An Essentially Complete Class of Decision Functions for Certain Standard Sequential Problems

Abstract

A sequential problem is considered in which independent observations are taken on a chance variable $X$ whose distribution can be represented by \begin{equation*}\tag{1}dG_\theta(x) = \psi(\theta)e^{\theta x} d\mu(x),\end{equation*} where the parameter $\theta$ belongs to a given interval $\Omega$ of the real line but is otherwise unknown. The problem is to test $H_1:\theta \leqq \theta^\ast$ against $H_2:\theta > \theta^\ast$, where $\theta^\ast$ is a given point in $\Omega$. Under certain assumptions the following class $A$ is shown to be essentially complete relative to the class of decision rules with bounded risk functions. The decision rule $\delta \varepsilon A$ if and only if after taking $n$ observations (i) $\delta$ depends on the observations only through $n$ and $v_n = \sum^n_{i = 1} x_i$ and (ii) $\delta$ specifies a closed interval $J_n:\lbrack a_{1n}, a_{2n} \rbrack$ for each $n$ and the following rule of action (a) Stop experimentation as soon as $v_n \not\varepsilon J_n$ and (1) accept $H_1 \text{if} v_n < a_{1n}$ (2) accept $H_2 \text{if} v_n > a_{2n}$. (b) If $a_{1n} < a_{2n}$ take another observation if $a_{1n} < v < a_{2n}$. (c) If $a_{1n} < a_{2n}$ and $v = a_{in},$ accept $H_i$ or take another observation or randomize between these two $(i = 1, 2)$. The Koopman-Darmois family of probability laws given above contains discrete members such as the binomial and Poisson distributions as well as absolutely continuous members such as the normal and exponential. It is interesting to note that the members of the class A can be obtained by starting with the sequential probability ratio test for testing some point $\theta^\ast_1 \leqq \theta^\ast$ against another point $\theta^\ast_2 > \theta^\ast$, namely, continue as long as $B < \frac{\prod^n_{i = 1} \psi(\theta^\ast_2)e^{\theta^\ast_2 x_i}}{\prod^n_{i = 1} \psi(\theta^\ast_1)e^{\theta^\ast_1 x_i}} < A$ and replacing the constants $B, A$ by two arbitrary sequences $B_n, A_n$ such that $B_n \leqq A_n (n = 1, 2, \cdots).$