On the Monotonicity of a Certain Expectation

Abstract

Let $\{X_n, n \geq 1\}$ be a sequence of random variables and let $P_\theta$ be a probability measure under which $(X_1, \ldots, X_n)$ have joint pdf's $f_n(X_1, \ldots, X_n, \theta) = L_n(\theta), n \geq 1$. Suppose $u_n = u_n(X_1, \ldots, X_n), n \geq 1$, are statistics such that $(u_n - c)(L_n(\theta') - L_n(\theta)) \geq 0, \forall (X_1, \ldots, X_n), n \geq 1$, for some constant $c = c(\theta,\theta'), \theta \neq \theta'$. For any increasing function $\psi$ and stopping time $T$, it is shown that $E_\theta\phi(u_T) \leq E_{\theta'}\phi(u_T)$, provided that one of the expectations is finite and $P_\theta(T < \infty) = P_{\theta'}(T < \infty) = 1$. The given result holds for a certain monotone likelihood ratio family and an exponential family in particular. This generalizes a result of Chow and Studden and provides a sequential version of a result of Lehmann.