v—

 

 

WEED RULES FOR THE 7 . E

SEQUENCE COMPOUND DEC£SSDN PROBLEM WITH ‘ "

. mxn COMPONENT

_ DiSsertation for the Degree of Ph. D. V
‘ MHCHIGAN STAYE. UNWERSITY
ROBERTJOHN-BALLARD _ ’
1974 — -

.-. .J.-._1-‘. a}.

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I" . "’ iv. Li!- I‘ £*¢ .1 t
g I.

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E. .r‘ A: LU. 1‘. 1!! 1:3 7

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This is to certify that the

thesis entitled

EXTENDED RULES FOR THE SEQUENCE COMPOUND
DECISION PROBLEM WITH tn X n COMPONENT

presented by

Robert John Ballard

has been accepted towards fulfillment
of the requirements for

Ph.D. Statistics and

Probability

degree in

 

 

é/yi/Z/

Major professor .

 

 

Date August 12, 1974

 

0-7639

 

ABSTRACT

EXTENDED RULES FOR THE SEQUENCE COMPOUND
DECISION PROBLEM WITH m X n COMPONENT

By

Robert John Ballard

For a sequence compound decision rule 4g = (91,...,gh),
where gh’ is a function of the first a observations, 1 s a s‘N,
let 5N(3Ӥ9 denote the compound (average) risk at state
Q a (91,...,QN). The usual standard for compound risk is R(GN)
where R is the Bayes envelope for the component problem (the

simple envelope) and G is the empirical distribution of com-

N
ponent states 91,...,eN. Much of the literature in compound

decision theory has dealt with the construction of rules satisfying

TEEN supa[l_zN(g,Q) - R(GN)] s o

for various components.
. k k)
More stringent standards for compound risk are R (GN ,

k - 1,2,..., where Rk is the Bayes envelope for a construct called

the Pk game and G; is the empirical distribution of the k-tuples

k _ k _ k =
3.1 - (91,---:6k), 9,2 " (62,---,9H1),---,9N_k+1 (eN-Hl’...,eN). The

k+1 standard is asymptotically more stringent than the k standard
whe R1( 1) = R(G )
re GN N .
We will consider the m X n component and demonstrate for

each R, a k-extended sequence compound rule ‘9, “a being Pk

Robert John Ballard

k .
Bayes versus an estimate of Ga based on the first a-k observa-

tions, k s a s N, which satisfies

___. k Pk
limN SUP&[3N(_Q,52) - R (GN)] s 0

-l S
with a rate of N / . Furthermore, we compute the envelope R

where the component is discrimination between N(-1,l) and N(l,l)
and use Monte Carlo methods to estimate the compound risks for some
k = 1 and k = 2 procedures and various .3 and N in order to
determine possible small N advantages of extended procedures. In
addition, we compute some Bayes compound risks versus strictly

stationary Q for various N.

EXTENDED RULES FOR THE SEQUENCE COMPOUND
DECISION PROBLEM WITH m X n COMPONENT

By

Robert John Ballard

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1974

TO MY PARENTS AND CINDY

ACKNOWLEDGMENTS

I wish to express my sincere appreciation to Professor
Dennis C. Gilliland for his guidance of the research for this
dissertation. His comments, observations and suggestions were
invaluable. Further, I am indebted to Professor James Hannan for
suggesting the problem and many helpful observations.

The financial Support provided by the Department of Statistics
and Probability and the National Science Foundation made my graduate
studied possible. I wish to thank them and Noralee Barnes who

demonstrated both patience and skill in typing this dissertation.

iii

Chapter

1

TABLE OF CONTENTS

INTRODUCTION ....................................

ASYMPTOTIC SOLUTIONS TO THE EXTENDED SEQUENCE

C OMPOUND PROBLEM ................................
1. Preliminaries .............. R ................
2. Estimation of the Empiric G ...... .........
3. Main Result ................ q ................
COMPUTATIONS . ...................................
1. Introduction ................................
2. Computation of R2(G) .......................
3 Simulation Study of the Risk Performance of

4.

Some Compound Rules .........................
Performance of Compound Rules when the
Parameter Sequence is a Strictly Stationary
Process . ......................... . ..........

CONCLUSIONS .....................................

REFERENCES ......................................

iv

OV-L‘

18

l8
19

24

36
42

43

Table

10

ll

12

l3

14

Values of

Values of R2(p,5) for p

6:

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Sequences

Estimated
Estimated
Estimated

Estimated

LIST OF TABLES

R(p) for P

Risks for N
of Type I

Risks for N
of Type II

Risks for N
Risks for N
of Type II ..

Risks for N

Risks for N

of Type II ..
Risks for N
of Type I

Risks for N

of Type II
Bayes Risk of
Bayes Risk of
Bayes Risk of

Bayes Risk of

= .l(.l).5
= 1(.1).5 and
= 20 and Parameter

:50

= 100

udb

edb

and Parameter

and Parameter

of Type I ............ . .......... ....

and Parameter
of Type I ...........................

and Parameter

for N = 50
for N = 50
for

for N = 200

oooooooooooooooooooooooooo

27

28

29

3O

31

32

33

34
38
39
40

41

LIST OF FIGURES
Figure Page

2
1 Plot of R (p,6) as a Function of 6 for
fixed p ...................................... 23

INTRODUCTION

The compound decision problem introduced by Robbins (1951)
involves N independent repetitions of a component problem. For a
compound decision rule ‘9 = (Q1""’3N)’ where $3 is a function of
all N observations, 1 S a s N, let RN(Q,§) denote the compound
(average) risk at state Q = (91"°"9N)' The uSual standard for
compound risk is R(GN) where R is the Bayes envelope for the
component problem (the simple envelope) and GN is the empirical
distribution of component states 91,...,9N. Robbins (1951)
demonstrated a "bootstrap" compound rule '9 satisfying the two-

sided (limit) version of
"T‘ R ,f _ R
(1) MN supfiLNm a) <an s o

for the featured example where the component is discrimination between
N(-l,l) and N(l,l). The term ”bootstrap" refers to the fact that
each $5 is component Bayes versus an estimate of GN based on all

N observations. Much of the literature in compound decision theory
has dealt with the construction of bootstrap rules satisfying (1)

for various components. Recently, Cilliland, Hannan and Huang (1974)
have shown that a large class of Bayes compound rules satisfy (1)

for 2—state, compact risk component.

The sequence compound decision problem introduced by Hannan

(l956),(1957a) restricts the class of compound rules to ‘Q where

l

$5 is a function of the first a observations, 1 S a S N. Hannan
(l956),(l957a) and Samuel (1963) were the first to propose sequence
compound rules to satisfy (1) for various components. The rules are
bootstrap in nature, Eb being component Bayes versus an estimate of
Ga based on the first a observations and possible artificial
randomization, l s a s N. In the meantime, many additional components
have been considered in the sequence compound problem.

The notion of using more stringent standards for compound
risk is found in Johns (1967, §4). He introduces a set of standards
Rk(G;), k = 1,2,..., where R1(G;) = R(GN) and the k.+ 1 standard
is asymptotically more stringent than the k standard. (Here we
have used notation similar to that of Gilliland and Hannan (1969)
who give the most general treatment of these standards. Rk is the
Bayes envelope for a construct called the Fk game and G; is the
empirical distribution of the k-tuples g: = (91,...,ek),

512 = (92""’91¢+1)""‘%:i-k+1 = (9N_k+1,...,eN).) Johns (1967) for a

two action component and fixed k gives a sequence bootstrap rule

g5 g5 being Pk Bayes versus an estimate of G: based on the first

a observations, k S a s N, which satisfies the two-sided version of
.___ k k

(2) limN Supfl[§N(g,Q) - R (GN)] s O

Swain (1965) in a thesis under Johns treats some squared error loss

estimation components, obtaining (2) with rates later improved by

Yu (1971). Swain calls the compound problem with standard Rk(G;)

the extended compound decision problem.

We treat the extended sequence problem with m X n component

and, thus, cover the original featured example of Robbins. We

demonstrate for each k, a sequence compound rule which satisfies

(2) with rate. For proving this asymptotic reSult we rely heavily

on Van Ryzin's (1966b) analysis of the unextended version of the
problem. Furthermore, we compute the envelope R2 for Robbins'
featured example and use Monte Carlo methods to estimate the compound
risks for some k = l and k = 2 procedures and various 3. and

N in order to determine the possible small N advantages of
extended procedures. In addition, we compute some Bayes compound

risks verSus strictly stationary .Q for various N.

CHAPTER I

ASYMPTOTIC SOLUTIONS TO THE EXTENDED SEQUENCE COMPOUND PROBLEM

1. Preliminaries.

 

Consider a component decision problem with _m states
@ = {1,2,,,,,m} indexing .9 = {P1,P2,...,Pm} where the P1 are
distinct probability measures on (Iufi). Let U be a finite

measure dominating .9 such that
(3) fi = dPi/du ‘5 K, i E O

for some positive finite constant K. There is no loss of generality
m

in making this assumption since we may always take u = Z P. and
i=1 1

K = 1.

Suppose the action Space is 6': {1,2,...,n} and the m x n
loss matrix L has non-negative finite elements. A (randomized)
c I o I I *
component dec1sron rule, m, is aIG-measurable mapping into a’,

the (n—l)—dimensional simplex of probability measures on at That

n
' = ... h» r. 2 O, l s ' s , d * , = 1. Th
13, m (m1, ,mh) w ere P] J n an Ll QB e
risk of cp at state i is

n
(4) R(i,e) = j(.2 L(i,j)mj)dPi

J=1

L t P = P X P X...X P and = , ,..., where
e _ 91 90 $2 (91322 EEN)
‘ k

for each a, $5 is a Ea-measurable mapping into (i. The risk

incurred in the a component decision by the sequence compound rule

4

Q is
n
(5) Rates) = f<jE1L(ea,J-)ea,j)dz
and compound risk at N repetitions and state .g is
N

-l
(fimg) =N 2: R (3.59)
a=1 0

<6) EN

In order to motivate the construction of rules satisfying

(2) we describe the Pk decision problem. In our case it is an

k k k
m X n problem with states 1 = (i1,...,ik) 6 ® indexing product

distributions P k E 9k. The loss matrix Lk is mk X n with

1
Lk(ik,j) = L(ik,j). A (randomized) decision rule m in the Pk

k * k
problem is a 6’-measurable mapping into a'. Letting R (ik,m)
denote its risk at state ik, the Bayes risk of m versus a prior

k
G on ® is

(7) RkCG,<p)

k k
ZR (1 ,<p)G k
k 0

l

[H

n
k k k k
P . .
,[jileq ).E L(1kaJ)fik(§ )Gikldu (a)

3:. __ _

f. (X )f, (x ) ... f, (X ) and uk is the k-fold
11 l 12 2 1k k

III

R
where f k(>_<_)
i

’ th
product of u. Let Li denote the j“ column of Lk’ for 1 S j S n,
k
f denote the m X 1 matrix of densities f k and Ljf denote

k .k i .
the m X 1 matrix with components Lk(l ,j)f k' Letting ( , )k
k i
denote the usual inner product in Em -Space, it follows from (7)
that a Bayes rule in the Pk problem places all its mass on the j's

which minimize Aj(§3) E (Li£(§¥),6)k. A particular version denoted

by (pk{G} is

1 if j is the smallest integer such

that A,(§F) = min A (5%)
J lsth

<8) «he; x l =

0 for the other j

The Specification k = l in the Pk construct gives the
component decision problem. For Simplicity of notation we abbreviate
k = l by omission whenever it is both possible and convenient.

Theorem 2 of Gilliland and Hannan (1969) suggests that for

the compound problem the sequence rule which plays P Bayes versus

k
k
an estimate of G , k s a s N, may satisfy (2). Let x = (x ,...,x )
a ‘0 1 a
and Ev = (xv,...,xv+k_1) for a,v = 1,2,... . The sequence compound
rules .Q that we investigate have
kak k
9 = G ;x , 2 k
< ) QQCEQ) <9 i a 7,140.11 01
6k nk mk
where for each a 2 k, Ga = G (xa) is a E -valued estimator of
a
hk
Gk. When G is a function of x} k only we refer to ‘g of (9)
a a a-

as a delete bootstrap rule.

Since the Pk construct is itself a finite state, finite
action decision problem, many of Van Ryzin's (1966b) results apply
to the analysis of extended rules (9). Consequently, notations,

a preliminary lemma and the main theorem to follow are patterned
after his work on the k = 1 case. However, we first treat some of
the problems related to estimation of higher order empirical dis-

k
tributions G
O!

2. Estimation of the Empiric G:

For general results on the estimation of finite mixtures
see Hannan (l957b),'Teicher (1963), Robbins (1964, §7) and Van Ryzin
(1966a, §3). Here we discuss the estimability of mixtures of the
Special finite class of product measures 19k and the Structure of
the dual basis estimators.

k
The class ‘9 is estimable if there exists a function

k .
= O I O h . .
h- (hl , o o . 3 1, , m9 . . o am) on I WIth components In L1(|J'1£)
such that E h_= w for all mixtures Pm of .9k and h is said
w _
to be an unbiased estimator of .9k. (Em denotes expectation with

reSpect to the mixture Pw.) Such functions provide kernels for

. . k .
unbiased estimators of G Since
a

k
(10) s kh k = 5(gk, 3k) for all g3, 3k 5 a

Q L

where 5 is the Kronecker delta function. From Van Ryzin (1966a,

k k
§3),‘9 is estimable if and only if ‘9 is identifiable, if and only

k
if 3k 5 {f klik E @k} is linearly independent in L1(u ).

1
Remark 1. 3 = {f1,...,fm} linearly independent in L1(u)
implies 3k linearly independent in L1(uk).

Proof: Suppose

(11) z a f = o a.e. u

k l .
Almost every 5 -section must be 0 a.e. u, so that from

f k = f k—lfi and the linear independence of 3 it follows that
i i k

k-
(12) z a k_1 f k-l = o a.e. u 1, i e e .

By induction on k it follows that a k = O for all 1k 6 @k.

i _
If h_ satisfies (10), then

1 a-k‘tl

(l3) ha(xa) (a - k.+ l) iEI hflii

Ill

is an unbiased estimator for 6:; and if its components are in
L2(uk), then it is consistent. Henceforth, we take 3 to be
linearly independent which assures the existence of h satisfying
(10). From Robbins (1964, §7) and Van Ryzin (1966a, Theorem 1) if

h_ satisfies (10) with components in L2(uk), then

(14) hk=f +g
* k

where the f form the (unique) dual basis of S , the subSpace

,k

k i k k k
of L2(u ) spanned by 3 , and g k I.Sk for all i E O .

* * a l

*
Remark 2. f k = f f ... f, for all ik where

. i i i
x * * l 1 2 k
[f1,f2,...,fm] is the (unique) dual basis of S, the subSpace of
L2(u) Spanned by 3.
Proof: The dual basis of Sk is the (unique) Subset of

k
m element from the span of 3k with elements satisfying (10).

* * e k .
The products f, f, ... f, are in the Span of 3 Since
1 i i
f1,f2,...,fm are in the Span of 3 and satisfy (10).
From any unbiased estimator h = (h1,...,hm) of .9, one
. . . _ k
obtains an unbiased estimator h — (h11 ..l’°°°’hmm...m) of .9
by taking
k ,k k
(15) h (i ) = h, (x )h, (x ) ... h. (x ) for all 1 6 ® .
ik 11 l 12 2 1k k —

We call such an estimator a product estimator. Our theorem concerns

an extended compound procedure based on an unbiased bounded product

k
estimator of '9 and Remark 2 shows that such an estimator is pro-

* * e * * *
1f1 .,f f ... fm). (In general, the class

1". m m
of unbiased product estimators of ‘9k does not exhaust the class

*
vided by 3; (f

of all unbiased estimators of .Qk.)
The following remark can be applied to determine the rank
of the covariance matrix of a product estimator.

Remark 3, 'Let h = (h .,hm) be unbiased for «9 ‘with

1,..

components in L2(n) and let V(i) denote its covariance matrix
under Pi’ i E @. Let h_ denote the product estimator (15) and

k ' . k
let V(i_) denote its covariance matrix under P k’ L E @k. Then

1
Rank V(i) = m for all i 6 ® implies Rank.V(iF) = mk for all

k k
i E ® .

Proof: We use the fact that the rank of the covariance

matrix is the dimension of the Span of the centered components in

k k

L2. Let ‘i 6 ® and
16 Z a. (h. - E h, ) = O a.e. P .
( ) k 11‘ 11‘ iklk 1k
1 _ _
Taking E k-l expectation gives
i
(17) Za.(Ek_h_)(h. -E,h,)=0a.e.Pk.
,k 11‘ i likl Jk 1k Jk i
l _
. = .k-l ,k-l .
Since E,k-lh,k-1 6(i ,1 ), it follows that
.L l
m
(18) z a (h. -E, h.) =0 a.e. 9,.
j=1 ik 1,j J 1k J 1k
Since V(ik) has rank m, (18) implies a k-l = 0, j 6 ® .

i ,j

lO

3. Main Result.

 

We will now State a lemma which is used to obtain rates for
our decision procedure. Let §_E (X1,X2,...) be a sequence of
k-dependent random variables with means zero and finite variances.
(Here k-dependent means that the variables (X1,...,X ) and
(X ,...,Xt) are independent for all 1 s r < s s t and s - r 2 k.)

S

We will use the notation
n 2 h __
= , B = E s , F = s s B ,
sn 351x], n ( n) n(x) Pr[ n x / n]

2
¢<x> = <2n>‘% jfm e't lzdt

AS a corollary to the proof of Theorem 1 of Egorov (1970)

it follows that if there exist constants a b > 0 independent

k’ k
of n such that

n 2 l

(19) \an S ak,

(20) B 2 b n, n 2 1

then there exists a constant ck > 0 independent of n and the

distribution of X. such that

(21) SuleFn(x) - §(X)\ S ck(bkn)-1/5, n 2 1

From this result, Lemma 1 easily follows.
Lemma 1. Using the notation and conditions (19) and (20)

from above,

-L -l/5
2
Pr[d s Sn s d + a] 5 (2n bkn) a + 2Ck(bkn) , n 2 1

for all real d and a 2 0.

11

In this section we consider the extended compound rule 9

of (9) with C: = h _ , a 2 k where

a k
_1 a-k+l k
(oz-k+l) Z h(x,) azk
. -i
i=1
(22) 13(5) =
CY CY
Q a < k
and h_ is defined by (15). We take h = (hl’ .,hm) to be an
unbiased estimator of ‘9 ‘with
(23) max \h‘ SH<co a.e. p,

lSjSm
For convenience in establishing the asymptotic result (2)

for the extended delete bootstrap procedure

— k - o k o
(24) 9J5.) — cp awe”). gm), 0: 2 k.

we consider the average risk over the decisions concerning

ek,ek+1,...,eN. i.e..

N
k -l
(25) SHE’S) = (N - k+ 1) 2: R (9.49)
a=k a
. . - k . .
With 36 of (24), the independence of ha-k and Ea-kfil implies

k k k -
(26) Ra(-Q’59) - ‘g‘; R (gadkfl, (p {ha-kp’ k 5 01 s N

where here and throughout this section E. denotes expectation with

= XP x...x , .
reSpect to P. P81 82 P8N measure on (x1 x2 xN)
Using the notation of Van Ryzin (1966b) we let e k

k i
Em -vector of all 0's except 1 in position i, and let

denote the

k -
p k(w) = Rk(1 ,m), 1% E @k. Since xk is independent of h
i

QI-k‘l'l CY-k

aEd h. is unbiased for 19k we have

12
k ..
(27) Ra(_Q,gQ) = geek , 9(cp {ha-k]))k
"ow-H1

k k —
= §.(t_1(>_<a_k+1): m {ha_k}))k, a 2 k .

Using (27) in (25) it follows that

N k k - k -
aikym’ia-W)’ m» {ha-kl) - m (ham,

(28) gay = (N - k+ 1)‘1

-IN
+(N-k+l) )3

k _
1. seeim). .o_(cp than), -
(Y

The left inequality of Lemma 3.2 and equality (3.4) of Van Ryzin

(1966b) applied to the Pk decision problem, together with (28)

gives
(29) Rk . A + E Rk '
where
-1 N k ' ..
(30) AME) = (N - k+1> z 2 were. 4&1). Lire“) - LJ _f_(£‘))
a=k j<j' a k

kl-

k- k k - k k- k k - k k k
{Na-(1101*; Y. )CPj:(ha; Y.) + ((30101; imjmha‘k; 1)}du (x)

k . - k
and y - (y1,...,yk). By the unbiasedness of hN for GN
k - k k — k k k _

m (hN) - R (GNU s th - 6N, p_(.-,o (61,)»k - 0

so that by (29)
k k k

(31) was) - R (0N) sANqa)

We now proceed to Show that if the kernel h = (h1,---,hm)

has full rank covariance matrix under Pi’ i E Q, then AN(3) =

N-l/S

O( ) uniformly in ‘3. Thus we establish (2) with rate for the

procedure (24).

13

k

Fixing y_, j and j and letting 3 denote the mk X l
. j k j' k . .

matrix Lk 2(1 ) - Lk f(y ), we see from definition (8) that the

factor in curly brackets in (30) is bounded by

a-k+l k a-k‘tl k
(32) [-2 (N(x.). 2) < S s 0] + [o < s s -3 (h(x ),g) ]
i=a-2k+2 ‘1 k i=a-2k+2 "1 k

where the square brackets denote indicator functions and

q-2k+l

(33) S = 1:1 (hflxi), 2)k -

A calculation Shows that

k
(34) (E(£1)’ 3>k = Zi,lzi+1,2 "' zi+k-l.k
where
, f , = 1, ,k-l

(h(xe) (yy)) Y

35 Z =

( ) aw
(h(xa), U(Yy)) » Y = k ’

- u
f = (f ( ),.--.f ( )) and u( = LJf - LJ f . With
(yy) llyv ,n yY yY) (yY) (yY)
H H denoting the Euclidean norm in Em-Space, (34), (35) and the

Schwarz inequality for Em applied to the Z5 y’ it follows that
k-l

k 2
(36> when, an s m“ Hkllu<yk>ll __r__I (my)

3 1

In order to apply Lemma 1 to the sum S of k-dependent
variables we need to investigate the variance of 8.

Lemma 2. Suppose h has full rank covariance matrix V(i)

2
under Pi’ i 6 ® and let N2 = min N. where N? is the minimum
isism 1
eigenvalue of V(i), l s i s m, (necessarily positive Since V(i)

14

is positive definite). Then for all 3,

k-l

(\u(yk)\\2 -n1 llf<yj>l12
J-

- 2
(37) Varfl(S) 2 [g~—E35ilqi k

where [x] denotes the integer part of x.

- 2
Proof: Letting r = [QT-Bil], we write
a-Zk-l'l r r
= Z ... Z = - _ -
(38) 8 .§ 1,121+1,2 i+k-l,k .2 8(3) + (S .2 8(3))
i—l j=l J=1
where
jk
(39) S(j) = 2 Z, Z, ... Z, _ , j = l,...,r.
i=jk-k+l i,l i+l,2 i+k l,k

By letting " denote conditioning on all xi except xk,x2k,...,xrk

we have

V§r(S(j))
1

V§r(S) =

r
J:

Defining

= z ...z ...
zJ’ (zjk-k+1,1 jk-k+2,2 jk-l,k-l’zjk-k+2,l ij-1,k-2ij+1,k’

. z z ... z
’ jk+1,2_na2,3 jk+k~lfl91xk

U(yk) \

f(yk_1)

f(y1) ka

and 6j = sz, it follows that

~ . 2 l 2
Var(S(j)) = Var(6j, h(xjk)) 2 x usjn

15

Hence,

(40) Var (S) 2 i2 r EH6 H2 .
3 =1- j

J

Letting Subscript jk+k—l on E and Var denote conditioning
th
x. "——
on all xi except jk+k-l and 6j,i be the 1 component of 6j,

we have for j = 1,2,...,r

E2

2
l = l
(41) alajl EEjk+k_1ll6J-\l
m 2
= g. z E, _ (6. .)
i=1 Jk‘i'k l 3,1
m
2 E_iEI varjk+k-l(6j,i)
m
2 2 2
= E Z ... Z V Z
— i5113191) jk+l,2 jk+k-2,k-l ar< jk+k-l,k)
The right hand side of (41) is bounded below by
2 2 2 2 2
l l ...
(42> w Hume) New.) 2W>
2 2 2 2 2
= A HU(yk)H Hf(y1)H §(ij+1,2).-.§(ij+k_2,k_1)

IV

2 2 2
k H”(yk)H Hf(y1)H Var(ij+1,2)...Var(ij+k_2,k_1)

k-l
2
l N (1H),) \l2
' l

1:

IV

2 k-l
x ( )Hu(yk)l
where use is made of

Var(Z ) = Var(U(Yk), h(X )) Z NzHu(yk)H2

B:k B

and

Var-(ZB ) = Var(f(yy), h(xa)) 2 xzqf(yy)uz for Y = 2,...,k-l .

’V

Thus, (40) - (42) imply (37).

16

BY (36) the bound (32) is seen to be zero if Hu(yk)“ =

Hf(yj)H = O for any j = l,2,...,k-l. Otherwise, Lemma 2 implies

 

(43) Varfi(S) 2 bk(q - 2k+l), a 2 3k
where
k-l
2 2 2
knuwu (I (New)
_ i=1
«4) bk“ keen :>o.

Further, using (3) and (36) and defining D = max ‘L(i,j) - L(S,t)\,
i,j,S,t
we have

\(Eofii‘), ekl s ak
where

ak = mkaD Kk .

Hence, conditions (19) and (20) of Lemma 1 are satisfied for Sn = S.
Using (32), Lemma 1, (36) and (3), it follows that for

a 2 3k the a, j, j' summand of (30) is bounded by

(45) 31(a — 2k+l)-% + B2(a - 2ic+1)'1/5
where
-t 3 g m3k/2H k
B1 = 2(2n) Znik (k+1)] M[U(I)] K
m4k 5 Hk 2k 5D 3 5 k 3k 5
82 = 4ckm / Hi / / [k(k+1)]1 /5[U(1)] K /

For k s q 5 3k, we bound the summands of (30) by B3 = ak[u(I)]k

which together with (45) gives

17

N
(46) AN(3) 5 (N ‘ k+1).1(n){(2k*1)3 +'3 E (d-2k*1) %
2 3 l
a=31¢+1
N
+32 2 (a-2H1)-1/5)
a=3k+l
Noting
N _ p“
2 (a - 2k+l)p s f: xpd S (p+l) 1N
a=3k+1 X
for -l < p < O, we can state
-1 -% -l/5
(47) ANQ) scln +C2N +C3N
= 2 n = 2 n = n
where C1 k( k+l)(2)B3, C2 (2)kB1 and C3 5/4(2)kB2 .

The above analysis establishes the following theorem.

Theorem. If the bounded unbiased estimator h = (hlso--,hm)

is such that V(i), the covariance matrix of h under Pi’ is of_
full rank for all i 6 ® and 'Q is the sequence procedure (24)
then there exists a positive constant Ck independent of 3

such that

k k k -1/5

CHAPTER 2

COMPUTATIONS

1. Introduction.

 

The target for k-extended procedures is Rk(G:). We showed
in Chapter 1 for the compound problem with m X n ‘component that
this standard is achieved asymptotically by certain extended
sequence compound rules. Gilliland and Hannan (1969, Corollary 1)
showed tha t Rk+1(GI:+1

k k
R (GN) so that these extended rules will have asymptotically lower

) is asymptotically more stringent than

risk than the unextended sequence and set procedures constructed by
Van Ryzin (l966a,b) for the same component.
In this chapter we use Robbins' original component where

m = n = 2, L(1.1) =L(2,2) = 0, L(1,2) =L(2,1) = 1, P =N(-1,1)

l
and P2 = N(l,l). We calculate R2 and compare it with R1.

Furthermore, we compute the compound risks for four unextended and
four k = 2 extended sequence compound rules for various N and
Q to determine possible finite N advantages of extended procedures.
Finally, for a selected unextended rule and a selected extended rule

we compute the Bayes compound risks with reSpect to various distribu-

tions on g, for N = 50, 200 to study the risk behavior of the rules.

18

l9

2
2. Computation of R (G).

Consider the F decision problem based on the Robbins'

2
component. Let f1 and f2 be the usual normal densities of
N(el,l) and N(l,l) with respect to Lebesgue measure n. (In

Chapter I u was chosen to be a finite measure only for convenience

in establishing the asymptotic result.) Then the Bayes rule (8)

versus G = (p1,1,p1,2,p2,1,p2,2) is

2 o o ‘
¢1{G, (x1,x2)} — 1 if and only if A1(x1,x2) s A2(x1,x2)

where
2
A1(X1,X2) = f2(X2) g pi,2fi(x1)
1—1
and
2
92(X1’X2) = f1(X2)iE1 pi,lfi(xl)

By defining

pi,2f1(x)
l
2
z 2 pi .fi(X)
i=1 j=l ’3

NJllP1l9

p(X)= i

it f0110WS that A1(X17X2) S A2<X17X2) if and only if P(X1)f2(xz) ‘

(l - p(x1))f1(x2) which holds if and only if x s c(xl) where

2

c(x is the cutoff value of the component Bayes rule versus

1)
l - p(x), p(x), namely c(x) = % log((l — p(X))/p(x)).

Hence

2
z p. j jR(j, d(x))fi(x)dx

(48) R2(G) =
1 j=l 1’

i

III"’ll\J

where

20

R(l, d(X))

l - Q(c(x) + l)

and

R(Z, d(X)) §(C(X) - 1)

2
We will compute R (G) for various C using the trapezoidal
rule for the integrations indicated in (48). Since the marginals of
2 . . . 2 2
ON are equal in the limit and R (GN) is the standard for compound

risk in the k = 2 extended compound decision problem, we will

2
compute R (G) for G which satisfy

(49) P1 2 + P2 2 = p2 1 + p2 2

’ 3 3 9

Letting p denote the common marginal probability (49) and defining
5 = p2 2/p for p > O, we obtain a convenient parameterization for

G satisfying (49),

G = C(Pyé) E (1 - p(2‘6)$ P(1‘D), p(1‘6)a P5):

2
and we will, henceforth, abbreviate R2(G(p,5)) to R (p,6). Since
each component of G(p,6) is linear in 6, aG(p,ol) + (l-a)G(p,62) =

2
C(p, a6 + (l—a)62) for O S a S l, the concavity of R implies

l
R2(p,5) is concave in 6 for fixed p. Furthermore, by Remark 1

of Gilliland and Hannan (1969), R2(p,6) as a function of 5 is
maximum at 6 = p; the maximum value being R(p), the k = l envelope
evaluated at the prior 1-p, p on states 1 and 2 reSpectively.

The k = l envelope R(p) is easily computed by hand and

is given in Table l to 5 place accuracy.

21

Table l — Values of R(p) for p = .l(.l).5

 

p R (p)

.1 .07006
.2 .11207
.3 .13875
.4 .15378
.5 .15866

Values of R2(p,6) for p = .l(.l).5 and 5 = 0(.05).l
were computed on a CDC 6500 computer using (48) and the trapezoidal
rule of numerical integration. These are given in Table 2 with
maximum column values underlined. The grid used in these computa-
tions was Sufficiently fine to guarantee that the error terms are
bounded by .005. However, for most values, we feel these errors
are less than .0001.

In order to more clearly understand the behavior of R2(p,6)
as a function of 6 for fixed p, we have plotted the values from
Table 2 in Figure l. The relatively flat nature of the curves
correSponding to Small p indicates that the extended rules will
not be much better than the less complicated unextended rules at
parameter sequences Q_ with a small proportion of states 2. (By

symmetry also at g with a large proportion of states 2.)

22

2
Table 2 - Values of R (p,6) for p

O
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90

.95

1.00

.06939

.06993

.07008

.06995
.06958

.06901

.06824

.06729

.06617

.06489

.06344

.06183
.06005
.05811
.05599
.05369

.05121

.04852

.04562

.04246

.03901

.10757
.10975
.11111
.11186
._11_2_12
.11188
.11123
.11019
.10876
.10695
.10477
.10220
.09925
.09589
.09212
.08791
.08322
.07801
.07220
.06570

.05831

.12474
.12957
.13312
.13569
.13744
.13845
ease
.13846
.13751
.13594
.13375
.13093
.12747
.12334
.11852
.11294
.10655
.09926
.09093
.08133

.07007

.1(.1).5

.11999
.12898
.13608
.14177
.14623
.14961
.15196
.15336
422.52.
.15336
.15199
.14970
.14647
.14226
.13702
.13069
.12316
.11429
.10387
.09154

.07658

and a = 0(.05)1

.07866
.09705
.11167
.12365
.13348
.14146
.14780

.15262

 

.14781

.14147

.13348

.12365

.11168

.09705

.07867

23

 

 

 

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24

3. Simulation Study of the Risk.Performance of Some Compound Rules.

In this section we compute and compare the compound risks of
various unextended (k = l) and extended (k = 2) compound procedures
through the use of computer simulation. We consider two different
unbiased kernels for estimating the empirical distributions in both
the delete rule (24) and the correSponding non-delete version where
hu-k is replaced by ha. Thus, a total of 8 different rules are
considered.

A bounded unbiased estimator is provided by the dual basis

of {f1,f2) in L2(u). In the example under consideration, it is

b(x) = (b1(x), b2(x)) where

b1(X) (A2 - Bz)-1(Af1(X) - Bf2(X))

1.20.) (A2 - 32,-1(Af2(x, - Bf1(x))

2 2 —1
= = = 9 = f = .
A If1(x)dx ff2(x)dx (q/h) and B j£1(x) 2(x)dx A/e
Estimator b has full rank covariance matrix under both
P1 = N(-l,l) and P2 = N(l,l). The second unbiased estimator we

consider is r(x) = (r1(x), r2(x)) where

(1 - x)/2

r1(x)

r2(X) = (1 +X)/2 9

which is the kernel function used by Robbins (1951). Estimator r

is unbounded and its covariance matrix has rank 1 under both P1

and P2.

The four unextended rules we consider are given by (9) with

k = l and C1 one of the following:
a

25

-l -1
(or-1) 2‘: b(x,) .0722

-1 -1
(07-1) 2‘: r(xi),0122

-1 a

a £1 b(xi) , a 2 l
- (1’

(17 2:1 r(xi) , a 2 1

We refer to these compound rules as udb, udr, unb and unr
respectively where, for example, unr denotes the unextended, non-
delete rule with kernel r.

The four extended rules we consider are given by (9) with

«2
k = 2 and G one of the following:
a

- - 2

(cr- 3)12<]1:313_(3C_.),0724
l
-l -3 2

(a - 3) Z? 5‘51) . a 2 4
-1 0,4 2

(CY ' 1) 21 2(5) 3 (Y 2 2
- -1 2

(oz - 1) 1:: Lei) , a 2 2

where b. and £_ are the product estimators based on b and r.

We refer to these rules as edb, edr, enb and enr respectively.
For our computations the compound losses for the rules edb

and edr are calculated as average loss over the last N - 3 com-

ponents; for udb, udr, enb and enr as average loss over the last

N - 1 components and for unb and unr as average loss over all

N components. In practice one might use the component minimax rule

as the initial segment of the sequence compound rule. In defining

the rule for the Theorem of Chapter 1 we took for convenience the

estimator of G: to be 9 (cf. (22)) in the initial segment forcing

all initial decisions to be action 1 (cf. (8)).

26

The behavior of our eight procedures will be examined for
N = 20, 50, 100 and 200 components and for two extreme types of para-
meter sequences.

Type I - Means of l occurring uniformly along the sequence
Such that 6 = 0. The proportion of these means will take values
p = .l(.l).5.

Type II - All means of 1 occur in a group after means of -l.
The proportion of means of 1 will take values p = 0(.l).5. In this
type of sequence 5 = l - (pN)-1.

Rayment (1971) has used these types of parameter sequences
in an investigation of the compound risk behavior of the unextended
delete sequence rule with C; = Ea truncated to the range of Cl.

One hundred simulations were made for each given 3' and N.
All eight rules operated on the normal variables generated in a simula-
tion. The estimated compound risk of a rule was obtained by averaging
the one hundred compound losses. These averages along with error
ranges of twice the standard deviation are given in the following
tables. Envelope values from Tables 1 and 2 are given to indicate
the unextended and extended asymptotic risk standards for each p

and parameter sequence type.

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Noo._H NmmH.
moo..H mooH.
Noo..H NmoH.
m.

mmNo.
moo._H momo.
moo. H ono.
ooo..H HooH.
moo. H mmoH.
mmmH.
moo. H mmoH.
moo._H mmoH.
moo. H NHNH.
Noo._H omoH.
m.

HH mazH mo moonwavmm umumEmumm ocm

HoNo.
moo._H ommo.
moo. H Nmmo.
moo. H Nmmo.
moo._H Nmmo.

NmmH.
moo..H ommH.
moo. H mmmH.
moo..H mmmH.
moo. H mNmH.

m.

mmmo.
moo. H HmNo.
moo. H HmNo.
moo. H NHmo.
moo. H. Home.

HNHH.
moo. H omHH.
moo. H mHNH.
moo. H HoNH.
moo. H HmNH.

N.

ommo.

\‘T
O
O
+|

ommo.

+l

coo. ammo.

+|

qoo. ammo.

moo. H oNoo.

MONO.

+l

moo. mmmo.

+l

moo. mmmo.

+|

moo. mono.

+l

moo. mmmo.

Noo.

moo.

Noo.

Noo.

Noo.

woo.

woo.

woo.

oom n 2 How mxmmm UmumEHumm n

O

+l

NHHo.

+|

Huao.

+|

mmoo.

+l

Hmmo.

+|

«moo.

+l

mmoo.

+l

mmoo.

+|

oooo.

oH manmfi

.o
AH on

use
now
pom

pom
Hove

NC:
2::
no:

no:

majm

35

The following observations can be made from the above tables.

1) When R(p) and R2(p,6) are nearly equal and N > 20,
the unextended and extended procedures appear to have similar be-
havior. However, at points where R2(p,6) is considerably less
than R(p) and N > 20, the extended procedures are significantly
better. The results for N = 20 are somewhat inconclusive except
when the parameter sequence is of Type I and p = .5 the extended
procedures are a great improvement.

2) The performance of the nondelete rules appears on the
average to be better then the delete rules at 20 and 50 components.
But for 100 and 200 components this advantage seems to disappear.

At p = 0 for N = 20, 50, 100 and 200, the delete unextended rules
have uniformly lower estimated risks.

3) Generally, the behavior of the rules based on the dual
basis kernel is the same as the behavior of the rules based on Robbins'
original kernel.

Observations l and 2 are cretainly consistent with the theory
and intuition. When the extended envelope is significantly below
the unextended envelope, one would expect the extended procedures to
be better. Further, it is consistent that the advantage of nondele-
tion would become negligible as the number of components increases.
The last two observations seem to indicate that Theorems 4.2 and 4.3

of'Van Ryzin (l966b) may be generalized to the extended setting.

36

4. Performance of Compound Rules when the ParameterfiSequence is a
Strictly Stationary Process.
Gilliland and Hannan (1969, Theorem 3) Show that if
Q = (91,92,...) is a Strictly stationary stochastic process then any
asymptotic solution of the k-extended sequence compound decision prob-

lem £2 = (521,522,...) satisfies
‘7" “R k k
11mN O_N(g,m)dc(g) s R (G*)

. . . k
where C denotes the measure on infinite sequences g. and 6*
k
denotes the mar inal on , = . , ... i = 1 2 ...
8 a1 (91’91+13 ,ei+k-1), , ’

This theorem serves as the motivation for our next set of
calculations. We modified the computer program used in the above
computations so that the sequence of parameters is generated by a

Markov process. The distribution of the initial parameter is

Prle1

ll
N
1...:

ll

“U

l
,.a
L..J
II
,.a
I
’U

Prie1 -

and the transition probabilities are

PrlZeM1 = Zlei = 2] = 6

PrieH4_= llei==2] =1.- e

Prfew1 = Zlei = 1] = p(1-6)/(1 - p)
Pr[9i+1= llei = 1] = 1 - p(1~6)/(1 - 13).

It is not difficult to Show that this process is strictly Stationary.
In our calculations we compared the Bayes performance of

udb and edb for 50 and 200 components. One hundred simulations were

37

made for each case and both rules operated on the same hundred samples.
The estimated risks of the two rules were obtained by averaging the
one hundred compound losses. These averages, along with error ranges
of twice the standard deviation are given in the following tables.
From these tables we observe that for 50 components the un-
extended procedure, udb, performs better at every (p,6)-value while
for 200 components the extended procedure, edb, is significantly
better at many (p,6)-values. Further, the estimated risks for edb
indicate that its convergence to or below R2(p,5) is relatively

Slow.

38

000.
NHo.
HHo.
HHo.
HHo.
oHo.
oHo.
oHo.
NHo.
oHo.

NHo.

+| +| +l +| +| +l +| +| +l +1

+|

ano.
onH.
mNmH.
mama.
come.
waa.
mHmH.
mNmH.
HmoH.
Hmom.

Hmoa.

coo.

«Ho.

Nao.

oHo.

HHo.

Hao.

mao.

HHo.

NHo.

NHo.

HHo.

+l +| +I +1 +l +| +l +l +I +l

+|

ooo.

mHo.

oHo.

mHo.

HHo.

oHo.

Hao.

moo.

woo.

woo.

moo.

mNHo. moo. H mmoo. moo. H .88.
mooH. 29 H mmNH. NHo. H SS.
83. NS. H NomH. mHo. H omoH.
HNNH. HHo. H NmmH. mHo. H mNmH.
SS. HS. H mooH. 39 H HmmH.
HomH. oHo. H mNmH. HHo. H mmNH.
38. H8. H 83. oHo. H mNmH.
mNmH. oHo. H omoH. oHo. H mNmH.
mmmH. HHo. H mHNH. oHo. H mNmH.
momH. HHo. H mmmH. moo. H Si.
83. H8. H mmNH. oHo. H mmmH.
m. N.
z NON no: mo mem momma omumeHumm - HH mHomH

+| +| +l +l +l +1 +I +I

+|

mNHo.

mamo.

NHmo.

qmmo.

mmmo.

momo.

memo.

mooo.

oomo.

ommo.

Homo.

39

~00.
NHo.
HHo.
NHo.
HHo.
mHo.
NHo.
NHo.
NHo.
NHo.

oHo.

+1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+l

Nomo.
quH.
Noam.
mHNN.
«mmm.
comm.
omHN.
oooN.
mmmH.
oomH.

omHH.

moo.

«Ho.

mHo.

HHo.

NHo.

mHo.

HHo.

NHo.

mHo.

NHo.

«Ho.

+l +l +| +| +l +1 +| +| +| +|

+|

Nmmo. moo.
HmmH. mHo.
mmNH. HHo.
mmHN. mHo.
mmNN. mHo.
mNHN. HHo.
mmmN. NHo.
HmoN. NHo.
moHN. mHo.
NmmH. NHo.
NHNH. mHo.
z pom mom

.H oHNo. moo.
.H mmmH. NHo.
H mmmH. mHo.
.H NomH. mHo.
.H ommH. HHo.
.H mmoN. NHo.
H.mmmH. HHo.
.H mNoN. NHo.
H mmoN. oHo.
H HmmH. NHo.
.H mmNH. HHo.
m.

+1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+|

memo.
mmHH.
mama.
Noam.
mama.
coma.
mooH.
mNmH.
whoa.
HmmH.

mmmH.

moo.
mHo.
NHo.
oHo.
HHo.
HHo.
mHo.
NHo.
oHo.
ooo.

oHo.

mo xmmm mmzmm woumemumm 1 NH manmh

+1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+|

mmmo.
oomo.
Numo.
ommo.
ammo.
qoaa.
omHH.
omHH.
HooH.
mmoa.

oooa.

40

Noo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

+l +| +| +| +| +| +| +| +| +l

+1

mmoo.
cqu.
mmmH.
mmmH.
owna.
ooNH.
mHNH.
mmmH.
mama.
mmmH.

ooNH.

Noo.
moo.
moo.
moo.
moo.
moo.
coo.
qoo.
moo.
moo.

moo.

ooN 1 2 so mm:

+1 +1 H H H H H +l H H

+|

mmoo.
mmmH.
mama.
HmmH.
mama.
mama.
mmmH.
mama.
omoH.
mNmH.

mNmH.

Noo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

+1 +1 +1 +1 +I +1 H H +1 +1

+|

mmoo.
HmmH.
quH.
mmmH.
mama.
HOmH.
mqu.
oomH.
mmqa.
HmmH.

onH.

Hoo.
moo.
moo.
moo.
moo.
moo.
moo.
moo.
qoo.
coo.

moo.

+1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+|

qqoo.
mmoa.
moma.
mama.
mmNH.
mNmH.
mmHH.
ooNH.
mHNH.
mmNH.

NONH.

Noo.
moo.
moo.
moo.
moo.
moo.
qoo.

«oo.

«00.

qoo.

mo xmmm mommm woumEHumm 1 ma maan

+l +l +1 +| +| +| +| +l +l +|

+|

mmoo.
oNno.
Hmmo.
NONo.
mmmo.
ummo.
ammo.
omno.
Hmmo.
ammo.

Hmmo.

41

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

moo.

+1 +1 +1 +1 +1 +1 +1 H +1 +1

+|

mNHo.
ooqa.
mmmH.
mHNH.
ommH.
mama.
HmmH.
HmmH.
ommH.
mmNH.

Nmmo.

Noo. H mHHo. Noo. H 88. Noo. H 88. moo.
moo..H mmNH. moo. H mmoH. Noo._H mmmo. Noo.
moo. H NNmH. moo..H mNNH. moo. H mmoH. moo.
moo. H momH. moo. H HmmH. moo. H mNHH. moo.
moo. H HmmH. moo. H HmmH. moo. H NmNH. moo.
moo. H NmNH. moo. H NmmH. moo. H mmNH. moo.
moo. H mmNH. moo. H mNmH. moo. H mmNH. moo.
moo. H NoNH. moo. H mNmH. moo. H momH. moo.
moo. H ommH. moo. H ommH. moo. H mmmH. moo.
moo. H ommH. moo. H NNmH. moo. H mmNH. moo.
moo..H mmmH. moo. H HmmH. moo. H mNNH. moo.
m. m. N.

oom u z

pom mom mo mem mmmmm oommeHumm - mH mHmmH

+1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+|

NHHo.

mmmo.

mmmo.

omno.

mono.

nmno.

mmmo.

mmmo.

ommo.

ommo.

mmmo.

CONCLUSIONS

In this thesis we have shown that the product estimators of
GS’ are natural and logical extensions of the estimators of CN in
the unextended problem. Further, using these estimators, we pre-
sented a sequence extended delete compound decision procedure which
satisfies (2) with a rate of N-lls.

Our computer simulations for the featured example in Robbins
(1951) Showed that the k = 2 extended envelope is Significantly be-
low the Simple envelope for many parameter sequences. For these
sequences, the four extended rules investigated have estimated risks
that are significantly better then those of the correSponding un-
extended procedures for as few as 20 repetitions of the component
problem. The calculations indicated that the risks of extended rules
converge relatively rapidly to R2(G§). From a comparison of extensive
tables of component by component loss not published here, one can
conclude that for most parameter sequences the average number of errors
made at component a has reached the envelope value for a 2 50.
However, it takes some time to average out the large number of errors
made in the first few components. Our last set of calculations showed

that the extended rules perform quite effectively when the parameter

sequence is being generated by a Markov process.

42

REFERENCES

REFERENCES

Egorov, V.A. (1970). Some limit theorems for m-dependent random
variables. Litovsk Mat. gp. 10 51-59.
Gilliland, Dennis C. and Hannah, JameS~F. (1969). On an extended

 

compound decision problem. Ann. Math. Statist. 40 1536-1541.
Gilliland, Dennis C., Hannan, James and Huang, J.S. (1974).

 

 

Asymptotic solutions to the two state component compound
decision problem, Bayes versus diffuse priors on proportions.
RM-320, Dept. of Statist. and Prob., M.S.U.

Hannan, James F. (1956). The dynamic statistical decision problem

when the component problem involves a finite number, m, of

 

distributions (abstract). Ann. Math. Statist. 27 212.

Hannan, James F. (1957a). Approximation to Bayes risk in repeated

 

play. Contributions pg the Theory pf_Games, 3 97-139.

Ann. Math. Studies No. 39, Princeton University Press.

 

Hannan, James F. (1957b). Unpublished lecture notes. M.S.U.
Johns, M.V., Jr. (1967). Two-action compound decision problems.

Proceedings pf the Fifth Berkeley Symposium pp_Mathematical

 

Statistics apd_Probability, 1 463-478. University of
California Press.
Rayment, P.R. (1971). The performance of established and modified

compound decision rules. Biometrika 58 183-194.

 

~~

Robbins, Herbert (1951). Asymptotically subminimax solutions of

compound statistical decision problems. Proceedings pf the

 

Second Berkeley Symposium pp Mathematical Statistics and

 

Probability» 131-148. University of California Press.
Robbins, Herbert (1964). The empirical Bayes approach to statistical
decision problems. Ann. Math. Statist. 35 1-20.

~~

 

 

Samuel, Ester (1963). Asymptotic Solutions of the sequential compound
decision problem. Ann. Math. Statist. 34 1079-1094.

 

43

44

Swain, Donald D. (1965). Bounds and rates of convergence for the
extended compound estimation problem in the sequence case.
Tech. Report No. 81, Department of Statistics, Stanford.

Teicher, Henry (1963). Identifiability of finite mixtures. App,
Math. Statist. 34 1265-1269.

Van Ryzin, J.R. (1966a).:~ The compound decision problem with m X n
finite loss matrix. Ann. Math. Statist. 37 412-424.

Van Ryzin, J.R. (l966b). The sequential compound~3ecision problem
with m X n finite loss matrix. Ann. Math. Statist. 37
954-975. N"

Yu, Benito (1971). Rates of convergence in empirical Bayes two-action
and estimation problems and in extended sequence-compound

estimation problems. RM-279, Dept. of Statist. and Prob.,
M.S.U.

   

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