The Annals of Mathematical Statistics

Asymptotically Optimum Properties of Certain Sequential Tests

Seok Pin Wong

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Abstract

Let $X_1, X_2, \cdots$ be independent and identically distributed random variables whose common distribution is of the one-parameter Koopman-Darmois type, i.e., the density function of $X_1$ relative to some $\sigma$-finite nondegenerate measure of $F$ on the real line can be written as $f(x, \theta) = \exp (\theta x - b(\theta))$, where $b(\theta)$ is some real function of the parameter $\theta$. Consider the hypotheses $H_0 = \{\theta \leqq \theta_0\}$ and $H_1 = \{\theta \geqq \theta_1\}$ where $\theta_0 < \theta_1$ and $\theta_0, \theta_1$ are in $\Omega$, the natural parameter space. We want to decide sequentially between the two hypotheses. Suppose $l(\theta)$ is the loss for making a wrong decision when $\theta$ is the true parameter and assume $0 \leqq l(\theta) \leqq 1$ for all $\theta$ and $l(\theta) = 0$ if $\theta$ is in $(\theta_0, \theta_1)$, i.e., $(\theta_0, \theta_1)$ is an indifference zone. Let $c$ be the cost of each observation. It is sufficient to let the decision depend on the sequence $(n, S_n), n \geqq 1$, where $S_n = X_1 + \cdots + X_n$. We shall consider the observed values of $(n, S_n)$ as points in a $(u, v)$ plane. Then, for any test, the region in the $(u, v)$ plane where sampling does not stop is called the continuation region of the test. A test and its continuation region will be denoted by the same symbol. Schwarz [4] introduced an a priori distribution $W$ and studied the asymptotic shape of the Bayes continuation region, say $B_W(c)$, as $c \rightarrow 0$. He showed that $B_W(c)/\ln c^{-1}$ approaches, in a certain sense, a region $B_W$ that depends on $W$ only through its support. Whereas Schwarz's work is concerned with Bayes tests, in this paper the main interest is in characteristics of sequential tests as a function of $\theta$. In particular, it is desired to minimize the expected sample size (uniformly in $\theta$ if possible) subject to certain bounds on the error probabilities. Our approach, like Schwarz's, is asymptotic, as $c \rightarrow 0$. It turns out that an asymptotically optimum test--in the sense indicated above, is $B_W \ln c^{-1}$ if $W$ is a measure that dominates Lebesgue measure. Such a measure will be denoted by $L$ (for Lebesgue dominating) from now on. Thus, Bayes tests, as a tool, will play a significant role in this paper. In order to prove the optimum characteristic of $B_L \ln c^{-1}$, some other results, of interest in their own right, are established. For any $W$ satisfying certain conditions that will be given later, we show that the stopping variable $N(c)$ of $B_W(c)$ approaches $\infty$ a.e. $P_\theta$ for every $\theta$ in $\Omega$. This result together with Schwarz's result that $B_W(c) \ln c^{-1}$ approaches a finite region, leads to the following results: (i) for $B_W(c), E_\theta N(c)/\ln c^{-1}$ tends to a constant for each $\theta$ in $\Omega$ and (ii) the same is true for the stopping variable of $B_W \ln c^{-1}$. Furthermore, it is shown that for $B_L \ln c^{-1}$ the error probabilities tend to zero faster than $c \ln c^{-1}$. Consequently, the contributions of the expected sample sizes of both $B_L \ln c^{-1}$ and $B_L(c)$ to their integrated risks, over any $L$-measure, approach 100%. Moreover $B_L \ln c^{-1}$ is asymptotically Bayes. The last result can be shown without (i) since it is sufficient to show (ii) and to apply the same argument used by Kiefer and Sacks [3] in the proof of their Theorem 1. But we show (i) because of its intrinsic interest and present a different proof using (i). Kiefer and Sacks assumed a more general distribution for $X_1$, constructed a procedure $\delta^{'I}_c$ and showed that it is asymptotically Bayes. Our $B_L \ln c^{-1}$ is somewhat more explicit than their $\delta_c'I$. We would also like to point out that an example of $B_L \ln c^{-1}$, when the distribution is normal, is very briefly discussed in their work. We shall restrict ourselves to a priori distribution $W$ for which $\sup (\mod W)H_0 = \theta_0, \inf (\mod W)H_1 = \theta_1$ and $0 < W(H_0 \cup H_1) < 1$. The phrase "for any $W"$ or "for every $W$" is to be understood in that sense. Any Lebesgue dominating measure satisfies these conditions and also the following type of $W$ that will be used: the support of $W$ consists of $\theta_0, \theta_1$ and a third point $\theta^\ast, \theta_0 < \theta^\ast < \theta_1$. Such a $W$ will be called a $\theta^\ast$-measure, and the corresponding $B_W$ denoted by $B_{\theta^\ast}$. From Schwarz's equations for $B_W$ it follows readily that $B_L \subset B_W$ for every $W$. In particular, $B_L \subset B_{\theta^\ast}$. As a consequence, the statement about the error probabilities as well as others concerning $B_L \ln c^{-1}$ in the last paragraph, remain true when $L$ is replaced by $\theta^\ast$ or any $W$. Those geometric characteristics will be dealt with in Section 2. We shall also show there that $\partial B_{\theta^\ast}$, the boundary of $B_{\theta^\ast}$ (which consists of line segments), is tangent to $\partial B_L$ at some point, and that if $\theta^\ast$ is such that $b'(\theta^\ast) = (b(\theta_1) - b(\theta_0))/(\theta_1 - \theta_0)$ then $\max_{(u, \nu)\text{in}B_L} u = \max_{u,v)\text{in}B_\theta^\ast} u$. Let the ray through the origin and with slope equal to $E_\theta X_1$ intersect $\partial B_L$ at $(m(\theta), m(\theta)E_\theta X_1)$. In Section 3, after proving $\lim_{c\rightarrow 0} N(c) = \infty$ a.e. $P_\theta$, we show $\lim_{c\rightarrow 0} N(c)/\ln c^{-1} = m(\theta)$ a.e. $P_\theta$ and $\lim_{c\rightarrow 0} E_\theta N(c)/\ln c^{-1} = m(\theta)$. It is shown in Section 4 that $\sup_{\theta \text{in} H_0 \mathbf{\cup} H_1} P_\theta$ (error $\mid B_L \ln c^{-1}) = o(c \ln c^{-1})$. The main results are given in Section 5. We first show that after dividing by $c \ln c^{-1}$, the difference of the integrated risks of $B_L \ln c^{-1}$ and $B_W(c)$, for any $W$, tends to zero. It follows from this result that $B_L \ln c^{-1}$ asymptotically minimizes the maximum (over $\theta$ in $\Omega$) expected sample size in $\mathscr{F}(c)$, a family of tests whose error probabilities are bounded by $\max_{i=0,1} P_{\theta_i}$ (error $\mid B_L \ln c^{-1}$). The precise statement is given in Theorem 5.1. A sharper result under a stronger hypothesis is given in Theorem 5.2 which states that $B_L \ln c^{-1}$ asymptotically minimizes the expected sample size $E_\theta N$ for each $\theta, \theta_0 < \theta < \theta_1$, among all procedures of $\mathscr{F}(c)$ for which $E_{\theta_0}N/\ln c^{-1}$ and $E_{\theta_1}N/\ln c^{-1}$ are bounded in $c$.

Article information

Source
Ann. Math. Statist., Volume 39, Number 4 (1968), 1244-1263.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698250

Digital Object Identifier
doi:10.1214/aoms/1177698250

Mathematical Reviews number (MathSciNet)
MR229344

Zentralblatt MATH identifier
0162.49805

JSTOR
links.jstor.org

Citation

Wong, Seok Pin. Asymptotically Optimum Properties of Certain Sequential Tests. Ann. Math. Statist. 39 (1968), no. 4, 1244--1263. doi:10.1214/aoms/1177698250. https://projecteuclid.org/euclid.aoms/1177698250


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