows thet %m (t... 67 > (HAUL
$3 (5 5) 5 (5—5)(5: 5 Ff]

are independently distributed es chi- -squeres with 2 end 2r-2 degrees

75.
[2(5 m5 M55555 5)
i

 

ad

 

,__ 555-55—- 5)
5*‘(515)+(55)(545)

 

402-

hes the well-known F distribution with 2 end 2r-2 degrees of
freedoe. The stetistic Z is cleerly equivelent to the above likeli-
hood retlo test. Thus the a - level test function for the eforeeention-

ed hypothesis is:

(53(5) {00 i 0.2le

otherwise 3

where b is detereined from the F - teble. The use of this test func-

tion is equivalent to the decision rule: eccept ii that 60 < t. < 60 + £9— ,
there
:65 -L)5 (mafia-t)
KL: a-“ H... ...5-< -..,. . , m def- '3 9"." by

?L*\

55-5 - -——1—- 75—5 LL
CR" M71256 9( )

559-:- e — (5—3)?

 

end p.d.ff. of t' is given by

'11 _.
95—5 ,LQG

O , elsetdiere.

5(5):

lie eey note here thet t' end u are independently distributed so thet
f(t', u) :- f(t') f(u) . The power function is derived now.

4.3-

Thereietions 6560<t'<60 +£2<co gives

55 5+“

(5(3):.5 -355 5(5)_(r ®;(()55,]55:( ((5)5 M”),

/

-(h-(
there O< :2 ,1" + i ) {ouch follows froe the fect thet

00 5 5-5

95(255 :05

Cesebll' 6>6 .
00 G55 5%
m5-5-(5-55) '- 63 .

 

—_—. ”655%“; )1[ ng a)
O<£%l(G*G+ o1) 1PM ”M 75-10 HOE: 59]
So ')

 

 

x. F-‘l__
were iIPixln Uzi) _ 3C. fl xOle

[(19) _ flb5fl441

 

4.1.-

thich is the fore in which the lncowlete cal-ea Function hes been tabu-
lated [5].
llaving derived the power function we show now that the likelihood

ratio test is m_L_iam.

5m)

1A0
E
9»
i

For 6560

(3(3)::- \(I- o<)*€_ 7/05".

For 6260 , wewrite

.590 I J70+- Eli”
5-5-52.) 6.
(\9 -- 5~\:5"r5((§- (“SEQ-$5]
":2: \ 5 g [5:6 ~— LLfiW“
ELG‘ifl-Gb)

Differentiating ,9“) with respect to c,
# (>0 55(5— <5 )— ~55]
W) = 95‘ 857%) H5955
6:6,.

4.5-

Recalling that 6 2 Go , the integral expression for P'(G) is
clearly positive which ieplies that HG) is a monotone non-decreasing
function of 6 for 6 2 so and we have shown earlier that for 6 < ‘0 ,
HS) 20 hence it follows that for any 6 , HG) 2a and so the like--
lihood ratio test is unbiased.

The likelihood ratio test for the alternative 6 d 60 , 0 unknown,
is not W. But when we restrict ourselves to a class of unbiased tests
there does exist a llil’ unbiased test. Such W unbiased test function
is derived by leaking use of the hint given in Leheenn ([2], Problem i2

ii), page 202). This is

LIL 0/ /Y(( )(L: 6,)

" ww‘h’“ rm~

\

((5):? L ELL L/‘L “LL L)

f I, otherwise
K
where as shown earlier 2 has F distribution with 2. and 2r-2 de-

(rf\
{ b

)

grees of freedom.

it any be noted that our W unbiased test function and our unbiased
likelihood ratio test function for the ll: 6 - 60 versus A: G d 60
are identical.

For testingthe ll: 6-60 and 0-0 versus A: 6<G and

0 0

0 < 0 the W test based on the first r(5 n) ordered observations

0 3
is obtained by eeans of the lieyean-Pearson le-ee. The test function is

given by

4.5-

0 L) gift-“Gd +(WfiXfE-GQ7/C;

Wt): ,,

.... Si -
when under the null hypothesis éL ? (tfh GJL‘MJIthéoX/

has chi-squere distribution with 2r degrees of freedoa. the proof that

otherwise ,

the above test function indeed provides the W test for the ll: 6 :- 6

and 0-00 versus A: 6<Go end O<Oo

sleilar arguents of lleyean and Pearson [6]. it way be rewarlted at this

0
follows directly from the

stage that we have obtained earlier the exactly sane test function for
testing the H: 0- 0' against A: 0- 02 (< O.) asst-ing G to be
lam, say 60 .

A likelihood ratio test based on the first ”(5 n‘) and '3“ n2)

observations out of two ordered samples of sizes nI and n respec-

2
tively dram randoaly fro. exponential failure laws,

W): 'é‘f’LLL—GL 2 L768

, otherwise

for testing ii: 6' II 62 assming O , the canon scale paraweter un-

known has been derived by Epstein and Tsao [7]. ilotlng

<...<t Ind t (t

tli.< ‘12 ir' 2| 22

<eee<t

-u7-

2r are the first
2

r' (5 n') and r2 (5 n2) observations of randoe sales of sizes n'

anan

given by

So if 0<w<c
fit) -

I l , otherwise
\.

respectively, an equivelent likelihood ratio test function is

where

// 1 h.- \ . “I '

{31+ 7‘1 2) ”it LJ'III if
'A§“':: LL,
with

2.. 3‘1;

...— \

L-K :1

\
I

“s I l L g ' . I
. ' W C,

\ 1 -. / ' k“: . 3;. .
~ u 1-!- t' “" -. E u
/ L L t " ' "f L ‘ n
I r / e e K..— ( t l 1‘.

The statistic w hasen F distribution with 2 and Zr. +2r2 -h

degrees of freedow. Paulson [it] has considered the sane hypothesis and

has selected another equivalent (differing by constant only) likelihood

ratio test function for this hypothesis men rI - n' and r2 . n2 .

Further-ore he has shown that the likelihood ratio test for this hypoth-

esis is unbiased and has expressed its power function in terns of the in-

co-piete Gare-a Function. For rl S n' and

Sn

2 , the Paulson's

for. of the test function is given by

 

l , otherwise ’

 

c
were c'n r'd-rz-T and w isthesa-easgivenabove. Let

r - rI + 212 . For the a - level test function we have a - (”cg-(”2)

mich follows froe the relation,

Smfiw) 0W -.-. o<

6‘ (hr-Q.)

low we proceed to derive power function for the above test function
and show that it is unbiased by following Paulson [Ii].

I"i“ii ' ‘2i’ " tii > ‘zi

writing 2 - _ ,
' "2(‘2i " ‘ii’ “L tzi > tii

we have £3- - which gives the acceptance region for the null

IA GIN

hypothesis as: O

(Law-3 -- %~
'“je :3 CL 7 0 ~

0 , otherwise.

Zsc'u , idierethep.d.f. of u isgivenby

4.9-

The p.d.f. of Z is derived by observing that the probability that .2
lies in any interval is the sin of the probabilities that nz(tz‘ - t")
and n‘(t” - tz') lie in that interval end by then using standard
methods for finding the distribution of the difference of tee rando-

variables. for the case ll - 62 - 6' z 0 , the p.d.f. of Z is

_. 3‘: m: __ ..Z
9 M19 3 a]

 

5%: ‘;—_ r 91H NM] 0 g 2473;
K 1;“ t + {TL} 19.. J -2
L T L?) “‘ "‘7‘"‘T'.A”s . f: 97
w L i' is, [22:3
: LLZL‘é—Z éw

Likewise for the case ll 5 0 , the p.d.f. of Z is

, M2» it. '1}
L §LL(2)——:E..: .--........ mfimiv—f Ni” C”. 2:]?
4*(329‘) ’ (MM 9611. ’
J L 1.8” M. P d”) £12535)“ LL
W1C + ’Ywfi T

iLLi’J”"‘“

(M HALL) 9 ,‘i’ilhé'/ .. {00

The power function, P01) , for the case ii 2 0 , is

(Djri-J

 

 

 

 

 

 

A ”it“ "h ‘9.
PW):— Li“ SAMSDCIREU) “LE“ [4% 5;(?)7/{L:i3013
O O 0 CU
(X) (j ILL.
.~.' <
+ ‘LLLS j 14) ELL/”LL
1’1.“ rm
Upon integrating out and s‘iqiiifying, the power function becoees
Yup»
FL I...— __.__ ’5: ' L?" . 1/ j , “,ab H1
wi/ * L K .L m 2 US
..._‘ . LYL'ML' Ll / L— J
Live w '1 a 2 :LIL'ULQ]
O< (g \ ‘....1 /L ) (\9
ml ““2. -1
- “L m 9112‘ ' H @5151:
-115 ‘5 /. 1--) 1W; "' axe ;
and“; K4“; vhf /' '_ _'

The pouer function for the cese H«5 0 is

41H 71%.": —-)1.H
WK) \{ngf WWW: UM KL 5-,¥(2)f("></‘a/%
O 0 leK
DO (6“&( 7
+8 O’KK'K ) 8.2 (%)]((’U) C/KK K SP
q

Agein on integrating out end simplifying, we get

 

 

WK 1%
I :r 4. g \ r "Y? ‘ H K
:7... ‘~ - * —-- K ’
PKKK)::O\ -K ”)filKhJ) 33‘
’WA V¥V\l. L’ I~
w 3+ ‘7
--/ /\ ”" “9” \ ( TM ‘1‘ __ ...3’} KKK—£855.91
©< { ”1%.? 7*” h f \V\—. J?» ) (OK 9 ——i
AVA "A i ’\ I]
f 1“ 1’ 9‘1'; ‘ ...HCYH“ MICK)'K
m, 6 / W\\ \ I K h...) . ___..... K
‘__ - ..«-~ " .-. -— ) (Z (9 K

-52-

To show that NH) > 0 than H i 0 , It I: sufflclant to than: that
tho darlvativa p'(fl) > 0 HI.“ I! > o and P'Oi) < 0 M ll< O .
0f coursa P0! - o) a a . For II > o , ua write POI) aftar lutagrat-

lug w.r.t. Z a:

Mfi'ml L 6'“
fig "' TS em
M f. ..é - ‘ _:;~" ‘
.. S: 1...; -..,-...... 3M1; )E‘e’ s/ flujam
’M‘fiML 4442\4

than [£00]: - m) - Ha) . Upon dlffarantlatlng and slwlifying

neat

M H
LT Ml ”MCI“ _€9(A
’ ‘\‘ WHEWLw-m “a I 99.96.4919: .42, fivflk
PM: (“W )90

 

M H ‘—

Both integrals are cieariy positive, so P'(ii) > 0 then H > O . Sin-
iiariy we can show that P'(ii) < 0 men ii < o . Therefore, it
follows that the test is unbiased.

Part 2.

In Part L we have shown that for testing the ii: 6 - 60 against
A: 6 i 60 asses-ing O unknom, an equivalent likelihood ratio test

function is:

o if 0525»
“0-5

; i , otherwise
\.

there

fiXfi G?)

x Z "'0; t) (Swat)

hasan F distribution: with 2 and 2r-2 degrees of freedom This

test is W unhiasad. how we propose a new statistic denoted by 'r,n

which is aneiogous to Carlson' s statistic h: [8] to test the above
hypothesis. This statistic is defined by

-51.-

{—6)},

; r ‘ , or for convenience in writing/>_ — TL"

...39' J»
2m K'N'h h tfi‘LN‘

-

g 19. p
’97)?“ J:
A great advantage in choosing this statistic stems from the fact that it
requires only two observations naeely, the first and the rth one to
test the hypothesis. Hoover, reconnendation regarding the use of this
new statistic depends eainly on its possessing satisfactory properties
of a good test function. Superficiaily, this statistic has properties
sieiiar to Student's t test in that it is homogeneous of degree zero
in the variable (t' - 6°) , and the nuserator and denominator are in-
dependently distributed. In the present discussion we hope to derive
l) the p.d.f. and c.d.f. of st," , ii) the expectation of sf,“ ,
iii) Two fluent-Recurrence formulas to cospute variance etc. iv) the
power function for this test function, and v) soee properties of this
test function.
hherever we shall consider it necessary to use sr n to avoid aw-

)

biguity we shall use it, otherwise we shall write s for sr n . incl-
’

dentaiiy, it is easy to see that if we were to replace t2, t3, ..., tr-l

by tr in z , it would reduce to

1* NN’ SNf (0‘6) In H,“ a

‘

m‘m--- -.-——~-——

" ._ (m\){th~{\] 7—34 AIW
0n the basis of our new statistic s , an a - levei test function

for the ii: 6 - 60 versus A: G d to , assuming 0 unknown is de-

fined as

(0 if ossSc
Nth”) ‘
i if s>c ,

- K.

-55-

so
were c is determined fro- the relation jfls) ds - a ; f(s)
8.
being the p d f. of s under null hypothesis nhlch we proceed to derive
We

Thejoint p.d.f. of x. and xj(i<j) , the ith andthe jth

ordered sanie observations out of an ordered sanple of size n drawn

free any continuous p.d.f., say f(x) is given by

LVN \ l I’ L9.
(L1): (”537/ ' j XII “Fag/I" FNJI]
}(xupfi .gr\_;: ( 1/):IV(*(XL/ (1%)

U , otherwise

I

—oo (IL 91*” 40:?
where fix) is the c.d.f. of 2: those p.d.f. is f(x) . In the present
case

I 1". .. f- \
(/ 7;: 9 X > U?
jNX‘:

Replacing x by t as we are dealing with tine neasurewent and writing
1 I I, I

j - r we get the following joint distribution of t‘ and t1
under the null hypothesis.

w 537% Q>4N%}+/7§ *Go)‘

l
a.)
———- -—— e

QM.) (T @191 96

£16.39): '

0 , otherwise.

linking the transforeations, u -.t' - so and v . tr - t' , we
get the following joint p.d.f. of u and v.

\ r: ”... -‘N
m} —g[flk+(/NIH)V-]

_. ..... “Maw, _............ ‘C 0
mm N (w)! 91 fi—
( W N ‘2' - -
..{NL N, —- LL, 0 70
J -’ . “ «(h-{:96 ) /
g 9

k 0 , otherwise.

9‘

Clearly we have here f(u, v) - f(u) f(v) with ranges for u and v
independent of each other, hence it inediately follows that u and v
are independently distributed, an observation wade in the beginning of
the 2nd part of this chapter.

 

.../32 9‘90- LL 9;...
“Chi,
r/NN -
‘ m ("M-“73‘6— u~ - °
W :\ )1M’fij 9 -—5 fi‘l
II’I * 4‘99 I 7 23070
910, otherwise. I

This finally gives the p.d.f. of s under null hypothesis as
DO ~~~C’VN/3+m )HN—J ‘9' __ U“ 71"2

5'“): ”@ij ”-9132 0139 . .(‘N‘mt ’5)0IU“

 

I
O

mlch after integration can be written as

 

 

26.9.2692. I): \ :for s>0
3%): C: w W 4/ , I Ighhwhexzy -

‘\
O , otherwise.

incidentally, f(s) ds - l proves the following interesting lemna.
O
-. 32-2 / ‘
r~.__ \ r. .-.“ 1"“ \ N
M3 (’\Y\~l>(M A I/ 1!, [9‘2 ILfld/ N, _
. / ha?! I . ’9 J ",‘i ‘7'.“f
fi 9 \ q A
“ u
When r - n , f(s) reduces to

which agrees with Carlson‘s [8] expression (2.5) where O 't-‘i

\ *
v cw" , and f(s) - 9“ (ha) .

To compute a percentage points of sr n we derive an expression
5

for F(s), the c.d.f. of s, so that for given no we have the rela-

tion, l - a0 . F(so) from which we determine so .

 

 

 

C-XO U‘ ‘ I3 U-
“ (M4) I. ( -"é (Ill ’m)‘ —5(YN,$+”VN~I2+) 1_
*(Ibl)! {my}; 50— I : S; "5‘9 0M :I‘OI/

_
”-..—..-. .

" Qv-l) N. (ENE; .79

0

(MIN ' “NI: 1—2 -(YN~IN*‘ I9: in»: 9
. (Ir—£9 *8 (”I 9W“

Emma—71H, 92—4)
is (FN—h—N—N, 77:1) I
I

where B(p, q) - [x94 (l-xlq-I dx is know as a Beta-function.
Again r e n , gives 0. F(s) - l - (n-l) Mnse l, n-l) , which agrees

 

:N—

s . 4% ‘K
with Carlson s expression (2.6) where s h" and F(s) .1 61. (hu) .
iiow we derive the expression for expected value of s . This is
obtained as

 

. , -’— 9(7‘I+’NNN~I>+N) NI‘ N~ M-
rfli I 'sJ H --_J. \\
...... __ 9 N‘ t K . . T" ,
Ejhm —_Q1’1)I.<TN h)! N9 lI I/J.€9 - . 9/ (99996.6 JIM/’9
0 o
30 N V \ U‘
f 9”“ (M-IN‘*-‘ ‘I‘ -~ 'g I‘t-Z-
"9 W792“: )N93fi I“? 9 9 j“ (Ft 9 am“
“MN 2I (M 999-) ON
-1!
o

by waking the transforeetlon y - e , we write the above integral as

cw '
”m “Tm: 3N If “2'9?

19M“?
Using the standard integral given in [9], namely,(

7—”
1.11990“ “MM I919” ““7059“ 70
OWL
writing (l-vl'9'2 - (l-vlii-vlr'3
$999939
N N» ,
II. 0 I

/-\
l?
V
(+—
\

 

- 50-

Letting k - t + i in the first expression occurring in the above

 

 

 

 

 

 

 

parentheses,
E KKK‘ IK‘ K‘IKXKKIm k
KIYTN: I710, L)! ri Lé\ qu H/) (”VII—Ir a)
2:3 ‘1
-—- ‘/ (fl—3 \("Qtffw (M—IH—t—H) I .
é H J
=0

lieu renaming the index i: as t and colabining the sins,

. C '9 l J $1-3 ’ _ In t I
E ’ghm ZN: ZJJIIKWIQ [ ifififi take) 1% 41+“)

\
j
+ —-l h.“ M- Jm 9 —)1+K ‘~
(I M“) (A )JI

Using thef identity INK—'3 + 71’3 71-2
nous 1 kt~l t Z ’1‘ 9

E [g -- (Tji' [Q \) {Raj/MIM— -)—- "‘/pN (WI-I‘M)

I‘Lfl‘ M #an‘L‘lj‘. (V: I").

 

+ :3 FAQ-Nit) QO-N-Kwtw)

-6l-

This is a sieple direct fore for calculating the expectation of sf n
)

Carlson's [8] expression (m3) is a special case of the present expres-
sion for Esr n then r e n . The variance and the higher accents of
i

'r n can he coveted by weans of either of the two recurrence forwuias
I

establ ishod below.

Luz: For it>l and k+l<r5n, the kth mtof ’r

can be expressed as follows:

I . w___ ‘ K_
U K KKKjKNN-I [UK N__; I WWII“
IQ,“ FDKKP“) / i

 

 

 

K/\7\~\) ‘ (In—A >KH K“\ L SKJ‘]
— ”"1"". ""‘"" ""‘” ' I e
... ’Y‘ K\—-) \ m 21”")WI —//T-C)’Y\J
_i’____roof:K CO 0.0 g
f . , .er- r‘ ,L—Z
E )3 ”IL ...-— K '\ /K{‘%(WHL)§/Ii/I-fi)AMI
fiIVI:\h“1—)TW"}L 61) J J \
‘9 o
k.“ 90 ~1- ~°Kz+ _I It).
04} K1?) R f GUM )L\_~c 9) 009
..-...» __..»...m‘m-” ”WM”
(MNICMImKJO ’J‘
lberevar convenient we shall denote integral of the following typ.
<30 -—-L"YI‘J‘ U:
9 h . f
gfi wfl<\_ _f‘ 9/) .H. S(AA)X}
3-K " 03/
O

liow integrating by parts the integral 5(n-r + I, r-2; k) under the

~62-

assmtionof k>l and k+i<r<n, weget
6L’s-L
U \ __W K Wh 21f" (”LL n+2 fl“? K 9

mm LL,LmJu n L€LW L:
"' '99 SL99, 9 29. 2 99)]

 

 

 

Noting that _ ~ 9‘ “ ‘/
L<~L

m W—L9NJL'L9L nk—I
SCM“}L+2, L1“3 ‘J. k”):< ) . --- 999:5 J

(m )‘L /L\_L}L a... fi~L’/n
CM, R M>JL§L ”(CL 9.< L

hfi L P/ L ‘“

SQVLA 32:“) h 2 KMK/J1Lrnij9L/L19hbz E/‘fifli 9

we finally get after simliflcation,
E L K KLKY‘VLH‘L LE K4 Lg 1K4 L .
2m “ m0“) Ail—L” 7L "" 7W '

Toshowanother relationship between (k-l)th end kth counts of s,

.. ..9 L'L-}~+1‘ U’ - l L’YL—f’L‘H) U— __ ..U:
‘\ t \ 9
uewrite ‘96, UK / :% [L‘Q-‘t 9)]
in the integrend of S(n-r+2, r-3; it-l) which gives
K—Z
Lm—9L' K‘ 9 V"

 

Efib ...—.-.. LLfi. )SLWL }»+Lh“3 K)—

")(h‘l )‘LLT UL?” L.“
‘ m L S/YL- )L-i-yL Ll'z.‘ t"L< L
ilotlng thet C ) <99: /‘

end
($7 ~2) ! (imam)! '7“- "’ K-‘J

«2mm J-J J= “fir; :er Em

we flnelly got efter slmllflcetlon,

Q.E.D.

Thls lnvestlgetlon polnts out e sllp ln Cerlson's lease ([8], p. 52).

The correct expresslon of hls le-e ls found to he

EL: “' MTV”) 1 ;(’T;\{/KF EX“ \—- E V K— 00:)

' ‘] [fl—l

ln eddltlon to thls, hls Table 2. does not seen to record correct m-erl-
cel velues. ln pertlculer, the correct velue of Eh: z . o.l96 egelnst
the recorded velue 0.l3l . Further work shows thet nuerlcel error ls

not due to the typogrephlcel error In the fondle.

Illustration: co-put. Es: 5 by two formles estebllshed ln tho
1
above lens end check thee by dlrect eveluetlon of Es: 5 froe lntegre-
I

tlon. The flrst formle glvos

2. 4"- - ' l'
r'; :-—- i f; *x -- E1; p\
[”9453 S L.“/"§,5' “/Q,:_;

"l

. ' * ‘1. ’

“:2. ELLE-5(erl—JxJJJB)” t/Z’m3 rim/:1),

5 5 ~'

l
l

N)'i\
O‘HDG

F” (- v~ 7

*

The second fomle glves

» "J
fl» 5* Z r- : r. Y ‘
E a) ,, w: W: I “'92.“: [3,- , ..- 1: //J/ ..
‘1J5 ? 7 .fiJ‘? ‘JD-J ;
8 ‘ ’«J (' ’3 ff fl h - ' A f; «\T
=——»--.- :3!an «- ..r law—w JJ'
5 L.“ T; K ‘ ’ 7 ‘ ‘4
: églgiM2v~J£mjl°
The dlrect conutetlon glves

 

VJ"?
be.
P
I l
p
L\
Q)
L/D
r J
h
\J

2 l,\
:%,§,£-'h2——317/h3!'

t!
r<i~9* J:
U‘ DO U}

”l

m

it/J V
rt;

L/J

k?

F

\l/

L‘sJ

hence we heve e perfect check.
Now luedletely we get

VMJ --fiéJJJJJ-JJmJJ-—_-(2J J J
t ...,)5

:QZQGO-

He now proceed to derlve the power functlen for the test functlon

-6
(0 If 03 H 3c
r l

llt) -

l , otherwlse

4g-

wlch hes been suggested eerller for testlng the ll: 6 . 6° egelnst
A: 6 d 60 essulng O to be «alum. Hrltlng u for tr - t' , the

ebove test functlon ls equlvelent to

0 If ‘03‘.5‘"*‘o

Mt) -

l , otherwlse.

llereue recell thet t. end a ere Independently dlstrlbuted. The
p.d.f. of t. end 0 ere

-1 __
’TJ B<tJ G) 9tx~ G

ig‘fl

0 , otherwlse

l I

JJJJ

'\,~ ,‘ 12L. C” .\
Qh*|)l ‘- < vagi'l} 9 ‘ ... Lg \J'e .( I \
_EQL): '2": (Jf'f- ) 7 A70

JJuJJJLfl-QTTU
, otherwlse.

low we ere reedy to derlve expresslons for power functlon by conslderlng

two “80‘.

count 6560 . CH9 ]

P<g>.—~.J-—J0 COOL fit it 0/“

MC, ,.
gm“ C”) (00 (\_-p 9 \L—(ujctn
: "" Jo /J x
_ | fleJLfiTa-LG-QJ L8 (m hJ thflm DH4~JP+J~ M J)
.* {Ya—2gifTwfiji

VJ. 'r— % ‘\.
"' (“(l—&*C~a(m Co) . when
.41 Q1 I‘L’i
O<"‘ B<T+ngxmdflm .2 wlch fellas frm the expresslon of cutle-
"‘ Ei’h- ‘J,h-J)

tlve dlstrlbetlon functlon glven eerller. lt eey be noted thet the power
fmctlen for thls slqle test functlon Is the seee under Cese l. es for

the lllnellhoed retlo test. Further-lore for 6 5 G

o It Is cleer thet

'(6) 20 CM "60) a a e'

 

Cesell: 626° .
(Co [- .‘M ”1
29(3): J~J Jim) JJ JtJJtJM
6—6”, L. G
50 _D/ 44.67..
:3 \-—_ g—JL(\L)[J_£ 9 o /A)O<t(
G‘Go ‘J
”E”
:- J—E (”fl-.MJ ‘< ‘   \.
Q. j ) ‘3JQ‘GMZ
+O<“€. L ai'YJ‘TvJZ'V‘l‘M' IND

-67-

ls the lncmiete lete function whose veiues ere tebuleted in [20]. This
gives N60) . a es expected. llere for the Cese ll., the power function
of the single test function is different fro. the power function of the
corresponding likelihood retio test.

M -M/Cu+F—C,
< CC "ék JJL>IJDCJJQJL

S 7“ (Pi
/ . “‘“" v' +CJ‘GJ
WC»): 1Q» { “f. 9 )LIJJJ kL
J J _ JL
61-69
__E..._.

which is positive for 6 2 60 end N60) ea , hence NS) 20 for

626 endeerllerweheweseenthetfor 656°, Haze so, it

0
i—edletely follows thet the sleple test is unblesed. it eey be eon-
tioned thet the lmiete lete function cen be expressed es couletiwe
binoelel probebilities.

...; that we have shown thet the mm. as: function is unblesed;
it my be interesting to mute suiteble power function tebies end
grephs to point out differences between the slqle test function end the
likelihood retio test. in life testing situetions, ll: 6 2 6° egeinst

A: B < 6 is of interest end for this situetion it ls'cieer thet both

0
the likelihood retio test end the siqie test function ere equelly good

in terms of the power and the unblesedncss of the test. Beceuse the
likelihood retio provides “0 unblesed test it follows thet in the ciess
of unblesed mu the pownr of the likelihood retio m: is miforeiy
better then thet of the single test.

for counting the eo-eets end especielly the verience of this siqle
test function we heve estebllshed two recurrence foreules eerller. lie
need not estebllsh such recurrence foreules for coeputlng eo-ents of the
iilaeiiheod retio stetistic, 2, since it hes e well-known f distribu-
tion with 2 end 2r-2 degrees of freedoe. ilotlng thet

Z70

 

f(z): (:2 hi?) )h )

, otherwise

we heve, Elk - (r-l)” I(r-k-l, k-o-l) , weild for r > low: , wdwich
cen be further slqpllfled to

{_C’I)

 

 

w) “\
E2 1—1} , where (h\‘“‘/ is e Ilnoelei coefficient.
( < J mm)?"
This gives E2 - Ei- : E Z: -,__‘; r ‘ j end
fax/JUVB)
jib—9% J
'.'(z) . _,,, _, , .. ...- “__..-.-._.—.-———--—- --—---

N w "'" l 2. Z...
My J J J_ JR)

wdwich is independent of sewie size n . fro- the woeent recurrence
foreuies for the slqie test function it eppeers thet its verience is not

Independent of can-pl. size, n .

-69-

iiow we propose two siepie test functions for testing the ii: 0 e co

egeinst A: 0 d 00 es'st-ing 6 to he know end unknown respectiveiy

end derive their power functions. For 6 low, sey Go , we define e

simle test function es

/
.o if c'ftr-G 5:

o 2
9m Iii
i , otherwise;

end for 6 «alum we define enother siQie test function es

fl) 0 if c'Str-t'sc2
(-

i , otherwise.

Thep.d.f. of tr forthecesewhen 6 is innate, is givenhy

mi fwé‘thm"? (t)? 6Q
3% )2 @Jhw‘f _JJJZJ \J-J
h x([_.t a h m))
0, otherwise. t

7G

Letting tr - Go - x we get 71 o
m I “{5W-fi“) I .... 2‘; J 9H
...,...LH -.-,“... , __ J
fix): wmhnefl < * “a 13,? ,7
0 , otherwise. ‘ 7 ‘

thder nuii hypothesis we have

 

-70-

......fl?‘ % (“Ev/x ... x. W!
( Qi \(MJ UI§J€ ." \~__€ 90
7(1): C1,
0, otherwise. /J
Letting-C 0:}, weget
n1
UCQQZ- CLfl‘M-fi) .2910“ We?) M'jf/(l
0, otherwise.
\
liedeternine cI end c2 frontheeouetion
Ci
r‘e “67°
4 01 :_
J J [v 3’ (5)
__ CL
*6 9'0 __5:
“‘9
[ we.
in conjunction with \2‘ q 8% é’)0(& 1O ”-2- (6)
09 J 6
-€/ E; _J 69::3 9o
The reietion (6) yields
~(m—3JH) .. 9- §i~l
Q0
6‘ —€. 1-— ‘C c
“<M‘h‘l‘) --—2; _C 31%
2 61 _C’ 9 33.7 .

 

-Ikilrir UP. a: I» ”r
. c

-71-

The reletlon (5) cen be reduced to the incomplete Bets Function es

0(1‘—-.‘_ GEM Hbfl+ 5"‘7‘ /QI-/1_!

The numerical solution for c' end cz can be cerrled out by successive

epproxinetions or graphically. The power function is now given by

J _ g r M7 —.- ‘ c h
{)(9) :3 {... 1 - g‘ Lrfl‘rfi-‘rl °) 77:" J "I“ C‘. {7“}711‘) /Cfij

Hhen G is unknown, the p.d.f. of tr - tI - u is given by

, _ "J
@__:D ‘ "" (T‘.'}J+y ZS ..., ,1) $7 2'
J J «:— we: ,
“LOX __ i/ ' ~ M
O
J 1’ 7
f 0 , otherwise.
K
_LL
Letting ——f 9: ‘Jj‘ we get
/ 605‘) 1 FY.“ 5'3.» 0,7,, (l.
- .— l__,-_,,...._-.. --— "—2:- ‘\ (\fi ‘1') )
807) " k7“)! CY ”HJ‘. U2 1

0 , otherwi se.

liedeterueine c' and c2 from the equetion

I

2090“)" -- ‘—c>< "’

in conjunction with —- ——

{“‘fi (f, “2!

 

I J r J.

’ ,Egé} \ :T'< qv) (J'J : -::- C) (S3)

L J -32. —.,’ 5: 5.0

The relation (8) gives ~€ 9
a! 97.. ”—2. J.
~(2—W‘2: - 95—; J" l {2%) :2: _ a: l
are <y—~<: /’ = 2:. (Jae).
/

The relation (7) can be expressed in terms of the inconpiete Beta Func-

tion as

I a L, J3
-€ '° ‘6 90 N
The power function of this test is given by Y .1
J \‘J - , _, ”I T W\-}-'\rl‘)‘1”‘ l J
E<e>:\_1 Cl‘mh-rljhldfii-ggL L) ‘1
L' 1‘:

~63;

iV. PERCENTILE ESTIMATORS FOR PARAMETERS 0F EXPOiiEilTiAL FAILURE LAN

The problem of obtaining percentile estinators for parameters of ex-
ponential failure laws and investigating some of their properties is
taken up in this chapter. A percentile estieetOr for the shape paraneter
of the Heibuil law’is derived in the next chapter. its expression being
unwieldy, it has been thought useful to take up soeewhat exhaustive in-
vestigation'on percentile estimators of exponential laws first.

He write p.d.f. of exponential failure law as

“left—6, .
$.09: ‘éflc 9‘ ))JC\/Gé(‘00/Q&)£

56(0/29)

0 , otherwise.

For a given cunulative probability p, the percentile flip is obtained

frm Tb
J‘s : G fit) all?

W1): Gr- 61~x<1~kfl7 $602)).

 

1- i Khulna,”

-- 7. -—-

 

-74-

Corresponding to population percentile 7“? , we denote samle percentile

by t p and obtain the following percentile estieators for the paraneters
of the above failure law.

Case i. if 0 . 00 is know», a percentile estl-ator for 6, denoted by
g , is given by

w. t%+9021m(l—U 3Com? 2400.

Case ii. if G - Go is known, a percentile estieator for O , denoted

by x , is givenby

1 :2. (GO—ft) l: LAU- bJJ-nlgvhmg {36051}.

Case iii. than both 6 and O are unknown, percentile estieators for
B and O can be derived fro. the equations

tP] . 6 ‘ °L(I'Pl) 9
tPZ " 5 " OE‘U'PZ) a
where p, and p2 both belong to unit open interval and are chosen in

such a manner that r)... . [np'] <‘L2 - [npzl ; n being the size of

randow seeple dram fron the above exponential population and [np] , as

usual neans the largest integer in up . Clearly F. < ti: would nean

pI < p2 . The above two equations give the following percentile estiea-
tors for O and G .

x- a(t - t ) where a - [gnu-p) - anU-p )1" >0 and
p2 pi I 2

-bt +i-bt where b--a£nl- >0.
9 p' ( ’92 (92)

lumen G - 0 , the above failure law reduces to the con-only used
om-paraeeter exponential failure law in which case the percentile esti-

eator for G boils down to

:1 ~— JEXDEJJmUflaflfl—I {at J>E(OJ1).

iieeay note that for 05p<l , wecanwrite ”gnu-p) -:-:-p' .

iiow we proceed to derive the samling distributions of the above per-F
centile estieators of G and O and investigate soee of their properties.
Following era-er ([lO], pp. 367 ff), the p.d.f. of tP , denoting for

sieplicity by t , in a random- saeple of size n is found to be

m1M<m~ (ta J—Je— (at-51) \V’
(W ‘Jc'j‘ *“Ce (‘1: “"2 /)3

LOL. , otherwise

 

 

where . [hp] and np is not an integer. if np is an integer we

are in the indetereinate case and tP may be any value in the interval

-75-

(tnp' ‘npd-l) . Since g-t+00£m(l-p), wehave

F.3— - EH”? MO 9)]: :<: N *Et

there d . Go QM“ -p) and the expression for it1 is obtained free

f(t) either by direct coeputation or by eeans of characteristic functions.

Free the results of Chapter li., it follows that

 

l.»
__ '1 __.- K Hal 3’ [QWXM “#2: 93”
E Je____m _ R: (l: / \
3 Mew-9'. 4: >‘w / m +92.“

which gives

 

E23C1+E7SLMNN )+9§”(N*L

w. .. 2: ,_
>13 6° @Wf’

Hence an unbiased estieator of 6 based on percentile estleator g is

H,

' l T l
4&1“ 6Lo [rffl—lu-kt

V

 

 

-77-

Recalling that t/k - [np] - np-q where O < q <i , it is easy

to see that

\

__._....-.-.-...-'. is asmtoticall C i to “VI/“(h ) ““1
--:-""" AA" H+ L Y q“. p
L“:- 0

I
p? 7* is asmtotically equal to 7}?)- , which
Lzo Q W“); n P

 

 

iqu that for large n , expectation of g approaches 6 and variance

02

of g approaches -2 TE;- . lnvollng Era-Jr's Theoree
n

([iO], p. 369), the above unbiased estieator of 6 based on percentile

estieator has asymtotically noneal distribution with wean G and var-

2
9° .2.
lance -n- l -p .

Now in order to obtain einieue variance unbiased percentile estieator

of 6, wekeep n fixedandchoose p such that for large n, the

2
O \
variance , ~52 TEF- is einieue. Clearly, any PEG; , 9?)

provides such a percentile estieator of 6 . This neans, ck- [np] - l ,
the first (or the snallest) saqle observation out of an ordered saeple
of size n drawn fro-I exponential population yields einieue variance
unbiased percentile estieator of G . lie note that the waxin- likelihood

estieator of G is the smiles: saeple observation.

 

 

 

k L/ ‘ C
__ h - I 3‘ v- 3 \i ",f}: (”4 t I,
El - EECittqu )1! "" C Z 1X 2;}, J M» i
3.77: {,2 3 /
there c I - “(i-p) > 0 Finally,
K1 K H
K m‘symxw K£( _L\
EX: 7‘17 (”A {UK-l)! (C6) -_,__ ‘v )(‘DQqFtHyK-H
This gives
M
- E1 Site ZT/‘H‘ii and Var ...;ng i (n-rbi‘. ”‘2-
L10 L20

 

--1
Therefore, it is clear that I [a is an unbiased
“‘Wl

L30

estlwator of O with variance equal to

Pk -2.

9 L:QMML)1E. c— th

relating to i'YLV‘VC and § _

l
t = o ’ (:6 Q? “+91,

 

Using asmtotic results

 

 

 

-79-

recalling that c - - Luau-p) we see that expectation of x , a

percentile estieator of O , approaches 9 and its variance approaches

02

.AndlkiC"Theo 10,.353
n(|-:)L:q_- (l-p) nvo ng raeers ran ([ l p 9) t

 

follows that unbiased percentile estisator of 9 has asyeptoticaily

nomal distribution with seen 0 and variance

2
lei-5%,,- [m 2(i-p) . He nay now atteept to choose p such that

the variance of unbiased percentile estleator of 0 is nlnieun. Con-
sidering expression for such variance as a function of p , and setting
its first derivative with respect to p equal to zero we find the equa-
tion, 2p +£‘h(i-p) - O . ilow p0 , the solution of this equation would
insure elnieue variance of unbiased percentile estleator of O . by
iterative procedure we obtain p0 - 0.797 . Hence ll,- [npo] gives
the appropriate ordered seeple observation which we should take to fore
unbiased percentile estleator of O in order to have an assurance of
elnleua variance.

For Case Ill., percentile estieators of O and 6 have been de-
rived earlier. They are x - a(t - tp ) with

 

P2 l
4 l
a . I'ljn-l 1;:J- > 0 and g - M” + (i-b) tpz with
[Ml-pl) 1
[3" “(1.?2) l W" P' 6(oa i) : P26(oa i) :

(UH . [np'] < [an) fikz (I-plylns p, < 92) ~ "0"“9 "m hi

.30-

and 2 are integers, for convenience, we replace ‘ by i and
by j , satisfying i (j . The Joint distribution of order stat stics
t. and tj(i<j) in a randoer sale of size n drawn from exponential

population ls given by

111% :6: }Ht‘ G)+(A1+)(t 6)]

(LLJIQ "1"11/pz)19.7- .

{,(t 91: c.)— at???“

. :71 C1 7
G (£101,400
1 ‘ <1

o , otherwise.

Now the suitable expression for percentile estimator of 9 in terns of
ith and jth order statistics is x - “t1 - ti) were i <j and
a as already defined above. The p.d.f. of x is derived free f(tl, t1)
by integrating out t1 over tj >641:- , after expressing t' in

terns of x and t1 . Thus, we get

-8l-

 

 

 

 

 

 

1 K: o fm—o (21“ m:_. (Tin 1.1+),
1 (me-L+)
\\ 0 , otherwise 17 0.
""3 gives rth eoeant of x as ‘ ‘
.... __ ...1 .
E11“- W‘ 721087;” {:1 k 1“‘Va+‘).
(1“)![1‘1’91m291 Kzo ’Mzo \ K A W
K‘W‘ __ j”
19:) (M'WV‘; L"1+)_fl:*)
(max-H)”

’11 -l "’1
where a - an 7:;— . Since x is a linear function of ti
2

it is clear that we can obtain eoeents of x in terns of

joint wants of tI and tJ. . Thus

r K_ K r. \E K K \< L—ymEtT“ 719W
El .0. :(1k—tcj ,QWZO(M> K. (1

By this foreuie we get

.32-

 

 

?" \ r} k.- " _
E1. 2 a(fij1‘ Etc):a'91:Y\I-31+K—Z“C+R 1
i g1<:.o (:0 ,J

prl :: 03—1:\/0)Ltc+ Vai‘w—121— 2601/ (t0 t&)1 .,

where

VwfiCL. =

 

and so Var t can be written

“(M (+97— 9 J

“1:, 1\/1.—.

by replacing i by j in the expression for variance of ti ; and
b

_ 1 1 *
$110311 9 gQfl‘H‘iJL

which also follows from Sarhan [ll]. Asymptotically expectation of x

 

approaches 9 and its variance approaches

7% (1 13.11%) 1% (“-1”)

The above asymtotic results follow directly from the expressions given
for expectation and variance of x . Alternatively these results could
have been derived frola lira-Jr's results ([iO], pp. 369). Furthemore,

Craee’r's result would iwly that x , percentile estleator of O , has

L m: I: "7"“
m WU Ln (PT-*7}

An unbiased percentile estieator of O is clearly seen to be

X My. w Egg—- may]
:(t— 4%: Tfi— —Z—-—~—]'

those variance can be written as

91 [iTm— _CP {92— L-wvljx
X [:m—k’i—Q‘K "Zm—— c-x-K

L. K10

 

 

 

 

 

Again invoking cn-ér's Theore- ([lO], pp. 369 m the following is true:
This unbiased percentile estleator of 0 has asyeptotically the sane
noreal distribution as the biased estleator x .

An appropriate expression for percentile estieator of 6 in terns

of ith and jth order statistic is: g - btI + (l-b) tJ mere

 

. .31.-
XMU‘l‘L
”'D‘W)

waking the transforaation g - htI 4- (Ha) tj , tJ - tJ and integrat-

] >0 and l<j. Uederlvep.d.f.of g by

 

ing out tj over t‘| > :b‘ . This gives
iqf &—L\
-1 ,
(lew LQ-L‘Ly‘ (r )LLQKZOWO 1% 0;}

:gm SJKJM‘C §[(—:———+J,+C-W A? 9]
WT M“ (KM-W) 276..

‘L

Since 9 is a linear function of ti and tj , the foreula for the

 

 

rth eoeent of g is given by

E? :EEL’C +(Wt1

:;(ULLL L») MEL t:,

where product Ila-ants of t' and t] of various orders can be ceeputed.
Thus

E}: LoEt +LL L) FELL
1G+9{b:%“’rK '34 "UK :owlx- FF]

 

.35-

and,

VOL} ;__.. S- M L. LL~L)"VMLK+2Lo-L)Whig)
_L)<Zb)9 20:0“ ”LAVIJL JVLJLSL mmi~d‘+)l-

 

Asywtotically expectation of g approaches 6 andvariance of g

approaches

 

.. 'l
)1le ...,... B:[\_flm(l bL—X .

Sting—L? 31+
7?; ~~ )/*L>,+( IMO—’92)

0n the basis of expectation of 9 it is clear that

L‘,
of“ 9 [b 20 m—L.+\<+ (“‘9 :WT‘iJ J

is an unbiased percentile estieator of 6 , if 0 is idiom. in case

0 is unknown we need to replace 0 by the unbiased percentile esti-a-

tor of O . For 0 lam, by use of eradr’s moore- again 9 has

asymptotically noreal distribution with mean 6 and variance

 

 

:37 [3:1 + M" Mm W‘L‘L-iEDLHMLH‘JWM

(‘”F'JL*'FL)

' ‘ . i'tf‘Ifl

 

 

which is tha sauna as giwn oariiar except that b has boon axprossod in
torus of p' and '2 .

lion ua prooood to dorivo joint distribution of x and g from
joint distribution of t, and t1 . Tho porcontiia astinators,
x - a(tj - t.) and g - be. 4» 04:) t1 for o and e rospoctivaiy

can ba convoniantly axprassod as

I
x-(t ~U£fi4 (:4 . '“d
-1 "P
g - [t.£fl('-Pz) ‘ tJLW(I‘P')]J/n (T79?)

Making the abova transofrnations wa got tho following joint distribution

of x and g from joint distribution of t and t1 .

mew(E—$3 __X
( G)‘.Q -L¥-\)'(M6)*61
a f-‘gbr IMO (VU’ _Q+(m FR} GIN-1&6 ”-6181
__.L .. ,— .. C"

X [use 933,150 962]]

 

3L (1);):

-—J
L-L—l

_%[?—1M\—h}-GJ fih’WOfi-CJT ,

Nah: -- —Q

I” “$761+it (MUWZIVW

K 0 , othorwiso.

1-22‘3-Er

 

-87-

. Frol- tho joint distribution of x and 9 it does not soon convaniont
to obtain oxprassion for covariance betuaan x and g . Houavar, sinca

both 2: and g are linaar functions of ti and tJ vacanconputo

covarianca of x and 9 byknouing covarianca of t' and t Thus,

1 .

CMxyfi) —.:. Cw [Mtg—t), Etc-\- (1—4,]th
: (10%)) Vault? —- OLA/om JUL. + 0 (LL—ny/(tkfij) ,

ilotlng that in tho prosont case axprassion for Cov(t‘, tj) - Var tI ,

the above axprassion reduces to

cm (x) 3) 2 cm») UM”. - Vent}
’L. F} U L

0

 

 

which is asymtoticaily equal to

L b.- I
£“(“%’w)/ml(%>% [Tibfika-X.

 

Z

V. UEIBULL FAILURE LAUS

The probability density function (p.d.f.) of the 3-parameter licibull

failure law is given by

i I
-i - - (t-G)
salt-g). . 9 , t 2 G€('¢h m) s90e(0, m) .
f(t) - j

0 , otherwise

 

where 6 is the location parameter, imam as guarantee tine, O , the
scale paraneter and m the shape parenter. lhen e e l , the 3-para-
eeter Heibull law reduces to the Z-paraeeter exponential law. when G
is known we have the Z-parameter Helbull law and when both 6 and O
are blown we have the l-paraneter iieibull law. Even here the shape para-
eeter, e , being unknown, presents rather a difficult problem of esti-
nation.

in the present investigation we shall work with the 3-paraueter
liaibuil law. The results will, of course, rewain valid for special cases
of this law. it will also be possible to obtain some more interesting
results in special cases.

The rth noeent of the 3-paraneter Heibuil law is given by

A

+\~

(’1’ fr (hf-K L4.
‘?7. \-.-- TL , w K
H. : LU 6‘ 9 V7?

:0

 

-89-

 

 

 

 

_ fir“: +‘J‘4V(%+;JW J KM *JFW J3r J“...
' {N -———-\ --X‘ {HQ}

where ‘62,}5 aarndV,“ esthe econd, thidandthefouthcent i

We nay note that the variance of the 3-paralaeter Heibuli randoe

 

~90-

variabie (r.v.) is a function of O and e and its B. and 82 func-
tions of a . Recalling that B' and 62 are neasures relating to the
shape of a frequency curve it see-s appropriate to call u the shape
para-eter of Heibuil law. The following relationship between the rth
went and the rth power of the first accent can be of use in investigat-

ing properties of the 3-paraeeter Heibuli law

8

(51-9

(3 Mewréi*);)

Lani: if P.(J-j)-

o , otherwise

 

in f(%+9 __ '75-
Et 2J3 .. _t
B [fig-“+9 C J
__.Ethz i“ (EJGWJBgJ—(éw

 

 

 

-9]-

Pros the above recurrence relationship, we have the following inter-

esting results.

/ (3”
F 7.; \J
For r-i, EB VJ ‘ -i
‘ l 754")
3’ ...lr'?‘ (‘,“..\-1
writing FEE; )* 7%:(6; an; YKTY-..~‘)_1 a; f ($)we get,
”hm“ ..
for r-i, E immijffi _m. in general, for any r>i,
B VIKLA‘ "
\va

 

x’W
22.3 — w —' .21.“--- _____ , were
\ 1 "" .
«(... +) H 1
J— “ 3: B \"JI'f‘w
\
3(a, b) - git." (i-x)b-' dx for a > O , b > O ,
O

is a Beta function.

Proof:
‘ ‘i \t ‘ . ‘ .‘w‘, _L.‘
3:. -1. (.5. .. ti. .. 3H1“; Mm)
w w m». H -— M,

3‘0“ 8(0, b) - m . Writing F<LL):\‘( 2-

“(a + b)

-92-

and repeating this process several tines we finally get

45., .. ~ Ar(--’--)
m“ " i“ 1‘: ma 1-,)
andweheve r&<-qu+\): <%;(\:‘;;)ar, hence

rQAH)
WW)

 

A
W?

 

i

C}
if.“ \afirii‘ J—

L\_‘ ’Yfi) W

 

—
——.

 

Q.E.D.

iiow we can write the above recurrence relationship as

a». 3% >7.
Et : EB§38(%51§) fifift} ‘

 

 

-93‘

The quantities B. and 82 can be expressed in teres of Beta function

 

 

5' -<&,a-)[s-=-u<§.&)+uga-.s) -(§..1-.)]’
«(m-w» M
82 _° “bl-XE“ - ”.2 a (My ungasw -:-;'- iJJ
II: °<&-:L’:->E~(-'v%>]’

(Ll-w] W" HJJ

For the Z-paraweter iieibuli law uhose p.d.f. is given by
"N.—\ __ Lt“
w 1? *6 9 , t 70 (Q 6)“ 50/va
0 , otherwise

wehave

r

aha: fig...)

This gives

an: ((5.0, ....

Whig—(31H) -t" 3.9]

in this case the recurrence relation between ments is. relatively simple.

in fact,

 

 

war-ms“)-o*r'<%*'>-3f;1.)”33229-«0'.

This is a special case of Lane l . liere we define

I if j I r
P(J - j) a
O , otherwise.

The above recurrence forlauia yields the following identities-

°r'-__. W J ’7— Mm :s EtL-z: 7““ :(‘JQ' J
H (W J few-tag)“ rh m) V

 

iiow without loss of generality, assueing G - 0 and O a i , the

p.d.f. of i-paraeeter Heibull law is given by l

 

vat-lg“) 'F2(&*') .
11» recurrence formula gr?“ uh - M 2:339
Rh": VCR? +9 : rh(;) T (5“) — “(7%?

which is identical with the recurrence fornula established earlier in

 

(El/h

case of the Z-paraneter lleibull law. This recurrence fornula is also a
special case of Lane l with degenerate probability law at J - r .

liow we proceed to discuss the problen of the estinatlon of the para-
neters of Ueibull laws. It is clear fron the functional representation
of Heibull laws that if e , the shape parameter of lieibull laws is
lam, the transfornetion, u - (t-G)' reduces the 3-paraneter Heibull
law to l-paraneter eXponential law. with 6 also know, the above trans-
foreation being a parameter free transforeatlon causes no difficulty in

getting the naxlnun likelihood estlnator (n.l.e.) of 0 , the scale

 

-95-

parameter of the Ueibull law, based on the first r(§ n) ordered observa-
tions out of a random senile of size n . in fact, such m.l.e. of O is

found tobe
r

E: (c, - e)” + (n-r) (z, - a)"
lull

o>

r,n r

it possesses all desirable properties of a good estimator, namely, con-
sistency, unbiasedness, sufficiency, coqleteness and asymptotic normal-
ity. The proofs are exactly the same as given by Epstein and Sobel [l]
and [l9]. in case 6 isunknoenbut m innownwesuggest that c be
estimated by the smallest sanpie observation (m.l.e.) which in life test-
ing case is the first saqale observation. The m.l.e. of O is now found
to be

r

Z (t' - t')" + (n-r) (tr - t')"

 

I’

it nay be added that the m.l.e. of. O in case of the scale parameter
lieibull law is a unique minimu- variance mbiased estimator which follows
from a theorem of Lainenn and Scheffe ([3], p. 6i). in this case a
single-observation minimum variance unbiased percentile estimator of O
can be obtained in exactly the same manner as has been explained in
Chapter iii. of this work. then m . l and 6 0- Go , the l-parameter
Heibull law can be i—ediately reduced to the l-parameter exponential law.

 

in this case the parameter of the Helbull law can be estimated most ef-
ficiently by the maximum likelihood method. And a single-observation
minimum variance unbiased percentile estimator of the parameter has been
the subject matter of discussion in Chapter iv. then m’ - l and C
mluaoima, the 3-parameter Heibull law becomes the Z-parameter exponential
law. The most efficient estimators for the parameters of the Z-parameter
exponential m based on the first r(5 n) ‘ observations have been found
by Epstein and Sobel [l9]. but again if we wish to derive estimators of
C and 0 based on only two observations, percentile unbiased estimators
for them have been obtained in Chapter iv. and we can insure minimi-
variance for this type of estimation by proper choice of caulative
probabilities. However then m , the shape parameter, is mluaovm and
we are interested in getting good estimators for all the 3-parameters of
lialbull law we face several difficulties. The likelihood equations to
obtain m.l.e. for C, O and m fail to provide explicit solutions for
{ them. no [l2], assuming 6 - o , proceeds to derive m.l.e. for O
and m on the basis of the first r(5 n) ordered observations .fm a
random saaple of size n . His likelihood equations, namely,

I'

A i E G ’3.

0-? t'+(n-I’) tr
III

and

 

 

g_ i :wfiwtc + (W86: Lit/x
__ w ___ an . ‘

. .- ‘IL‘IZ‘. m.

 

“1::- + 2k”: L'
"M Cr;

clearly reveal the need for use of the successive approximation method.
Of course, the similar situation will arise in case of the 3-parameter
Helbull law. liere we shall first estimate 6 by the smallest sqle
observation which is the m.l.e. for C and then the m.l.e. for O and
m can be obtained in the above manner.

Duggan [i3] has worked out the moment estimators for G, O and m
of the Helbull law. Again so do not have explicit solutions for 6, O
and m . iloumver his table seems to be convenient for cosputing such
moment estimators. iiis nmerical examle based on the data pertaining
to life of 310 automobile batteries provides negative estimate for 6 .

The recurrence formulas for moments of weibull laws established
earlier appear to throw more light on obtaining mnt estimators for
the parameters of iielbull laws. in case of the l-parameter lieibull law

our recurrence formula is:

h: I: \ :hwlflflm ‘E‘kh: ____
H“ W”) Wig) U EBWM

 

Equating population moments to sale moments we get, for r . l ,

n Hun-unhi- [(1)

For r-z,

 

- 2
ii) 3—; (saqle estimate) - -—2;'-TE—-T:: 2W five-33).,
f“ M
?e-:'- Zt' and Zuni-:3 , andsoon.
(1‘ L:\

Thus for every r we have an equation in m which provides moment as-
timator for m , the shape parameter of the I-parameter lieibull law.
This raises a problem of investigating the effect of properties of mo-
ment estimator with respect to (w.r.t.) r , the order of moment an in-
vestigation which we do not intend to take up at the present time. while
investigating this problem it seems fruitful to consider the consequences
of directly comuting moment estimator for the shape parameter from the
expression of the rth moment since Et' :- [(5" l) -£- [(5) .
For instance, if the 2nd moment is found to provide a better estimator
for the shape parameter than the lst moment, in that case the moment
estimator should be oowuted from the equation, ? 2 - £- WE)
relatively simler expression to handle than lil) mentioned above.

From the p.d.f. of the 3-perameter Heibuli law we obtain the

F‘ __...? x

-100-

following expressions for median and mode.

. l
A(median)-6+O; (1/172) , and

(6+O;(l-:-u1a'
W ., .3...)

This gives the following expressions for median and mode for the l-para-

td'ien m>l

meter Helbull law.

l L
A'(flfll2). and -<l-%)' men m>l.
0 anymm
Equating population median to senile median we get
i

tmed (mle median) - (EAZ) " idlich gives 3 - b/QJ’WZ
t
med

This is indeed a simle estimator for m whose exhaustive investigation
should be taken up on a subsequent occasion. Equating population mode
to sample mode does not provide such an explicit estimator for m as we

have with the median. iiere we have

t“. (whistle). (l-;): .

Since the recurrence formula for the moments of the 2-parameter
lieibuli law is identical with that of the l-parameter lieibull law we

shall obtain the moment estimator of its shape parameter in a similar

-l0|-

fashion and then proceed to derive moment estimator of its scale para-
meter. As for instance, if m is the moment estimator of its shape
parameter then one moment estimator of the scale parameter, derived from

the expression of its first moment, is found to be

A
.... m~
t__ “_- 4.. ti:

’3 (moment estimator)
I \ -- Va _L__
NET *9 F < ~4: )

in case of the 2-parameter Veibull law,

l l
Ahedian) I 0 3 ( fill 2) i- ‘ and

.5 ("5);- .. ....

"\ (node) -
. 0 , otherwise.

Equating population median and mode to senile median and mode we get

2—3. W)
This gives, m ‘2’“.«1 - 1%.“) - MEET) -an - o .

/\
Solving the above equation for m we have 0 I .

From the relation (i) it is clear that the median and the mode of the
liaibull law are quite apart provided m is less than two and away from

-|02-

one. in this situation it seems reasonable to use sample median and
seepie mode to estiaate the parameters, 0' and m of the lieibull law.
llien m is large it is not desirable to obtain estimators from seaple
median and mode for the parameters of iieibuii law. because in that case
median and mode are very close to each other.

Recognizing the fact that the moment estimators are usually not as
- good as maximi- likelihood estimators and furthermere realizing that both
moment and maximum likelihood estimators have failed to provide equations
explicitly solvable for the estimators of the parameters of lieibull laws,
we proceed to present some other estimators for the parameters of lieibull
laws in formula form so that it may be possible to improve these estima-
tors by following the technique of generating Mil estimators from them.
in Chapter "A, we have talnen up the problem of deriving percentile esti-
mators for the parameters of the exponential law and have investigated
their properties. There we have mentioned that the subject matter of
Chapter iv. has been the consequence of getting percentile estimator of
the shape parameter of Heibull laws. iiere we give such percentile esti-
mators of the parameters of lieibull laws.

Corresponding to the given cumulative probability p , the popula-
tion percentile (YD for the 3-parameter Heibull law is found to be

i l
'YP-G+0'[£.M(I-p)"l' .

in case of the l-parameter lieibuil lawwhere 6-0 and 0- l , we
have I

403-

TP-£m£(l-p)-l-Qm£ (11;) .

Equating population percentile to sample percentile we have

a
-

tp (sawle percentile) e by“ (‘L) which gives
A
m (percentile estimator) -

lie may note that p '% «runs to sale median in which case we

have shove: earlier that In. 2 , finch follows from the
t
mad

above percentile estimator than we put p . % .
The l-perameter Veibull law achits any positive known value of
scale parameter. if scale parameter is luiown 00 , the percentile es-

timator of the shape parameter is given by

[MONIZMM'
. 3... hi .

P

iiere 00 e l reduces this ’3 to the former C. which provides a check

on the accuracy of the expression. In case of the Z-parameter Heibuli

law we have population percentile ”(P given by
I I

’i’ab‘M;<'> .

p . 7:;

To obtain percentile estimators of 0 and m we choose two cumulative

-10“-

[an1 < [092] where n is the

probabilities p. and p2 such that

nuer of sawle observations. When papulation percentiles TIP and
i
’T’ are equated to sample percentiles t and t we get
I’2 ‘ Pi 92
WM my.)
“9 ‘ '9 A , -
A I 2 and ’éIt.,£m'(—'—>.
P. / "P

 

m I r
Mt... - M52

in case of the 3-parameter Heibull law let us pick up three cumula-

tive probabilities, namely, p' , p2 and p3 such that
[09‘] < [npz] < [”3] . Equating population percentiles to samle per-

centiles we get the following equations,

' l
tp - G + 0 . [ 13110-904] '
l

i l
t -6+O'[/f/V\(l-p)"l’
P: 2

l.
t -°+°'(£N\(l-r3)"l .
'3

These equations give

t - -i
P! ‘92 . I Jtn"'92ll
LEW-pp")

l
- [Mn-92W] "

l
- I ,EN.(l-p,)"l "

-105;

which provides estimator for m by the successive approximation procedure.

Nae

i l A
/‘ I.) -l ‘3' -l i; “'
0- ($3 - tpz) i JAM-93) - ibii-pzl j .
and

l
l ..
’3 - :9. ~13 El (hm-p941 9' .

it is clear that the above percentile estimator of m can be ob-
tained by successive approximations. This may not be convenient in
many instances. but, we'can derive a modification of the percentile
estimator for m in formula form if we use an indirect satisfactory
estimator for 6 . The smallest sample observation is the sufficient
statistic for 6 and can be used as its estimator. Denoting a satis-
factory estimator for c by c“ , modified percentile estimators for

m and O are

A {EMU-9,)" - QMQ/vc-(l-pz)"
a . Rap. - 6*) - flit/\(tpz - 6*)

 

and

>
'o

 

0 I '
Imil-p'rrw

 

-105-

respectively with t"' and tp2 as senile percentiles corresponding

to predetermined ciaaulative probabilities p' and p2 satisfying
[09'] < [092] '

iiow we present some other estimators for the parameters of lieibuli
laws. in case of the 3-parameter iieibull law, the expression for the
ci-ulative density function (c.d.f.) is found to be

I
flat) I l - e. 5 (x-C).

vdiich gives
I LOVE) 'XNW "‘ .QMQIAU-Hx) )4 .

ilotlng that, l - F(x) is the probability that an item will survive
beyond x , we call l - i-‘(x) - a(x) , the reliability of the item.

The equation,

QMQN» 840‘) - m XMix-G) - M0

is a linear function of (x-C) . it is known that any sepia distribu-
tion function of a continuous random variable obeys the uniform law on
the unit interval. For the sake of convenience, we denote Mullah)
by y . 0n the basis of saqale observations: t. < tz < ... < t" , we

define

 

-l07-

i!
sin. 0. 6) - Ziy, - aim/fire) 41M”: (2)

New. 11-0 . Iéj-o . %§-0 (3)

"we

yield 3 equations,

. in
if)”: (We) - £61m :Elfltrfi) . 3y, LN?"
C“ CH L‘ri

mZWh 4) - name - in

(1‘!

M
:2!“ )‘MTi "’ E i ’i
-/(M/0 3:76- ' i EFF
Lil (ii

i.:\

 

iiere it is easy to get expressions for m and O in terms of 6 .
The real difficulty is in obtaining estimator for C . lie can overcome
this difficulty if we use an indirect satisfactory estimator for 6 .
One such estimator for C has been pointed out earlier.

by means of the equations (2) and (3) we derive estimators for the

parameters of iieibuli laws under various situations.

 

i)

ii)

and,

-103-

l-parameter Heibull law:

a) Special Case: 6-0 and O-i .
n

E.— ’i Mt.

b) General: 6-60 and 0-00 .

A :- v, filmy-60) + 1m 00 :TX/Mtrco)
2: iv: (‘i'Go’

2-parameter Heibuli law:

a) Special Case: I

A :(y,-)1;A\:
it «Mr. WW

 

_ 409-

when n n

Zn, - ‘) 1m“, - ‘o’

A iIl

Z [Mia-60) - bait-60)]:

 

 

and,

iii) 3-parameter Heibull law:

iiere using 6* as a satisfactory indirect estimator for G ,

 

we have
Z (r, - a (mu, - a")
lIl
Z [Wig-6*) - Wit-6*”:
i-i
and,

 

 

«410-

in case of the zoparameter liaibull law we can derive estimators for
m and O in forlmlla form from another consideration as well. in the
field of life testing, the concept of intensity function, }\(t) (also
called force of mortality or hazard rate) plays a very useful role.
This is defined as

2“ ) f t probability density at t of a failure time ramda variable
. 4+ -
ii t

 

reliability function at time t of the item under consider-
t."-

7 mt...
iiow for the Z-parameter lieibuli law with c .. o , )\(t) - e .

This gives M“) I Mb) + (m-l) QM; - INC idlich is a linear

function in t . lie can convert sample observations to the data on in-

 

tensity function by fol lowing Lomax [l8]. Let us denote QMKR) by
z . 0n the basis of sale observations: t' < t2 < ... < t" , we de-

fine

ll
hie. oi - Z (2, - Mia) - (I-l) Mt, + 9m»? .

iiere %% I0 and %—% IO yield
2 (2, - 5) Mt,
II

2 (QM‘i 'W’2

A
mIl+

 

lm.’ ' “‘ "WP-A -

 

 

~ll2-

VI. lITEliSlTY FWTIOI: GEKRATOR 0F FAlLlIE LAHS

The statistical analysis of data pertaining to life, death or failure
time of inanimate and animate objects (e..g. i. length of life of elec-
tric bulbs, electron tubes etc. which are specimens of industrial produc-
tion and, ii. reaction time observed while determining the effect of
drugs on mice, rats etc.) and also fatigue of men, machines etc. can be
successfully conducted only mien we correctly know the probability density
function (p.d.f.) of the random variable (r.v.) concerned. The problem
of actual determination of the p.d.f. of a r.v. arising in the field of
life testing has not yet received due attention from the statisticians.
on the basis of eapirlcel evidence of Davis [l5], the exponential lair lms
taken as a good first approximation to the distribution of length of life.
Epstein, Sobei' and others have made useful statistical contributions
idlich are valid under the assintion of exponential'ity. Realizing the
limitations of this asswtion, some work has been done with the lieibull
law [l2]. The ergo-ants put forward in favor of the use of the lieibull
' law appear in observing that the intensity function, defined in chapter
V. of the r.v. representing length of life can change with time in con-
trast to the exponential law wdlose intensity function is constant in
time. Since several industrial products show aging effect, it becomes
. apparent that the intensity function of such r.v. must essentially be a

function of time. The matter does not seem to end here. The intensity

-113-

function, as a matter of fact, appears to be a very useful tool in gen-
erating a large nmaber of p.d.f.'s appropriate to life testing data.

The term, intensity function, is due to Ml [l7]. itwis synony-
mous to hazard rate or force of mortality in actuarial statistics. For
the sake of convenience to the readers we restate the definition of the

intensity function, )qt) .

ft ft
Ami-77%;) - 5-H- , provided t>c ,

where f(t) is the p.d.f. of a r.v. representing length of life of an
item and a(t) I l-F(t) is the reliability function of the item which
is the probability that an item will survive beyond a given time, t .
before we proceed further, it may be proper to list some of its siqlle

properties.
i) Alt) 3 f(t) since 0 5 F(t) 5 l . .
This ilmllies that K“) is always non-negative.

ii) The reciprocal of the intensity function is called llills' ratio.
it has been studied by lillls, Gordon, birnbaua, Des Raj etc. in different

colulections.

iii) A“) .ybelndependent of t; itmayincrease with t

without limit; it may converge toward a constant.

iv) A“) I {-13- with t > 6 , gives

-l "i-

a) f(t) I A“) e and
b) f(t) . - il'(t) where t >‘c .

The proofs for a) and b) are inediate and hence we omit them. Unless
otherwise specified 6 will, for convenience, generally be taken .as
zero in the following.

The intensity function of the 3-paramater lieibuli law whose p.d.f.

is
..i - -'- (t-ci'
ISC'G.) . O , t>G€(' O, m) ‘ O,IE(os 0) '
f(t) I
0, otherwise
is found to be

>\(t) - m ,

illen m . l, )\(t) .g. idlich is the intensity function for the 2-para-
meter exponential law. The sinlicity of the intensity function for these
failure laws, idlich have been found to agree well with wirical data in
many cases, and the appeal of the idea that the intensity function, an
instantaneous propensity to failure in an object with has survived to

time t, should be a siqle function of t, suggests that forms derived
from other siqlle assumtions about the behavior of the intensity func-
tion may find application in a wider class of cases than those covered by

~ll5-

the Ueibull distributions. One naturally considers using a polynomial

in t for the intensity function. if

 

p
)6” - E a! t' then
iIO p
.i ti'l'i
P _ he
'i t' e ' 1 " ' , t > o
iIO

O , otherwise.

Unless the polynomial is restricted, we have the trouble of too many
parameters to be able to tell without a large ni-aer of data whether the
fit is good because of the appropriateness of the form or because of the
Mr of parameters.

In some applications it is reasonable to assume that the intensity
function is a decreasing function of time. Ltmax US] has pointed out
that >\(t) I 5-1-1 appears to be more appropriate for the data relating

to retail, craft and service groups in business failure and Mg) . . .- bt

for manufacturing trades. Corresponding to A“) I 5%? we get

- l
§<l+§> (u ), t>o

0 , otherwise,

f(t) I

and corresponding to A“) I a a- in we have

 

'1 I»: at—

-ll6-

-[~~: (' --"")]

ae , t>0

f(t) I

0 , otherwise.

b t
it is clear that m I . «l- . is a linear function of t . be

noting X'Tt) by z we obtain sons estimators for a and b on
the basis of sale observations: t' < t2 < ... < tn in the following

mannen

In! m re 0 and M I 0 yield two

equations whose solutions are

Z; -Z:

 

 

-‘l7-

there 'z-m-l- Z 2' . Similarly from )Ut) Iae- M
iIi

we get/(MAR) I he - bt which is a linear function of t . iiere
n
let h(a, b) I Z“, mgMa «l- bt,)2 , where y I fiMXh)
‘ iIi
and t t t are l ob 7"
l’ 2, , n salmle servations. iiow "1—; I0 and
r)

T I 0 yield two equations whose solutions are

3’

b
_. A-

Q'ey‘ibt ’ and
n
E (t‘ " I) (V. 'y)

Ab . iIi

n
:5” - 't") z
iIi
where t and 'y' are arithmetic means.
Finally we generate failure law from the consideration of growth
cruves. iiere A“) I _ (a 1i) , which is known as
. l -l- e

a logistic function, gives

 

1.l +fi%: ’
(l + s““* at)
f(t) -
o , otherwise.
F rthermore, .' ..45‘51... I ia‘+ bt
“ Jr“! mm)

t . Hence letting

n
Na. bl-E in, «ax-ing2

 

Hhere u I J)Y\ ijt) and
l- Nt)
observations, :;%%E ..(l and
r\ _. A..
a:- u - D t , and
0
Eu -t)(uI w)
A in) '

 

-li8-

t>0

is a linear function of

-119-

where t and 'u' are arithmetic means.

t
The transformation u I _S . >\(x) dx is helpful in reduc-
o ,

ing unwieldy expressions of failure laws to relatively slqle forms pro-
vided the parameters involved in the intensity function are known. in
particular, with extreme value distributions which have possibility of
applications in life testing problems the above transformation may prove

of linense value.

 

~i20-

liBLimlfl

[l] benjamin Epstein and Hilton Sobel, ”Life Testing," Journal of the
American Statistical Association, Vol. #80953), pp. - .

[2] 35;. Lehmann, Testing Statistical Hygtheses, John liliey and Sons,

[3] O. A. S. Fraser, mramatrlc iiethods in Statistics, John liiley
and Sons, l957.

[ll] Edward Paulson, 'bn Certain Likelihood-Ratio Tests Associated with
the Exponential distribution,” Annals of Mathematical Statistics,
VO'O '2’ 09‘“), ”a 30'-3we

[5] ii. Pearson (Editor), Tables of the l lete cease function
biometric Laboratory, 533, i511.

[6] J. lieyman and E. S. Pearson, ”Sufficient Statistics and Uniformly
ilost Powerful Tests of Statistical Hypotheses,” Statistical ile-
search iiemoirs, Vol. l(l936), pp. ll3-l37.

[7] benjamin Epstein and Chia Kuei Tsao, ”Some Tests based on Ordered
Observations from Two Exponential Populations," Annals of llathema-
tical Statistics, Vol. zii(i953), pp. #58466.

[8] Phillip b. Carlson, "Tests of Hypothesis on the Exponential Lower
Limit,” Skandinavisk Aktuarietidskrift, l958- lieft l-z (pp. 10-90.

[9] b. O. Peirce, A Short Table of lnte rais Ginn and Co., l929 (page
65, formula Sis), 3d rev. :3.

(lo) ilarald (truer, llathematical lieth of Statistics, Princeton Uni-
versity Press, '37. ‘

[ii] A. E. Sarhan, "Estimation of the ilean and Standard Deviation by
Order Stagstics,” Annals of Mathematical Statistics, Vol. 250990,
"a 3'7'3 e

[l2] John ii. K. Kao, ”The iieibull Distribution in Life Testing of Elec-
tron Tubes,” Unpublished paper.

[l3] John A. Ouggan, "Utilizing the lieibuil Distribution in Life Testing,”
Engineering Memorandi- lie. 20, bendix Aviation Corporation.

[iii] b. Epstein, 'Estimates of lean Life based on the rth Smallest Value
in a Saple of Size n brawn fra an Exponential Population," m

University Technical 2%“ lie. 2, prepared under Oil Contract Nonr-
S 00 , IR , a Y '952e

 

Us]
[16]
in]

['3]
['9]

[20'
[2']
[22]

[13]
[2‘5]

[25]
[25]

l 27]

i231

-12i-

O. J. Davis, "An Analysis of Some Failure Data " Journal of the_
American Statistical Association, Vol. 1009525, W - .

Paul Gunther, "Techniques for Statistical Analysis of Life Test
Data," General Electric Raport iio. #566L278, Nov. 23, l956.

E. J. Gubel, Statistics of Extremes, Colubia University Press,

i958.

K. S. Lomax, "business Failures: Another Exaqle of the Analysis
of Failure Data," Journal of the American Statistical Association,
vol. 1.90955). mm

b. Epstein and ii. Sobel, "Some Theorems Relevant to Life Testing

from an Exponential Population," Annals of ilathematical Statistics,
Vol. 25095“); PP. 373-3“-

K. Pearson (Edi tor),

Tables of the i lete beta-Function bio-
metric Laboratory, London, '53}.
bateman iianuscrlpt Project, mgr Transcendental Functions, Vol. l,
iicGraw-iiill book Comany, inc., . ..
E. T. Copson, 232% of Finctlons of a ”lax Variable, Oxford
University Press, .

b. Epstein, "Simle Estimators of the Parameters of Exponential
Distributions than Saimiies Are Censored," Annals of the institute
of Statistical ilathematics, Vol. iii(iio. l, p. . - .

Charles E. Clark and G. Trevor liilllams, "Distributions of the lam-
bers of an Ordered Sample," Amals of hathematical Statistics, Vol.

Gunner blom, Statistical Estimates and Transformed beta-Variables
John Wiley and 5s, '33.

F. ii. David and ii. L. Johnson, "Statistical Treatment of Censored
Data," biomatrika, Vol. iii(lssli), pp. 228-”.

Charles E. Clark and G. Trover iiilliams, "Distributions of the hem-

bers of an Ordered SaqleuAn Addenda," Anals of Mathematical Sta-
tistics, Vol.30(l959), p. bio. W

ii. G. De bruijn, As totic iiathods in Anal sis north-Holland
Publishing Co. - item”, '33.

‘11!
‘vi.

rain
(3
”U

a‘

On

 

 

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