## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 5 (1973), 838-850.

### 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.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 5 (1973), 838-850.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342505

**Digital Object Identifier**

doi:10.1214/aos/1176342505

**Mathematical Reviews number (MathSciNet)**

MR345355

**Zentralblatt MATH identifier**

0274.62058

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62L99: None of the above, but in this section

**Keywords**

Sequential distinguishability countable family stopping rule sequential probability ratio test optimality Kullback-Leibler information measure asymptotic optimality

#### Citation

Khan, Rasul A. On Sequential Distinguishability. Ann. Statist. 1 (1973), no. 5, 838--850. doi:10.1214/aos/1176342505. https://projecteuclid.org/euclid.aos/1176342505