The Essential Completeness of the Class of Generalized Sequential Probability Ratio Tests

Abstract

Consider a sequential decision problem in which independent observations are to be taken on a random variable $X$ whose distribution is of the form \begin{equation*}\tag{1}dG_\theta(x) = \Psi(\theta)e^{\theta x} d\mu(x),\end{equation*} where the parameter $\theta$ lies in a given interval $\Omega$ of the real line but is otherwise unknown, and the measure $\mu(x)$ is either absolutely continuous or discrete. The problem is to decide between the hypotheses \begin{equation*}\tag{2}\begin{split}H_1: \theta &\leqq \theta^\ast\\H_2: \theta &> \theta^\ast,\end{split}\end{equation*} where $\theta^\ast$ is a given point of $\Omega$. Under the assumptions stated in Section 2 the class A of generalized sequential probability ratio tests is shown to be essentially complete relative to the class D of decision functions with bounded risk. A decision function $\delta$ belongs to the class A if and only if after taking $n$ observations, (i) $\delta$ depends on the observations only through $n$ and $v_n = \sum^n_{i = 1} X_i$; (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 $v_n \not\in J_n$. If $v_n < a_{1n}$, accept $H_1$. If $v_n > a_{2n}$, accept $H_2$. (b) If $a_{1n} < v_n < a_{2n}$, take another observation. (c) If $a_{1n} < a_{2n}$ and $v_n = a_{in}$, accept $H_i$ or take another observation or randomize between these two $(i = 1, 2)$. The problem considered here is the same as that treated by Sobel [1], and the foregoing statement of the problem and conclusion, as well as the assumptions to be given in the next section, follow his work very closely. The contribution of this paper is that the requirement of bounded loss functions made by Sobel is removed.