On Sequential Distinguishability

Abstract

Let $X_1, X_2,\cdots$ be a sequence of independent and identically distributed random variables governed by an unknown member of a countable family $\mathscr{P} = \{P_\theta: \theta \in \Omega\}$ of probability measures. The family $\mathscr{P}$ is said to be sequentially distinguishable if for any $\varepsilon (0 < \varepsilon < 1)$ there exist a stopping time $t$ and a terminal decision function $\delta(X_1,\cdots, X_t)$ such that $P_\theta\{t < \infty\} = 1 \forall\theta\in\Omega$ and $\sup_{\theta\in\Omega} P_\theta(\delta(X_1,\cdots, X_t) \neq \theta) \leqq \varepsilon$. Robbins [12] defined a general stopping time (see Section 2) as an approach to this problem. This paper is a study of this stopping time with applications to some exponential distributions.