Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests

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.