The Annals of Mathematical Statistics

Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests

R. H. Farrell

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Abstract

Let $R$ be the set of real numbers, $\mathscr{B}_1$ the set of Borel sets of $R$, and $\mu$ a $\sigma$-finite nonnegative measure on $\mathscr{B}_1$. Let $\Omega$ be an open real number interval (which may be infinite). Throughout we consider a Koopman-Darmois family \begin{equation*}\tag{1}\{h(\theta) \exp (\theta x), \theta \varepsilon \Omega\}\end{equation*} of generalized probability density functions on the measure space $(R, \mathscr{B}_1, \mu)$. We consider one sided tests $T$ of the hypothesis $\theta < 0$ against the alternative $\theta > 0$. In general, in this paper, $T$ will be a sequential procedure. Associated with $T$ is a stopping variable $N$ (mention of the dependence of $N$ on $T$ is usually omitted). $N \geqq 0. N = n$ means that sampling stopped after $n$ observations and a decision was made. In this context we consider $\infty$ to be an integer, and $N = \infty$ means that sampling does not stop. In the discussion of Section 1 we will assume that if $\theta \varepsilon \Omega$ and $\theta \neq 0$ then $P_\theta(N < \infty) = 1$, that is, sampling stops with probability one. The reason for the exclusion of $\theta = 0$ will become apparent in Section 1. We will be concerned with two events, decide $\theta < 0$, and, decide $\theta > 0$. The main result of this paper may be stated as follows. Theorem 1. Suppose $(R, \mathscr{B}_1, \mu), \Omega$, and $\{h(\theta) \exp (\theta x), \theta \varepsilon \Omega\}$ are as described above. Define \begin{align*}\mu_\theta = \int^\infty_{-\infty} h(\theta)x \exp(\theta x)\mu(dx), \\ \tag{2} \\ \sigma^2 = \int^\infty_{-\infty} h(0)x^2\mu(dx),\end{align*} and assume $\mu_0 = 0$. Suppose $0 < \alpha < 1$ and $0 < \beta < 1$ and \begin{equation*}\tag{3}\sup_{\theta > 0} P_\theta (\text{ decide } \theta < 0) \leqq \beta;\quad \sup_{\theta < 0} P_\theta (\text{ decide } \theta > 0) \leqq \alpha.\end{equation*} Then \begin{align*}\lim \sup_{\theta \rightarrow 0+} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N \geqq 2\sigma^2 P_0(N &= \infty); \\ \tag{4} \lim \sup_{\theta \rightarrow 0-} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N \geqq 2\sigma^2 P_0(N &= \infty).\end{align*} If $\alpha + \beta < 1$ there is a generalized sequential probability ratio test $T$ with stopping variable $N$ such that for the test $T$, \begin{equation*}\tag{5}P_0(N = \infty) = 1 - (\alpha + \beta);\quad (3) \text{ holds };\end{equation*} for the test $T$, \begin{equation*}\tag{6}\lim_{\theta \rightarrow 0} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N = 2\sigma^2 P_0(N = \infty).\end{equation*} For all tests $T$, if $P_0(N = \infty) > 0$ then $\lim_{\theta \rightarrow 0} \theta^2E_\theta N = \infty$. In Section 1, (7) and (8), it is shown that $P_0(N = \infty) \geqq 1 - \alpha - \beta$. Consequently the relations (4) and (5) of Theorem 1 are not vacuous. We were led to formulate Theorem 1 by a problem of constructing bounded length confidence intervals. The relationship is explained in Section 2. The proof of Theorem 1 is given in Section 3.

Article information

Source
Ann. Math. Statist., Volume 35, Number 1 (1964), 36-72.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177703731

Digital Object Identifier
doi:10.1214/aoms/1177703731

Mathematical Reviews number (MathSciNet)
MR157459

Zentralblatt MATH identifier
0156.39306

JSTOR
links.jstor.org

Citation

Farrell, R. H. Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests. Ann. Math. Statist. 35 (1964), no. 1, 36--72. doi:10.1214/aoms/1177703731. https://projecteuclid.org/euclid.aoms/1177703731


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