On the Asymptotic Behavior of Densities with Applications to Sequential Analysis

Abstract

$\{X_n\}$ is a sequence of random variables defined on some probability space, $\mathscr{P} = \{P_\theta, \theta\epsilon \Theta\}$ is the family of distributions of $\{X_n\}$ and $\mathscr{A}_n$ is the subfield generated by $(X_j, j \leqq n)$. It is assumed that $\Theta$ is a real interval, $X_n \rightarrow \theta$ a.e. $P_\theta$ and that, for each $n, X_n$ is sufficient for $\mathscr{P}$ on $\mathscr{A}_n$ while $\mathscr{P}$ is a homogeneous monotone likelihood ratio family on $\mathscr{A}_n$. Let $p_{\theta n}$ denote the density of $P_\theta$ with respect to some $\sigma$-finite measure on $\mathscr{A}_n$ and consider the sequence $\{R_n\}$, where $R_n = p\theta_2n/p\theta_1n$ and $\theta_1 < \theta_2$. Conditions are given for the occurrence of the following limiting behavior of $R_n : (1)$ there exists a $\theta_0, \theta_1 < \theta_0 < \theta_2$, such that $R_n \rightarrow 0$ or $\infty$ a.e. $P_\theta$ according as $\theta <$ or $> \theta_0$ and (2) $P_{\theta_0} (b < R_n < a) \rightarrow 0$ for $0 < b < a < \infty$. This limiting behavior of $R_n$ guarantees the termination with probability one of sequential probability ratio tests based on $\{X_n\}$. Let $q_{\theta n}$ denote the density of $P_\theta$ on the Borel field generated by $X_n$. In order to describe the results of this paper we introduce Condition $A_1$ which states, essentially, that $n^{-1} ln \lbrack q_{\theta n} (x)/K(n)\rbrack \rightarrow h(\theta, x)$ for some functions $K$ and $h$ satisfying very mild conditions. Condition $A_2$ requires $h(\theta, x)$ to possess, for each fixed $x$, a unique maximum at $\theta = x$. It is shown that under condition $A_1$, part (1) of the above limiting behavior of $R_n$ is equivalent to the statement that $h(\theta, x)$ satisfies Condition $A_2$ with $\theta_0$ being the solution of $h(\theta_2, x) = h (\theta_1, x)$. Moreover, under Condition $A_1$, part (2) is implied by Conditions $A_3$ and $A_4$, where Condition $A_3$ states that $q_{\theta_2^n}(\theta_0 + c/n)/q_{\theta_1^n}(\theta_0 + c/n) \rightarrow \alpha e^{c\beta}$ for some $\alpha, \beta > 0$ and all $c \neq 0$, while Condition $A_4$ requires $n^{\frac{1}{2}}(X_n - \theta_0)$ to possess under $P_{\theta_0}$ a limiting distribution which is continuous at 0. For the purpose of checking the above conditions we study the asymptotic behavior of generalized Laplace integrals. The results enable us to assert that a certain Condition B (stronger than Condition $A_1$) obtains, where $B$ states, essentially, that $q_{\theta n}(x) \sim K(n)C(\theta, x)e^{nh(\theta,x)}$ for some functions $K, C$, and $h$ satisfying mild conditions. The same techniques enable us to verify Condition $A_3$ also. The applications include the sequential $t-, F-, \chi^2-, T^2-$ and other tests.