In the sequential multinomial sampling case,a sufficient condition for a non-randomized sequential procedure to be complete is given,and also a necessary and sufficient condition for a randomized sequential procedure to be complete is obtained

Akahira Masafumi

Koike Ken-ichi

小池 健一

赤平 昌文

Sequential Analysis

KOIKE Ken-ichi

AKAHIRA Masafumi

Tsukuba Repository  (Tulips-R)

Completeness for sequential sampling plans著者 Koike Ken-ichi, Akahira Masafumijournal orpublication titleSequential analysisvolume 12number 3-4page range 211-218year 1993権利 (C) 1993 by Marcel Dekker Inc.　This is anelectronic version of an article published inKen-ichi Koikea & Masafumi Akahira,　Completeness for sequential sampling plans, Sequential Analysis, 12(3-4), 1993, pp.211-218.Sequential Analysis is available online at:http://www.tandfonline.com/doi/abs/10.1080/07474949308836280?journalCode=lsqa20#.UeXtRKzLN95URL http://hdl.handle.net/2241/119503doi: 10.1080/07474949308836280COMPLETENESS FOR SEQUENTIAL SAMPLING PLANS Ken-ichi Koike and Masafumi Akahira Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki 305, Japan ABSTRACT In the sequential multinomial sampling case, a sufficient condition for a non-randomized sequential procedure to be complete is given, and also a necessary and sufficient condition for a randomized sequential procedure to be complete is obtained. 1. INTRODUCTION In the sequential binomial sampling case necessary and sufficient conditions on the se-quential procedure for completeness are obtained (e.g. see Girshick et ai. (1946) and Lehmann and Stein (1950) ). In the sequential multinomial case, a sufficient condition on the non-randomized sequential procedure for completeness is given by ICremers (1990), but, in the sequential binomial case, the condition seems to be a little stronger than that of Girshick et ai. (1946). In this paper a sufficient condition for completeness is given so that it coincides with that of Girshick et 81. in the sequential binomial case. The condition is weaker than that of Kremers (1990). Further a necessary and sufficient condition for a randomized sequential procedure to be complete is obtained. I 2. DEFINITIONS AND NOTATIONS Suppose that UI , U2 , .•. is a sequence of independent and identically distributed k-dimensional multinomial trials, that is , for each i = 1,2, ... , Ui = (Uil , ... , Uik ) is a random vector with Uij E {O, I} (j = 1, ... , k), I:;=l Uij = 1, and, for P = (PI,'" ,Pk) with 0 <Pj < 1 U = 1, ... ,k) and I:;=IPj = 1, Pp(Uij = 1) =Pj (j = 1, ... ,k). In the sequential sampling a decision whether or not to sample U n+l is based upon U1 , . .. , Un for each positive interger n. The sample size may be a random variable specified by a sampling plan under consideration. So whenever a capital letter is used, it denotes a random variable (or vector). Since the random vector )[ N = I:~l Ui is a sufficient statistic for P (e.g. see Ferguson (1967)), we consider only estimators based upon _3CN , and assume that any decision whether or not to sample is also based upon X N. Then let the stopping rule <p be a sequence ( 7) - ( ( (1)) ((2)) ) <p ~ - <.po, <.pI X , <.p2 X , ... , where z = (xU), x(2), ... ) with <pj defined on the sample space of Xj and 0 ~ <pj ~ 1 for all j = 0,1,.... For each j = 1,2, ... , the function <.pj(x(j)) represents the conditional probability that a statistician stop sampling, given that he has taken Xj = x(j), and !.po is a constant representing the probability of taking no observations at all. To avoid the case where the sampling continues forever , we assun1e that the stopping rule is closed , that is, Pp(N < ex)) = 1 for all p. For a given stopping rule !.p, the probability mass function of _3C N is given by k Pp(XN = x) = c(x) IIpjXj, j=l where 0 ::; c(x) ::; N!j n;=l Xj! with x = (Xl,' .. , Xk)' The outcOIne of such a X N can be represented as a random walk in the k-fold direct product N~ of a set No of the all non-negative integers. The walk starts at origin and moves a unit to the direction according to the first trial's result. From the resulting point it again moves a unit in the same manner, and continues in this way until the stopping rule tells it to stop. The sequential procedure is said to be complete if XN is complete as a statistic. For the purpose of the present paper it is enough to restrict to the space N~. 3. NON - RANDOMIZED SEQUENTIAL PROCEDURE In this section we consider the case when a stopping rule is non-randomized, that is, in the above !.p, each !.pn takes on only the values 0 or 1. For a point x = (Xl, ... , X k) E N~ the sum :Z=~=l X j of its coordinates is called the index of x. The point x is said to be accessible if Pp(Xm = X, N 2:: m) > 0, otherwise it is said to be inaccessible. The point x is said to be a continuation point if it is accessible and !.pm (x) = 0, and a set of the all points is called a continuation region. The point x is called a boundary point if !.pm (x ) = 1, and a set of the all points is called a boundary region. A stopping rule is said to be bounded if there exists some constant c with 0 < c < co such that the index of any accessible point is less than c, and simple if the convex hull of the continuation region on each index contains no points except for continuation points. In order to get a sufficient condition for completeness of the sequential procedure, we have the following lemma. LemIna 1. If the stopping rule !.p has, as a boundary region, either or k k :L xjAj 2:: c, :L Xj = n}, j=l j=l k C={X=(X1, ... ,Xk) : :Lxj=m}, j=l where c, AI, ... ,Ak are constants, and m and n are non-negative integers with m > n, then the sequential procedure is complete. Proof. Without loss of genenality, we may assume that Ak = max {AI, ... ,Ak}. At first we translate B over C parallel by m - n in the positive direction of the axis of x k, and we denote it by Bf. Then the point of B' is inaccessible. Indeed, otherwise, we have points X = (x 1 , ... , X k) and Y = (Y1, ... , Y k) with 0 ~ Y j ~ x j U = 1, ... , k) such that on the index n and on the index m k k :L yjAj < c, j=l :L xjAj 2:: c + (m - n)Ak, j=1 k :LXj =m. j=l Since, by the assumption, Aj ~ Ak (j = 1, ... , k), we obtain (Xj - Yj)Aj < (Xj -Yj )Ak (j = 1, ... ,k). Taking their sum, we have k (1) ~(Xj - Yj)A j ~ (m - n)Ak. j=l On the other hand, if we translate the point Y in the positive direction of the axis of x k so that its coordinate is (Yl, . - . ,Yk-l, Yk + m - n), then it is contained in C - 8'. Hence we have k-l (2) ~ yjA j + (Yk + m - n)Ak < c + (m - n)Ak -j=l Since, by the assumption, (3) we obtain from (2) and (3) k c+(m-n)Ak ~ ~xjAj, j=l k ~(Xj - Yj)Aj > (m - n)Ak-j=l This contradicts the inequality (1). Hence the point of B' is inaccessible_ If we redefine X = 8 U (C - 8') as a boundary region, a stopping rule with X is essentially the same as the previous one. Suppose that for a point x = (Xl, .. _ , X k-l, X k) E 8 there exists a point x' satisfying x' = (Xl,'" ,Xk-l,Xk') E C - 8'. Since, from X E 8,2:;=1 Xj = n, it follows that k-l k-l k-l ~ xjA j + XkAk = ~ xjA j + (n - ~ xj)Ak 2: C, j=l j=l j=l hence k-l k-l (4) ~ xjAj - Ak ~ Xj 2: c - nAk. j=l j=l On the other hand, since, by x' E C - H', 2:;:::~ Xj + Xk' straightforward computation A I m, we similarly have by a k-l k-l L xjAj - Ak L Xj < c - nAk · j=l j=l Then this fact contradicts the inequality (4). Since for a point x = (Xl,'" ,Xk-l,Xk) E B there does not exist a point x' satisfying x' = (Xl,'" ,Xk-l?Xk' ) E C - B' , it follows that T = (Xl, .. " xk-d and X = (Xl,.'" Xk-l,Xk) are one-to-one. To show the completeness of the sequential procedure, let f be any unbiased estimator of 0 , i.e., Ep [f(XN)] = 0 for all p. Then it it enough to show that f(x) = 0 for all X E X. First we have, from L~=l Pj = 1, k-l k-1 (5) Ep [f(XN)] = Lf(x)c(x)(l- LPj)Xk IIpjXj x j=l j=l = L!(x)c(x){l + f= (Xt}- ~pj)l} IIp/i x 1=1 )=1 )=1 =0 for allp. Since (PI,'" ,Pk-1) moves over an open set, it follows from (5) that the coefficients of polynomials on (PI?'" ,Pk-l) is equal to O. Now we use an induction method. Since T and)( are one to one, there exists at most one X = (Xl, ... , Xk) E X such that ~~~: Xj = O. Denote it by x', and the constant term of (5) is only f(x')c(X ' ). Since C(X') > 0, we have f(x ' ) = O. Next assume that f(x) = 0 for any X = (Xl"'" Xk) E X satisfying ~k-l < l 1 Tak' "( 1/ ") V . h ~k-l "l h '-.Jj=l Xj _ -. 'lng any X = Xl,··· ,Xk E t'\.. wlt '-.Jj=l Xj = ,we see tat the coefficient on I1~~: Pjx/' is only f(X")C(X") since T and X are one-to-one. Indeed, we obtain from (5) k-l k-l (6) f ( x) c( X ) (1 - L P j ) X k II P j x j j=l j=l k-l L f(x)c(x){l + PI + P2 + ... } II pjXj Xl +"'+Xk-l"2:1 j=l Considering the tern1 of I1~~: Pjx/' in (6), we can get it from a product of the inner part of { ... } and I1J~: pjXj. Then it follows from the assumption that the degree of each term of { ... } rrJ==: pjXj in (6) is equal or greater than that of rrJ==: pjx/'. Hence the coefficient for rrJ==: Pjx/' is only J(X")C(X"). Since C(X") > 0, it follows that J(x lf ) = O. Thus we complete the proof. It is noted that the induction part of the above proof is similar to that of Kre-mers(1990). The following lemma is due to Lehmann and Stein (1950). Lemma 2. Let Xl, X 2, ... be a sequence of random variables such that for each positive integer m the set Xl, ... ,Xm admits a real valued sufficient statistic Tm = tm(XI , ... ,Xm)' Let :BI , :B2, ... , :By- each be a complete, closed, sequential procedure based on these sufficient statistics. Let :BI U :B2 U ... U :By- denote the sequential procedure according to which we continue taking observations until at least one of the stopping rules :B1, :B2, ... ,:Br telles us to stop. Then the procedure :BI U :B2 U ... U :Br is complete. Combining Lemma 1 and Lemma 2 , we get the following theorem. Theorem 1. complete. If a bounded stopping rule is simple, then the sequential procedure is Proof. It follows from the supporting hyperplane theorem that the convex hull of the continuation region for each index can be expressed as an intersection of a finite number of supporting half-spaces. Further, by the boundedness of the stopping rule, we can take a sufficiently large m which XN never fail to stop. From Lemma 1 and Lemma 2 we complete the proof. Remark. It is known that, in the sequential binomial case, the sequential procedure is complete if and only if simple (see Girshick et ai. (1946)). As a sufficient condition for cOlnpleteness in the sequential multinon1ial case, I(remers (1990) also stated that the stopping rule is bounded and the convex hull of the continuation region does not contain a boundary point. The following example sho'\;vs that Theorem 1 holds but not the I(remers condition. Assun1e that a boundary region consists of (0,2), (1, I), (2, 2), (3, 1) and (3,0) in the se-quential binomial case. Since the continuation points are (0,1),(1,0),(2,0) and (2,1), it / o is easily seen that the stopping rule is bounded and the convex hull of the continuation region contains the boundary point (1,1). Hence the Kremers condition is not satisfied, but Theorem 1 holds since the stopping rule is easily seen to be simple. 4. RANDOMIZED SEQUENTIAL PROCEDURE Let {qn} be a sequence such that qn 2:: 0 (n = 0,1,2, ... ) and ~~=oqn = 1. For a stopping rule i.p in the section 2 we assume that i.pn = qn for each non-negative integer n . Then the stopping rule is randomized . We have for any estimator f k (7) Ep [f(XN)] = L L qnn!f(x1,'" ,Xk) IIpjXj IXj!, n Xl+"'+Xk=n j=l where means that we take the sum with respect to the all possible combination Xl+"'+Xk=n with 2::=5=1 x j = n. Then we have the following. Theorem 2. A necessary and sufficient condition for the sequential procedure to be complete is that there exists a unique non-negative integer n satisfying qn = 1. Proof. Sufficiency. Since, in this case, XN is distributed according to the k-k x. dimensional n1ultinomial distribution M(n,P1,' .. ,Pk ) with a mass function nl II P: .: j=l ). for all non-negative integers satisfying ~J=l x j = n, it is easily seen that the sequential procedure is complete. Necessity. Suppose that qm, qn > 0 for m < n with non-negative integers m and n. It suffices to prove that unbiased estimator of 0 based on only the points over the indices m and n can be constructed. The number of the possible cOlubination (Xl,"" Xk) such that 2:5=1 x j = n is equal to (k+~-I). On the other hand, since ~J=1 Pj = 1 yields k k-l k-l II pjXj = (1 - LPj)Xk II pjXj, j=1 j=l we may regard the right-hand side of (7) as a polynomial in (PI,'" ,Pk-l). If the polyno-mial is identically equal to 0, then each coefficient must be equal to O. Since the number 'I I of terms of the polynomial with degree 7' is (k+~-2) in nJ==: pjXj , it follows that the total number of its terms is 2:;=0 (k+~-2). It is also seen that the coefficients of the polynomial of the right-hand side of (7) are linear functions of f(x}) ... ,Xk)'S. Since each coefficient of the polynomial must be 0, we may regard it as the simultaneous linear equations of f( xl, ... X k) 'So Then there exist non-trivial solution of these equations. In fact, since it is easily seen that Moreover the left and right sides of the above inequality are equal to the number of unknown variables f( Xl, ... , X k) and that of linear restrictions, respectively. Consequently we can get a non-trivial unbiased estimator of ° based on the points over the indices m and n. Thus we complete the proof. The above result may be applied to the following estimation problem. For each n = 0,1,2, .. 'J let 5n be an estirnator of a function g(p) of p. Then our purpose is to minimize 00 (X) L Ep [qn {5n - g(p) }2] under the condition L Ep [qn5n] = g(p) for all p. In order to do n=O n=O so, it is enough to obtain 5n minimizing Ep [{ 5n - g(p )}2] for each n under the condition 00 L qnEp [8n] == g(p), since, for each n, the randomized stopping rule rpn is equal to qn n=O which is independent of ){n' Hence it is seen that the above problem is similar to that in the case of a fixed size of a sample. In this case it is also easy to check the necessary and sufficient condition for the sequential procedure to be complete in Theorem 2. 8 REFERENCES Ferguson, T. S. (1967). Mathematical Statistics. Academic Press, New York. Girshick, M. A., F. Mosteller & L. J. Savage (1946). Unbiased estimates for certain bi-nomial sampling problems with applications. Annals of Mathematical Statistics, 17. 13-23. Kremers, W. K. (1990). Completeness and unbiased estimation in sequential multinomi-al sampling. Sequential Analysis) 9. 43-58. Lehmann, E. L. & C. Stein (1950). Completeness in the sequential case. Annals of Mathematical Statistics, 21. 376-385. 9 

Completeness for sequential sampling plans

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