L \IWHWIIHHHIWIHHHIIHIHHIWWWW "7348st [210”! o nun-‘llil - v; "*v 'Wfll‘ 4‘ .4... C.e..,,.1,, ram-r .‘v cum at: U ' i‘ #5 gvgfr-e-~,,1 u.- vi an. This is to certify that the dissertation entitled Second Order Sequential Estimation Of The Mean Exponential Survival Time Under Random Censoring presented by Girish A. Aras has been accepted towards fulfillment of the requirements for Ph.D. degreein _5.I.dtlS_LLCS' . Lr(:i>é;é1/Zéé;%ce4”' W Major professor Date W MS U i: an Aflirrmm've Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES m. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. SECOND ORDER SEQUENTIAL ESTIMATION OF THE MEAN EXPONENTIAL SURVIVAL TIME UNDER RANDOM CENSORING by Girish A. Aras 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 1986 ABSTRACT SECOND ORDER SEQUENTIAL ESTIMATION OF THE MEAN EXPONENTIAL SURVIVAL TIME UNDER RANDOM CENSORING by Girish A. Aras We study in this work a sequential estimator of the mean 9 of an exponential distribution when the data is randomly right censored. The loss is measured by the sum of squared error loss of estimation and a linear cost function of the number of observations. Without any further conditions, second order expansions are provided for the expectation of the stopping time and for the risk. Also the asymptotic normality of the stopping time is demonstrated. Sequential interval estimation of 9 is also considered. ACKNOWLEDGEMENT I wish to express my sincere thanks to Professor Joseph Gardiner for his guidance and encouragement in the preparation of this dissertation. I would like to thank Professors J. Hannan. H.L. Koul and V. Mandrekar for their careful reading of the thesis and Professor H. Salehi for serving on my committee. Finally. I wish to thank Cathy Sparks for her superb typing of the manuscript. TABLE OF CONTENTS Chapter Page 1 INTRODUCTION. . . . . . . . . . . . . . . . . . 1 2 PRELIMINARIES . . . . . . . . . . . . . . . . . 9 3 SEQUENTIAL POINT ESTIMATION . . . . . . . . . . 17 3.1 The Model. . . . . . . . . . . . . . . . . 17 3.2 Some Preliminary Formulae and Results. . . 17 3.3 Sequential Procedure . . . . . . . . . . . 20 3.4 Lemmas . . . . . . . . . . . . . . . . . . 21 3.5 The Main Theorems. . . . . . . . . . . . . 25 4 SEQUENTIAL INTERVAL ESTIMATION. . . . . . . . . 35 4.1 Sequential Procedure . . . . . . . . . . . 35 4.2 The Theorem. . . . . . . . . . . . . . . . 37 4.3 Concluding Remarks . . . . . . . . . . . . 38 REFERENCES. . . . . . . . . . . . . . . . . . . . . . 4O .I’ é/IL" :/ 534" > CHAPTER 1 INTRODUCTION In several survival studies pertaining to clinical trials. lifetesting. reliability and epidemiological investigations the estimation of the mean survival time 9 is of fundamental importance. This is usually based on the data gathered from a sample of n(21) units as in a reliability study or lifetest. An analysis of the estimator 3n constructed would now be necessary before its practical application in a given situation. However, it is often the case that an (with n held fixed) is very hard to analyse, but its salient features become more apparent "in the limit as n tends to infinity". The consideration of large sample sizes is often inappropriate in many longitudinal studies where ethical reasons. high per unit costs and monitoring costs preclude implementation of statistical procedures which require genuinely large sample sizes for their proper utilization. This leads us to consider some sequential or quasi-sequential schemes that may effectively reduce the sample sizes required for efficient estimation of 9. Generally this would engender substantial savings in costs and on-test time with a reduction in the loss of experimental units and without serious loss of sensitivity or efficacy of the statistical investigations. A common feature of several survival studies is that the lifetimes (or failure times) of the units under observation may not be completely observable due to the presence of censorship. This is typically the case in a clinical trial in which patients under treatment may be lost to follow up due to withdrawals from study. In some situations competing risks. other than that under study. curtails observation of the duration variable of interest. Suppose that the true survival time X of a specimen may be detered from complete observation by the action of a censoring variable Y. so that the only datum available to the investigator is (2.6). where Z = min (X.Y) and 5 is 1 or 0 according as X S Y or X > Y. The random censorship model assumes that X and Y are independent variables. Suppose X has exponential survival function F, F(t) = exp(-t9-1 ), t ) O and Y has censoring distribution G(G(') = P{Y > 0}). Both 6 and G are unknown and we wish to estimate 9. If c(>0) is the per unit cost of observation we place n items on test and record the data {(21.61)= ISiSn}. For an estimator an of 9 we measure overall loss incurred by Ln(c) = (en—e)2 + on and the preliminary objective is to minimize the expected 1038, called the risk Rn(c) = ELn(c). by optimal choice of n. We exhibit this by obtaining the expansion -1 2) from which the optimal sample size Nc may be taken as the -1/2c-1/2 Rn(c) = 92(E4‘5)-1 n + on + 0(n- integer closest to 9(E6) The corresponding optimal minimum risk is then RN . Since both 9 and G C are unknown. Nc and RN are not completely specified and c therefore we are led naturally to consider an alternative sequential procedure for estimating 6. We propose such a scheme with a stopping rule T(=Tc) and consequent estimation of 6 by RT. The performance of the sequential procedure is described by comparing its risk RT = E(LT(c)). with that of the optimal "fixed sample scheme" risk RN c We say that the procedure (T.6T) is asymptotically risk efficient if R /R a 1 as c a 0. Since R T N N c c = 2ch + 0(c-ll?). we will have that RT - RN = o(c-1/2). c The central thesis of this research is a careful analysis of the regret function R(9,G.c) = RT - RN . which c may be viewed as the additional risk incurred in using the sequential scheme given by Tc over the fixed sample scheme Nc' We shall obtain the expansion R(6.G.c) = Bc + o(c) where the constant B will be explicitly computed. This also shows that the procedure (T,8T) has bounded regret i.e. R(6.G.c) = 0(c). Additionally we obtain an expanion for the expected sample size ETc and the asymptotic distribution of an appropriately normalized version of Tc Gardiner and Susarla (1984) were first to consider the above problem. They demonstrated the asymptotic risk efficiency. Hence the present work is a second order extension of their work. The study of sequential point estimation of the exponential mean in the absence of censoring is taken up in Starr and Woodroofe (1972) and Woodroofe (1977). The stopping time Tc = inf(n 2 m: n ) X c—ll2 n } is considered. Woodroofe (1977) obtains second order expansions for ETc and the regret under the condition that m 2 3. To place our results in proper perspective. we present brief review of literature on sequential point estimation. Sequential procedures analogous to the one outlined here have been considered in the absence of censorship by several researchers beginning with the pioneering work of Robbins (1959) for the estimation of the mean of the normal population. Let Xn. n 2 1 be independent. identically distributed normal random variables with mean u and standard deviation 0. both unknown. Consider the loss function 2 Ln(c) = (X? - u) + on. for estimation of u. The risk Rn(c) = ELn(c) = azn-1 + en. The integer nearest to ac-l/2 say Nc. minimizes the above risk. Since a is unknown Nc is also unknown. Robbins suggested T0 = inf {n2m=n>c-1/2; as an alternative for No and conjectured that R(u.c) = RN - RT is 0(c) where c c - 2 2 -1 RT = E(XT - u) + cETc and RN _ 0 NC + ch. C C C Starr (1966) proved that the above procedure is asymptotically risk efficient if and only if m 2 3. Later Starr and Woodroofe (1969) showed that R(u.c) is 0(c) under the same condition. Woodroofe (1977) gave the second order expansions for R(u.c) and ETc' He showed that R(u.c) = (l/2)c + 0(c) if m 2 4. This paper is a landmark in the theory of sequential estimation in the sense that it developed and applied entirely new techniques--those of nonlinear renewal theory to obtain the necessary second order expansions. A formulation and a proof of a general nonlinear renewal theorem was given by Lai and Siegmund (1977. 1979). The above discussion strongly indicates the good performance of the sequential procedure for the normal case. But is this procedure good in general? Let P(X1 = 1) = u = l - P(X1 = O). 0 < p < 1. Then for m 2 2 - 2 RTc(c) - E(xT - u) + cETc c 2 I (xm- n)2 d? (X1 = 1. . Xm = I) ~ 1/2 and RN ~ 2(cn(1-u)) 4 0 as c 9 0. Hence 0 lim (RN (c)/RT (0)) = O and T is not asymptotically risk ceo c c efficient. To remedy this situation. Chow and Robbins (1965) suggested that the initial sample size should go to infinity at an appropriate rate as c a O. Ghosh and Mukhopadhyay (1979) exploited this fact and proved the asymptotic risk efficiency of Tc in the estimation of the mean (modified in view of the above fact) without the normality assumption. in the general nonparametric context. under the condition that the eighth moment is finite. Sen and Ghosh (1981) consider sequential point estimation of estimable parameters based on U-statistics under the condition that EIgI2+6 < a for some 6 > 0. where g is the symmetric kernel corresponding to the parameter of interest. Estimation of the mean is a particular case with g as the identity function. Note the drastic reduction in the moment condition from 8 to 2 + 6. 6 > 0. Chow and Yu (1981) proved asymptotic risk efficiency for the mean problem independently of the above two references under the condition that EX?+5 < N for some 6 > 0. Their result is a special case of the result of Sen and Ghosh (1981). Sequential point estimation of locaton based on some R-. L-. and M-estimators is discussed in Sen (1980). Sen's book (1981) has an excellent survey of the above mentioned article. None of these results in the nonparametric context go beyond asymptotic risk efficiency. Chow and Martinsek (1982) were first to show that R(u.c) is 0(c) for the mean problem under the assumption that E X16+6 < 0 for some 6 ) O. Martinsek (1983) obtains second order expansions for R(u.c) in the nonlattice case and bounds in the lattice case. under the condition that E X18+6 ( m for some 6 > O. In the nonlattice case. 2 + 222 z?) c + 0(c). R(u.c) = (2 - (3/4)E(zf - 1) where 21 = (XI - u)a-l. Thus if X1 is symmetric R(u.c) S 2 c + 0(c). That is. in the limit. one loses at most the cost of two observations when using the stopping rule Tc instead of Nc By way of contrast. it also follows that the regret can take arbitrarily large negative values as the distribution of the X ’s varies. even among symmetric distributions. To 1 illustrate this. let X X be i.i.d. with probability 1. 2... density function f. -5 HM = 2|X| [IXI21] where [A] denotes the indicator of set A. For M > 1 define x”I = x1[|x1| g n]. Then for each M. X1“. X2“... are i.i.d. and their common distribution is symmetric around zero. Thus R(u.c) = (2 - (3/4) log(M)/(1-M-2)2 + 3/4) c + 0(c). Clearly. as M tends to w. the coefficient of c in the above expression approaches -¢. The above example is due to Martinsek (1983) and it provides an answer to the question raised by Starr and Woodroofe (1972) and discussed further by Woodroofe (1977). as to whether the coefficient in the regret expansion can ever take negative values. Although Woodroofe (1977) got positive values in the gamma and normal cases. in general it need not be positive. and in fact for distributions with large fourth moments (as in the example above) arbitrarily large negative values can be achieved. In light of Martinsek (1983) there is a renewed hope that second order efficiency could possibly be established in other nonparametric problems reviewed above. The present work is one such example. In Chapter 2. we develop the necessary prerequisites of nonlinear renewal theory and moments of randomly stopped sums. Most of the results are taken from Chapter 4 of Woodroofe's monograph (1982) and Chow. Robbins and Teicher (1965). Hence proofs have been omitted. Chapter 3 is divided in to many sections. First three develop our model. In section 5 the main theorems are stated. Theorem 1 gives the second order expansion for ETC. Theorem 2 asserts the asymptotic normality of Tc and Theorem 3 gives the second order expansion for the RT' Proofs of these theorems are based on several lemmas. Some of them. which are of independent interest are stated and proved in section 4. Section 5 gives the proofs of the main theorems. In Chapter 4. a related but a different problem of asymptotic fixed width sequential interval estimation for 9. is developed. Second order expansion for the stopping time involved in achieved as a bonus from techniques developed in Chapter 3. CHAPTER 2 PRELIMINARIES Most of the results in this chapter are taken from Chapter 4 of Woodroofe's monograph (1982). Hence proofs have been omitted. Let (0.5.?) be a probability space. Let 3n. n 2 1 be an increasing sequence of sub-sigma-algebrae of 9. Definition 2.1. A random variable t is said to be a propep stopping gime (with respect to 3“. n 2 1) if and only if t is positive integer valued and {t=n) e 3n for all n 2 l. Qofinition 2.2. The random variables Xn’ n 2 1 are said to be independeotly adapteo to 3n. n 2 1 if and only if Xn is 3n measurable and 3n is independent of the sequence Xk k > n. for every n 2 1. e re .1. (Wald's lemma) Let Xn. n 2 1 be i.i.d. random variables which are independently adapted to increasing sigma-algebras 3n. n 2 1. let Sn = XI + X2+....Xn. n 2 1 and let t be a proper stopping time for which E t < w. If X1 has a finite mean u. then ES=uEti and furthermore E (st - cu)2 = 52 E c. if Xl has a finite variance 62 10 Definition 2.3. (u.c.i.p.) A sequence Yn. n 2 1. of random variables is said to be uniform continuous in probability if and only if for every 5 > 0 there is a 6 > O for which Max P {O$k$n6lyn+k - Ynl 2 e} < e for all n 2 1. Remark 2.1. If Yn’ n 2 l converges to a finite limit with probability 1 as n 4 m. then it is u.c.i.p.. Definition 2.4. A sequence Yn. n 2 1 of random variables are said to be stochastically bounded if and only if for every e ) 0 there is a c ) O for which P (IYnI > c) < e for all n 2 1. In particular. if Yn converges in distribution. then Yn’ n 2 l. are stochastically bounded. Example 2.1. Normalized partial sums. If X X 1. 2.... are i.i.d. with finite mean u and finite positive variance 62. then Yn = 6.1n1/2(Sn - nu) n 2 1. is u.c.i.p.. Lomma 2.1. If Yn' n 2 1 and Zn. n 2 l are u.c.i.p.. then so is Yn + Zn. n 2 1. If in addition Yn’ n 2 1. and Zn. n 2 l. are stochastically bounded. and if ¢ is any continuous function on R2. then ¢ (Yn. Zn). n 2 1. is u.c.i.p.. Theorem 2.2. (Anscombe's theorem). Suppose that Y1. Y2.... are u.c.i.p.; let ta. a ) 0. be integer valued random variables for which a—1 ta converges to a finite positive constant c in probability and let N8 = [ac]. a > 0. Then 11 Yt - YN 4 0 in probability as a 4 w a a If in addition. Yn converges in distribution to a random variable Y, then Yt converges in distribution to Y as a a 4 m We need vonBahr’s (1965) extension of the central limit which asserts: Theorem 2.3. Let Xi' i 2 n be i.i.d. with finite mean u. positive variance 62. and EIXlla < m where a 2 2. then a/2 -1 2 EI6 n'1/2(sn-np)|“ » P (1/2 + a/2) The convergence of moments in Anscombe’s theorem is examined next. The most general theorem available is by Chow and Yu (1981) which is as follows. jheorem 2.1. Let Y1. Y2.... be independent random variables with E Yn = O for all n 2 1. Assume that for some p 2 2. {IYnIp. n 2 1} is uniformly integrable. Let 3n be a a-algebra generated by {Y1.Y2....Yn} for each n 2 1. ’0 = {¢.n}. and let (M(b). beB} be proper sn-stopplng times with BC(o.w) such that {(b'1M(b))P/2, n beB) is uniformly integrable. Let Wn = 2 Y1. Then i=1 {lb’l/2 wu lp. beB) (b) is uniformly integrable. Following result is a part of Theorem 7 of Chow. Robbins. and Teicher (1965). 12 Thoorem 2.5. If X1. X2.... are independent with E Xn = O. E x: < m and t is a proper fin-stopping time with E t2 < m. then B s: < a where 3n = x1 + x2....xn. n 2 1 and 3n = o-algebra generated by (X1. X2.....Xn). The rest of the chapter is a review of linear and nonlinear renewal theory. Let S = X + X .....X . n 2 1. be a random walk and n 1 2 n for a 2 0. let Ta = inf(n 2 1: Sn ) a) be the time at which the random walk first reaches the height a. or m if no such time exists. Next. define Ra on {Ta( on} by Ra = STa - a. Thus Ra is the excess of the random walk over the boundary a at the time which it first crosses a. If u = E (X1) 2 0. then Sn 4 m with probability 1 by the strong law of large numbers so Ta < a for all a 2 0 with probability 1. It can be shown that Ta < w for all a 2 O with prob. 1 if u = 0. too. It can be verified that Ta is a proper stopping time if u 2 0 and fin = a-algebra generated by {X1.X2.....Xn). The following is a corollary of the classical renewal theorem. Theorem 2.6. Suppose that 0 < u < m . If F is nonarithmetic. then Ra has a limiting distribution H as a 4 0. where H(dr) = ET%_T P (Sr > r) dr r 2 O. T 13 and T = inf (n: Sn ) 0). Theorem 2.7. If Xl has a finite variance 62. then the mean of H is 2+ 2 w -1 _ p = E;———— - 2 k E( s ) 2p k k=l where — denotes the negative part. We have a following important corollary of Theorem 2.6 and 2.7. C r lar 2. . Suppose u > O. that E{Max (O.Xl)2} n. for every‘ n 2 1. The objective is to extend aspects of renewal theory to Z = S + § . n 2 1. under smoothness n n n conditions on §n. n 2 1. Define Zo = 0. $0 = (¢.D) and ’n = a {(X k S n}. n 2 1. Thus Xn. n 2 1. are k'sk): independently adapted to ,n' n 2 1. Next. let Ta = inf (n 2 1: Sn 2 a). l4 and Ra=Zt-a 820. These notations and assumptions are used throughout the chapter. efin t 2.5. The process §n' n 2 1. is said to be slowly changing if and only if: (i) i Max {|§1|,|§2|...|§n|} » o in probability as n n 4. w and (ii) 5“. n 2 1. is u.c.i.p. e ar . . Observe that (i) holds if §nln 4 0 with probability 1 as n 4 w. 1 11 Remark 2.3. If §n. n 2 1. and §n . n 2 1 are two slowly changing sequences. then 5n = §i + 5:1 n 2 1. defines another slowly changing sequence. Exam e 2.2. Let Y1. Y2... mean u and a finite. positive variance. then §n = be i.i.d. with a finite n(Y§-v)2. n 2 1. is slowly changing. Lomma 2.2. If (1) holds and N = N8 = the greatest integer in au-l. a 2 0. then ta < m for all a 2 0 with -1 probability 1 and t3 Na 4 1 in probability as a 4 m. In particular. §n n—1 4 O with probability one. implies t N.1 4 1 with probability 1 as a 4 m. a a Theorem 2.8 and Theorem 2.9 are generalizations of Theorem 2.7 and Corollary 2.1 in the nonlinear context. 15 Th rem .8. Suppose that X1 is nonarithmetic and that Sn. n 2 1 are slowly changing. Then Ra = Zt - a a has a limiting distribution H. as a 4 m. where 1 H(dr) = ETS-—) T P (ST > r} dr. r ) O. and T = inf (n: Sn ) 0}. That is Ra has the same limiting distribution as ST - a. a Theorem 2.9. Let An. n 2 1 be Fn-measurable sets. and Vn. n 2 1 be Fn-measurable random variables for which following conditions hold. 0 Q . (1) 2 P ( U Ak) ( m. n=1 k=n (2) gm = Vn on An. n 2 1. Max (3) {oSkSn lvn+k|' n 2 1} are uniformly integrable. Q (4) 2 P (Vn 5 us) ( m for some 5. O ( e < u. n=1 (5) E (Vn) 4 E (V) where V is some random variable. (6) p (ta 5 5 Na} = o(N;1) as a a a. e > o, where N8 = largest integer in a u-l. In addition. suppose Xl has finite. positive variance 62. and that Vn. n 2 1 are slowly changing and F is nonarthimetic. then E (ta) = p-1(a+p-E(V)) + 0(1) as a 4 m. where 2 2+52 w _1 _ p = E (s1)/2 E (sf) = E§E_' - kil k E (sk). Theorem 2.8 and a variant of Theorem 2.9 were first proved by Lai and Siegmund (1977. 1979). Hagwood and Woodroofe (1982) simplified the second theorem. Theorem 2.9 16 as stated here is a slight modification of Theorem 4.5. Woodroofe (1982). the proof being essentially the same. CHAPTER 3 SEQUENTIAL POINT ESTIMATION 1. The Model Let X and Y be nonnegative independent random variables with survival functions F and C respectively i.e. F(t) = P(X)t) and G(t) = P(Y)t) for all t 2 0. We assume that X is exponential with mean 6 and G(O) # 0. Consider Z = min (X.Y) and 6 = 1 whenever X S Y and 0 otherwise. Suppose ((21.61): 1 S i S n} is a random sample of size n. We wish to estimate 6 in presence of the nuisance parameter G. Consider the sequence of estimators 9n. n 2 l. of 9 given by G) — —_ 1 - n - Zn 6D [6D x O] (3.1) where the overscore denotes the corresponding sample mean and [A] denotes the indicator of Set A. The loss incurred in estimation of 9 by 9n is ‘ 2 Ln(c) — (9n - 9) + on. (3.2) where c is the cost per observation. 2. Some Pre imi ar rmulae and esults E 5 = P(X g Y) = 9'1 13 e'X/9 G(x)dx = b. I; P (Z>z) dz = I; F (z))G(z)dz = [3 e-)‘/9 G(x)dx E (Z) Thus. E(Z-96) = O. 2 Var (z - 95) = E (z - 95)2 = e E 5 l7 18 The covariance matrix 2 for the vector (2.6) works out to be 2 2 2 2 I; x e-)(/9 G(x)dx - 6 b 6-1[; xe-XBG(x)dx-6b 9-1 I; x e-XIG G(x)dx - Gb2 b(l-b) Denote by (e1. e2) be a normal vector with mean 0 and covariance matrix 2. Observe that by the strong law of large numbers 9“. n 2 1 is a strongly consistent estimator of 9 and by the central limit theorem we have. 47; (On - 9) converges in distribution to normal random variable with mean 0 and variance 02 = 62 b—l. Remark 3.1. Since P(6£ = 0) = bn 4 0 at an exponential rate as n 4 m and all our scale factors will be algebric powers of n.we shall suppress terms involving [5n = 0]. —-k cu— . . sup Lemma 2 1 For any k 2 1 E n 6n [6n g 0] a g (2b‘1)k + 2 nk P(6n g b/2) < m n=1 sup ‘ek ‘ _ sup -k - Eroof. E D an [an a 0] - E n D [an > b/2] sup ‘bk -1 ’ + E n 6n [n S 5n S b/2] l k ” k - g (2b ) + z n P(6n g b/2) < m n=1 Lemma 3.2. B (an - e)2 s azn-l + n'2 {-2 b'3 3(51 - b)(zl - 951)2 -4 2 2 -2 + 3b Ee2 (e1 -9e2) } + o(n ) 19 Proof. By Taylor’s theorem in two variables. we have — -2 — an - e + b-1(2£ - 9 an) - b (an - b)(§£ - 9b) 9b"2 (3g - b)2 + (A53 (35 - b)2(2£ — eb) - A1 A;4 (35 - b)3) where A1 lies between in and 9b and A2 lies between 6; and b. Thus E (6n - 6) =E Now we 2 -2 — _ _ an - b)(zn - 9b) b (2n - 93$)2 + E (b'2 ( 2 — 2 -3 ‘ 9b” (5n - b) - A2 (an - b)2 (E; - 9b) -4 - 3 2 Al A2 (an - b) ) 212(b’3 (3g - b)(§§ - 93g) (2g - 6b) eb'3 (2g - 93g) (3£ — b)2} 2 Eb’1 (E; - e 3;) (Ag3 (3; - b)2 (E; - 9b) -4 - 3 Alhz (6n - b) ) I + II + III + IV 11 n n n be easily checked that ID = 02n—1 -3 - - - 2 2 b E (an - b) (2n - can) n -2b"3 n'3 2 E (51 - b) (zi - 951)2 1:1 --2b’3 n’2 B (51 - b) (21 - 951)2 shall consider IVn. Let f(x.y) = 2b-4y2(x-9y)2 for any x.y real numbers. Let Pn denote the random variable f( 1/2 _ n1/2 (Eg-eb), n (fin-b)) 2O 2 -l - and Qn = 2n b (E; - e 6;){A53 (an — b)2(2£ - 9b) -4 ’ 3 - A1 A2 (6n-b) )-Pn. Thus n21V = E P + E o . n n n By central limit theorem. Pn converges in distribution to f(el.e2) and Qn converges to zero almost surely. Thus 2 -l ‘ - -3 - 2 ’ -4 ‘ 3 2n b (Zn-66D) {A2 (6n-b) (Zn-9b) - A1A2 (6n-b) ) converges in distribution to f(e1.e2). Now to conclude that n21V converges to Ef(e1.e2). we need to verify uniform integrability of {Pn+Qn. n21). which follows from the following facts. Since 0 < A-p< b-p 2 + 3;p [6; fl 0] by previous lemma. Agp is uniformly integrable for every p ) 0. Similarly A? is uniformly integrable for every p > 0. Also (111/2 (2g - 9b))p and 1 (n ,2 (En-b”p are uniformly integrable for every p > 0. Similar computations for IIn gives the lemma. 3. ue r r Using (3.2) and lemma (3.1) we have. 1 2 Rn(c) s E (Ln(c)) = azn’ + as + 0(n’ 1/2 ) For large n. Nc = nearest integer to (c- a) which minimizes the risk. Since a is unknown. Nc' the optimal sample size is unknown and thus one is naturally led to explore a sequential scheme to estimate 9. Define a 21 stopping rule c-1/2 “ Tc = min {n 2 n1c= n > an) (3.3) 1 where n = c‘2(1+a). a > 0. 1c ‘ - ‘43/2 - - an _ Zn 6n [6n fl 0] + [6n — 0] Note that an. n 2 1. is strongly consistent for a 4. Lemmas Let 0 < e < 1 be fixed. Let n and be the 2c n3e integer parts of Nc(1-e) and Nc(1+e) respectively. We may write Tc and Nc without the subscript c in the sequel. Also we shall freely write cml/2 a for N. Let 3n be the a-algebra generated by {(21.61). (22.62).....(Zn.6n)}. Romark 3.2. The terms involving the random variables [6N = O]. [6% = 0] are left out without any further indication since F(Efi = O) and P(6& = 0) go to zero at an exponential rate as c 4 O and all our scale factors Q will be algebric powers of c. (P(6T = O) = 2 P(6n = 0. n=m 1c ” - n1 1 T = n) g 2 P(6n = 0) = b c(l-b) .) n=n 1c -1/2 . IIsup _ = Lemma 3.3 'ln2m (an a)llp 0(m ) for all p > 0 and m 2 1. Ereei- Slan - aI 31-3/2“”n fl 0] (Izn —9b| + 9b-1/2lgi/2 _ b3/2l) 22 -3/2 - - _ an [5n a 0]|zn ebl -1/2 -3/2 - -l/2 1/2 —1 - + 9b an [an a 0] (5n + b ) (5n + 31/2 b1/2 + b) IE; - bl —.3,2 - - s an [an s cllzn - ebl + (39/2)(3 A b)’5/2 [3 a 0] |3 — b| (3.3) n n n -=3/2 - - s an [5n x 0]|zn - ebl + (39/2) (3;5/2 + b'5/2) [35 a O]I6#-bl (3.4) The Schwarz inequality. lemma 3.1. and the maximal inequality for reverse martingales give the lemma. Lemma 3.4. F(T g n20) = 0(cp) for all p > o and F(T 2 n26) = 0(cp) for all p > 0 Max ‘ Eroof. P(T g n2c) g P(n SnSn lan -aI > e 0) 1e 2c 3 P( "ax IE — ebl > n ) + P( “ax l3 - bl > ) n n 1 n Sn n 172 1cSn 10 (3.5) for some n1. n2 2 0. The above inequality is obtained by using (3.3) and a truncation argument similar to the proof of lemma 3.1. With the reverse martingale inequality. 1 = c2(1+a) 1c and (3.5) imply the lemma. 0 11a .1. For all p > o, {TCN;1}_p: o < c < I} is uniformly integrable. Epooi. Let k = (1-e)‘p [[(T/N)’p > k] (TN'1)'paP 23 g NP [ [TIN > l-e] d? = NpP(T < N(1-e)) S 0(1) as c 4 o. by Lemma 3.4. Lemma 3.5. {(TN-1)p: O < c < 1} is uniformly integrable for all p > O. opppg. TN" 3 1 + 0'1 (ST-l - a) + c1/2 0 + [T = nlc] n1c Lemma 3.1 and 3.3 imply the desired uniform integrability. Lemma 3.6. For all p > o. ((N'1’2(r - N))p O < c < 1) is uniformly integrable. Proof. By definition of T. we have -1/2 1 -1/2 A c aT < r g c aT_1 P(T > n10) + nlc P(T = nlc) +1. Hence -1/2 IN (T - Nllp -1/401/2( -1/401/2 s Haxilc -a)IP. Ic (0T 1 -allp 1/4 + (e nlc P(T=n::)}p + 1}. (3.6) Also -1/4 ‘ - /4 - - l4 - (c IaT-al)p S kp(c p a¥IZT - Gblp + c p b¥I6T -blp) where an = [6; i O] 6‘3/2. and bn = 9(En + 31/2 b1/2 + b)(61/2 + b1/2) 6-3/2b1/2 [an i 0]. Thus by the Schwarz inequality. E(c-1/4l;T _ 0|)p S kP {El/2a 2p E1/2(c -1/4|E& _ 9b“2p + E1/2b2p E1/2 (3% _ b)c-l/4)2p). E(c-1/4I2& - Gbl)2p = 0(1) by Lemmas 3.5 and 2.4 and Corollary 3.1 E a?” = 0(1) by Lemma 3.1. 24 The other term is treated similarly to obtain uniform integrability of {(c-ll‘llaT - al)p = O ( c < 1 }. (3.7) 1/4 (c “1aP(T = nlc))p=o(l). (3.8) Hence by [3.6]. [3.7]. [3.8] to prove the lemma. we only Furthermore by Lemma 3.4. need to show BIC-1,401,2(0T_1-a)lp = 0(1) for all p > o (3.9) Observe -1/4 A ~ - /4 - (c IaT_1 - al)p g kP(c p a¥_1|zT_1 -9b|p — /4 - + c p b¥_1|5T_l-b|p) and - l4 ‘ p ‘ -p/4 E p 2 -9b k c | 1-1 I s p(c -p/4 Elzf - Gblp + E T’p(zT - 9b)p c ) (3.10) By Theorem 2.4 the first term on the right side off (3.10) is bounded. on - p _ E _ p - ElzT Bbl - n=n Elzn ebl [T—n] 1c 00 g z Elzn - Bpr[T2n] n=n lo a p s nil E [Tsz E(lzn - ebl IFn_1) = Elz1 - GprE T. (3.11) Using (3.11). The Schwarz inequality and Corollary 3.1. we have that the second term on the right hand side of (3.10) is bounded. For all p > 0. E a¥_l = 0(1) by the Lemma 3.1. and similarly E b = 0(1). Hence the lemma. T-l 25 5. The Main Theorems. Let w = b1/29-1+(3l2)(6b1/2)-1(61-b)-(62b1/2)—1(Zi—6b) i n _ and S = 2 W . Let S denote the negative part of S . n i=1 i n n Let A be a 2 x 2 matrix defined as follows: (63b3/2)-1 -(3/4)(62b3/2)-1 A: -(3/4)(92b3/2)-1 (3/8)(Gb3/2)-1 Define V = (e1. e2)A(e1.e2)'. In the sequel no distinction in made between N and c-llza and Remark 3.2 applies. Theorem73.l. E Tc = N + a (p-EV) + 0(1) as c a 0. where 1 p = (1/2){(3/4) a- + (9/~'1)a-1b"1 - (I-Bb-3 I xe-XI9G(x)dx} on 1 _ - 2 K‘ E sK . K=1 36 Theorem 3.2. Tc = N1]2 (T-N) is asymptotically normal with mean zero and variance (9/4)9"2 - 4'19'2b + 9’4b’11 xe-X/e G(x)dx. Theorem43.3. R(9.G,c) = 30 + 0(c) where 3 2 B = -29’ b-2E {(z1 - 9b) - 3/29(5l - b))(z1 — 951) -4 -3 2 -49 b {E(e1 - 3/29e2)(e1 - 9e2)} —2e'2b'2 E(e1 - (3/2)6e2)2 +59‘4b‘3 E(el - 9e2)2 (e1 - (3/2)9e2)2 +39-3b-3 E(el - 9e2)2 ele2 --(15/4)9-10-1 E(e1 - 9e2)2 cg - 2a EV -3 -3 2 + 6 9 b Ee2 (e1 - 9e2) {el - (3/2)9e2} —3 -3 - 4b 9 E e2(e1-9e2) E(e1-9e2)(el-3/29e2) - 2b'29'1Ee2(a1-3/29a2). Remark 3.3. We note that in the absense of censoring, 26 the above results reduce to those given by Woodroofe (1977). The constant B turns out to be 3. 6. Erggfg. 1/2 c-1/2 A _1 _ . Let Dc = T(aT) -c and Tc = inf {n 2 nlc' Sn > }. with Sn' V as defined in the previous section. Lemma 3.7. As c 4 0. DC has a limiting distribution H. where H(dr) = (1:37)‘1 F(sT > r) dr. r > o, and T denotes the first ladder epoch of Sn' n 2 1. Thus Dc has the same limiting distribution as ST - c-(1/2). c ELQQL. Using Taylor's theorem for two variables. we have 1/2 -1 - 1431“)"1 = n(a'1 + (3/2)(9b (an - b) ) _ (92b1/2)-1(§g _ 9b) 1/2 -1 - 2 12 (an - b) + (3/8) A; - (3/2)Ai/2A;2 (3g - b)(2£ - 9b) 3/2 + A1 -3 ‘ 2 where A1 and A2 ‘ -1 respectively. Thus n(an) - Sn + 5n where lie between b and 3;. 9b and Z# -1/2 -1 - 2 sn = n{(3/8) A1 32 (an - b) 1/2 -2 - ’ - (3/2) A2 A2 (6n - b)(Zn - 9b) 3/2 -3 - 2 + Al A2 (Zn - 9b) }. The Wi's are independent. identically distributed. and non-arithmetic. Also E W1 = a.1 > 0 and {§n. n 2 1} is a slowly changing sequence. These follow from Example 2.1. Remark 2.1. Lemma 2.1. and Remark 2.3. Hence by Theorem 27 2.8. we have the result. Remark 3.4. Though (Dc: 0 < c < 1} may not be uniformly integrable. {(Dc 0T) 0 < c < 1} is. for all K ) 0. Observe that A ‘ -1/2 DcaT = T - aTc < T - (T—l)[T)nlc] “ -1/2 - on c [T=nlc] 1c ‘ -l/2 = l + (T-l - anlc c )[T=n1c]' Schwarz inequality and Lemmas 3.2. 3.4. 3.5 give us the necessary uniform integrability. -1/2 n Lemma 3.8. (1) En 4 0 in probability as n 4 w and (ii) En converges in distribution to V. ELQQL, (ii) implies (i). Application of the bivariate central limit theorem gives (ii). 1 _ Lemma 3.9. Let An = (ig > 2’ 6b and an > 2'1b) and Vn = fn[An] . Then (1) P( u Ag) < w. n 1 k2n (ii) Vn converges in distribution to V. "MB (iii) { Max IVn n21} are uniformly integrable. +k|' ogkgn (iv) 2 F(vng-np) < m for some 3. o < p < a'l. n=l Proof. Note that u A;={ u (Egzz‘leb))U( u (Eggz’lb)}. k2n k2“ k2n Thus P( u AL) 3 P(Max|§k-eb|4 z 16’194b4) an an + F(Maxlgg-bl4 > le‘lb)4). k2n By the reverse submartingale inequality. 28 -2 P( u AL) 5 n(EIZ§ - ebl4 + EIEg-bi4) = 0(n ) an where n is a constant. Hence (1) obtains. Since [An] 4 1 almost surely. by Lemma 3.8. Vn converges in distribution to V as n 4 m.. Hence (ii) obtains. Let a > 1. a - ‘ba/Z ‘-a — E Max (n+k) [ n+kflo] 6n+k Zn+k|6n+k OSkSn 2a -b| [An+k] _a/2E Max (n+k)alg -bl2a -1 -a -l S (2 9b) (2 b) O$k$n n+k -a/2 g (2'leb)‘“(2'1b) n'“ E Max (n+k)2“|3£+k-b|2a OSkSn k (2 lab) “(2’1b)'“’2n’“ E Max |2(5 -b)|2“ 0$k$n 1 By the martingale inequality. the right hand side of the i above inequality is bounded above by -a/2 -2 2 (2’leb)‘“(2'lb) (2a)'2“(2a-1) “n"“EI2n(51—b)|2“. 1 By vonBahr’s (1965) extension of the central limit theorem. (i.e. Theorem 3.2). E{(2n)1/2I3#-bl}2a » 2“ w-1/2F(2-1+a). Hence a "' "-a/2 J - SUP E Max (n+k) [ n+k#0]6n+k zn+k|5n+k n OSkSn The above inequality implies that 2 —b| “[An+k] < m. - -1/2 -1 - 2 , {Ogfign (n+k)[ n+k’035n+k Zn+kl5n+k-bl [An+k]° “21} is uniformly integrable. Dealing similarly with the other terms in Vn' (iii) can be obtained. Finally. F(vnsnn) s P(-(3/2)x}’23;2(3g-b)(§;-eb)[AnJ < -B) 29 1/2 -2 - - P((3/2)[An]hl AZ (an-b)(Zn-9b) > B) V\ P((3/2)[An]((eh)'2 + 2:2)Igg-bIIE£-9bl > B) V\ P((3/2)(9b)-2I(3$-blIZn-Gbl > 2‘15) + P{(3/2)2;2[An]IEg-bllag-9b] > 2’15) s P{(3/2)(eb)‘2l(Kg-b)(E§-eb)I > 2‘13) + P(3/2)(9b/2)-2I3#-blIEg-Bbl > 2‘13) By application of the Chebysev and Schwarz inequalities. we have 1/2 4 P(vn g as) s K E (3%-b)4 E1|2(2£-9b) = 0(n’2) Hence (iv) obtains. Proof of Theogem 3.1. Lemmas 3.9. 3.4 and Theorem 2.9 imply the theorem. Egoof of Theorem 3.2. Since sT + ET - D0 = e’1/2. we have - (ST-0-1T) N-llza-a N-1/2(ET-Dc) = N-1/2(T-c-1/2a). Since (Sn-a-ln)n-1/2a converges in distribution to normal random variable with mean zero and variance (9/4)9-'2 l 2 1 - 4' 9’ b + 9‘4h'11xe'X/°c(x)dx, the fact that N“ T a 1 almost surely. and Anscombe's theorem imply (ST-a—IT)N1/2a converges in distribution to above mentioned normal random variable. Similarly Lemma 3.8 and Anscombe’s theorem imply /2 that ETN-l converges to zero in probability. By Lemma 3.7. D n'l’z C converges to zero in probability. Thus N1/2(T-N) converges to normal random variable as stated in Theorem 3.2. 30 Remark 3.5. For the proof of Theorem 3.3. we shall use the following notation. Let P1 = G-lb-1/2(Zi-951). a. n _ s e 2 P and U = (9b) 1{2 -(3/2)e(5 -b)} - 1. Also n i=1 i i i i A n let S - 2 (51—h). It can be easily checked that the variance of P1 = 1. Lemma 3 10 E(3 -b)(2 -GE )2 ' ' T T T l 1 = ca-2{E(6l-b)(Z-951)2 - 3h” 9’ l Ee2(e1-9e2)2(el-(3/2)9e2) + 29' h"1 3(51-b)(21-951) E(e1-9e2)(e1-3/29e 2) + 9 Ee2(el-3/26e2)} + 0(c). Eroof. In view of Remark 3.5. - - - 2 2 -1* ~ 2 E(6T-b)(ZT-96T) = e b ET sT sT 2 -3 -3 . N2 2 -3 . ~2 _ a b E(T -N ) sTsT + a b N EST ST a 92b N'3 ET'3(N-T)(N2 + NT + T2) sTs$ + 92hN'3EsTs¥ e 92bN’2£(T’1N)N‘1’2(N-T)(T'2N2+T'1N+l)(sTN’1/2)(§¥ N’l) 2 -3 . ~2 2 -3 A + e bN E sT(sT - T) + e bN E STT = I + II + III. (3.12) Using the facts that T a;1 - cll2 = Dc and 3T 0-1 - 1 b”1 (if - 9b - (3/2)e(3}-b) - (3/2)e'1b‘2(zT-eb)(3}-b) + (15/8)a’1§}(3}-b)25'7/2) for some B between b and 3}. we have 1/2 A -l N‘1’2(N-T) = -N (0T0 -1) —N1/29-1b—1{Z&-9b-(3l2) 9(35-b) -1 -2 - - (3/2)e b (ZT-Gb)(5T-b) 31 + (15/8)a-12&(E&-b)2B-7/2}. (3.13) Theorem 2.4. Remark 3.4. Anscombe's theorem. B-7/2 < b-7/2 “-7/2 + 5T . Corollary 3.1 and (3.13) imply that -2 -1 -l 2 I = -3ca {9 b E(e1-3/26e2) e2(el-9e2) } + 0(0). (3.14) Computation of the term II is briefly scatched below since it is similar to the proof of the lemma in Chow and Martinsek (1982). Note that (S .5 ) and (Sz-n.$ ) are martingales. Also n n n n - ~2 * ~2 2 *2 ~2 2 2EST(ST-T) = -(E(ST-ST + T) - EsT - E(ST-T) ). (3.15) By Wald's lemma. Es: = E(61-b)2 ET (3.16) Theorem 1 and lemmas 6.8 of Chow. Robbins and Teicher (1965) imply . ~2 2 2 2 2 T ~2 E(s -s + T) = E(6 -b) ET + (P -1) ET + 4 E s T T 1 1 1:1 j-l 3 ~ 2 + 4EP1 ETsT - 2E(51—b)(P1-1) ET - 4E(6l-b)PlETST and ~2 2 2 T ~2 3 ~ E(s -T) =E(P - l)ET + 42 2 s + 4E? ETs . (3.17) T 1 1:1 1—1 1 T (3.15). (3.16). (3.17) and Wald's lemma imply that A ~2 EST(ST-T) 2 N E(51-b)(P1-1)ET + 2E(61-b)PlETST E(61-b)(P?-1)ET + 2E(5l-b)P1E(T-N)§T. (3.18) Now (3.18). lemma 3.5. (3.13). Remark 3.4 and Anscombe’s -7/2 7/2 - -7/2 theorem. B ( b + 6T and Theorem 2.4 imply that II = ca-292b(E(61-b)(ZI-951)2 1 + 26' b-1E(51-b)(21-961)E(e1-3/29e2)(e1—9e2)} + 0(c) 32 e ca-2(E(61-b)(Zl-951)2 + 29'16'1Ee1(el-ee2)E(el-3/2ee2)(el-eez)} + 0(c) (3 19) By Wald’s lemma. 2 3 111 = 6 b N‘ EST(T-N). Again. use of (3.13) and arguments similiar to (3.19) give us III = ca.2 9 Be2 (e1-3/2 9e2) + 0(c). (3.20) The lemma is proved by (3.12). (3.14). (3.19) and (3.20). The next lemma asserts that T-1S¥ and Dc are asymptotically independent. Lemma 3.11. P(T-1S¥ S x. Dc S y) 4 L(x)H(y) as c 4 0 for every x. y 2 0. where H is as described in Lemma 3.7 and L is the distribution function of a chisquare random variable with one degree of freedom. figooi. Proof is similar to the lemma in Martinsek (1983). Proof of Theorem 3. -1 2 ’2 e'1R(e.c.e)=(c a E PT '1 '1 2 —ET)+ 2(ET-N) - 2c b a E(E}-h)3¥ -1 - -3 ' 2 - -4 - 3 + 2c aEPT (A2 (6T-b) (ZT-Bb) - Alkz (6T-b) ) + c-l(b-2(3&-b)(E&-9b)-9b-2(3&-b)z-A;3(3&-b)2(2&-9b) + - - 3 2 Alhz (6T-b) ) (26'2h'2E(51-h)(zl-651)2 -3e‘2h'3Ee§(el-9e2)2)+o(l) q. I + II + III + IV + V + VI + 0(1). II and III are given by theorem 1 and lemma 3.10 respectively. 33 IV a 2 6’2b'3 E(e1-9e2)2e§ + 0(1) (3.21) by lemma 3.5. the central limit theorem. Anscombe’s theorem and by Theorem 2.4. Similarly v = 6’2h‘3E(e1-6e2)2e§ + 0(1). (3.22) Now I = E ST2{(N-1T)-2-1}. By Taylor’s theorem. I = -2 E 8%(N-1T-1) + 3 E S¥A-4(N-lT-l)2. (3.23) 1 where A lies between 1 and N- T. ~ E sT h'4(N'1T-l)2 ETN'1 (T(E&-63&)29-2b-1} A’4{N'1/2(T-N)}2 _1_1) ETN'1(T(§}-63§)29’2h’1)A'4(N1/2(3Ta -1/2 “ 2 + N DcaT) . Using (3.13). Remark 3.4 and reasoning similar to (3.21) gives us the convergence of the above term to E 6’2b‘1(e1-6e2)2(e1-(3/2)6e2)2a’2h’3. (3.24) Now consider the first term in the right hand side of (3.23). ~2 -1 ~2 “ -1 ~ -1 A E ST(N T-l) = E ST(aTa -1) + E ST N DTOT' (3.25) Using (3.13) we have. ~2 “ -1 E ST(aTa -1) ~2 - 1 2 - ~2 — b EST(ZT~9b)(5T-b) -1 -1 - - e b EST(ZT-3/29(6T-b)-9b}- 3/26 ~2 -1- - 2 -7/2 (15/8) E STa ZT(6T-b) p + (i) + (ii) + (iii) By Wald’s Lemma. ~ T E s¥( 2 u i=1 " -1 ~2 -l E(ST-T)(2 u1)N + E(ST-T)(2ui)(T -N m )T" i -1) 34 = I + II. 2 As in the lemma of Chow and Martinsek (1982). I can be evaluated as lN-IET(Eu¥-E(P¥-ul-l)2 + E(Pf-l)2) 1 I = 2- + 2E((ul+l)Pl)E(T-N)§TN’ Lemmas 3.4. 3.13 and the fact that -1/2 A Nl/2(T-N) = N1/2 a'l-l) + N DcaT. we have the ~ (CT I convergence of to 1/2 {E uf-E(Pf-ul-l)2 + E(Pf—l)2) -3|2 + 9-2b E(u1+l)P1E(e1-3/29e2)(e1-9e T -1 -1 -1 T 2 ui) N T - N E(N-T)(2 ui) = 1 2)' II = E §¥(N-T)( i l converges to E(el-(3/2)ee2)2e’2b'2-E(el-6e2)2(el-(3/2)ee2)2e‘4h'3. Thus (1) = 1/2{E uf-E(ef-ul-l)2 + E(Pf-1)2) + 2E(ml+l)l:1(el-(3/2)6ez)(el-eez)6'4h“3 + E(e1-(3/2)6e2)29’2h'2-E(tel-19e2)2(el--(3/2)6e2)29’4h’3 + 0(1). (11) = -(3/2)e’3h‘3(e1-6e2)2e1e2 + 0(1) and (111) = (15/8)6’1a'1E(el-9e2)2e§ + 0(1). Remark 3.4 and lemma 3.11 imply that N -l A EST N DTaT Putting together all the above terms we have theorem 3. = up + 0(1). CHAPTER 4 SEQUENTIAL INTERVAL ESTIMATION l. §eguential Procedure. It can be easily checked that n1’2(3n-8) 3 N(O.02(9)) (4.1) where 02(9) = 92b-1. For a given d > O and ae(O.1). in view of (4.1) let us take In = (9 - d. 9 + d) with n(d) n n defined by n(d) = inf{k 2 1: k 2 22/2 a2d'2) (4.2) where la is the upper 100a percentage point of the standard normal distribution. Since a is unknown. the specification of the 'optimal' sample size (4.2) can not be made. We therefore led naturally to construct a sequential procedure in which the sample size is a positive integer valued random variable N = N(d) and the desired confidence interval for 9 is IN = [9N - d. 9 + d]. The sequential N procedure (N’IN) is said to be osymototically consistent if for every 9 positive. 11 P 6 I 2 1 - . 4.3 d1: ( e N) a ( ) and is said to be asymptotically effioient if for all 6 positive lim E9{N(d)ln(d)} g 1. (4.4) dlo Following stopping time N is a slight modification of the stopping time defined by Gardiner. Susarla and VanRyzin (1985). 35 36 , -1 2 *2 16' k2(lat/2“ ) 0k} 1/2(1+A) for N(d) = inf{k>k where k1d = int (la/Z/d) A ) 0. They prove the following theorem. Theorem 4.1. (N.IN) is both asymptotically consistent and efficient. In fact P(GeIN) 4 l-a (4.5) and E {N(d)In(d)} 5 1 as d a o. (4.6) In the spirit of Theorem 3.1 and 3.3 one expects to achieve second order expansions for E N and P(9&IN). Theorem (4.2) gives the second order expansion for E N. The expansion of P(GeIN) remains an open problem. In their paper. Chow and Robbins (1965) illustrate a general methodology for the construction of the fixed width sequential confidence intervals for the mean of the population. Sen (1981) contains several refinements of these methods that have been successfully applied to obtain parallel results for the other functionals of the underlying distribution. Woodroofe (1977) gives the second order expansions for E N and P{GeIN} for the normal mean problem. So far this is the only second order computation available in the literature for the fixed width sequential problem. 37 2. The Theorem. Let -4 -1 3 39 b -36’ 1 b- 3 1 -2 -36’ h‘ 39 b and (e1.e2) as defined in Chapter 3 section 2. Let -1 W = (el.e2) B (e1.e2)'. 2 -2 2 Theoremo4ag. E N = a (d (la/2) + a - E W} + 0(1) as d 4 0. where p = 2’1(39'2h + 98‘2 - 46’4h'l )3 x eX/GG(x)dx) co 1 _ - 2 k“ E(Sk) k=l Proof. Using Taylor’s Theorem for two variables. we have “-2 -2 —3 - -2 - kak = k{a - 29 (2k - 6b) + 39 (6k - b) -4 3 - 2 + 3A1 A2(Zk - 9b) -3 2 - ’ - 6A1 N2 (Zk - 9b)(5k - b) -2 - 2 + 3N1 A2 (6k - b) } =Sk+§k. where s - g i 2 - a’2-26'3(z -9b) + 39‘2(6 -b) k ’1_1 i' i ' i i and A -4 3 ‘ 2 -3 2 _ “ Ek = n(3)\1 A2(Zk-9b) - 6A1 h2(Zk-Bb)(6k—b) -2 - 2 + 311 hz (bk-b) }. It can be easily checked that fk is slowly changing. Note 38 that £1 is nonarithmetic and E Xi = 0-2 > 0. further Xi's are independent identically distributed random variables. Hence by the Lai and Siegmund theorem (Theorem A-2 2 —2 2.8). we have the following result. Ud = NON - (la/2) d has limiting distribution M. as d 4 0. Where H(dr) = (E §,)-1 P(S'>r)dr. r>o and w is the first ladder epoch of SR. k21. Lemmas similar to 3.4. 3.8. 3.9 can be proved by exactly the same methods. Theorem 2.9 then implies the theorem. 3. Concluoing Remarks. It would be desirable to remove exponentiality assumption from the above problem and in the point estimation problem discussed in the previous chapter. Gardiner and Susarla (1983) is an attempt in this direction. They allow X to have any survival function P (not necessarily the exponential) and under fairly general moment conditions on F and G. they show the asymptotic risk efficiency of their procedure. The problem in this set up is very hard to work with since the estimator of the mean is an integral with respect to product limit estimator of F. Instead of a purely nonparametric approach. it may be easier to study robust procedures with respect to contamination of the exponential distribution. Repeated significance testing in the context of the censored model discussed in this dissertation will also be 39 of interest. particularly from the point of view of clinical trials and other applications in Medicine. It is hoped that the techniques developed in Chapter 3 would be useful to show 'bounded regret' in other sequential nonparametric problems such as procedures based on U-statistics and also on L-. M-. R- estimators of location. [1] [2] [3] [4] [5] [5] [7] [8] [9] [10] REFERENCES Anscombe. F. (1952). Large Sample Theory of sequential estimation. Proc. Combr. Philos. Soc. 48. 600-607. Aras. G.A. (1986). Sequential estimation of the mean exponential survival time under random censoring. Journal of Statistical Planning and Inference. To appear. Chang. 1.8. and Hsiung. C.A. (1979). Approximations to the expected sample size of certain sequential procedures. Proceedings of the conference on Recent developments in Statistical methods and Applications. Taipei. Taiwan. R.O.C.. 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