## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 40, Number 3 (1969), 828-835.

### Hypothesis Testing with Finite Statistics

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables drawn according to a probability measure $\mathscr{P}$. The two-hypothesis testing problem $H_0: \mathscr{P} = \mathscr{P}_0 \operatorname{vs.} H_1: \mathscr{P} = \mathscr{P}_1$ is investigated under the constraint that the data must be summarized after each observation by an $m$-valued statistic $T_n\varepsilon \{1, 2, \cdots, m\}$, where $T_n$ is updated according to the rule $T_{n+1} = f_n(T_n, X_{n+1})$. An algorithm with a four-valued statistic is described which achieves a limiting probability of error zero under either hypothesis. It is also demonstrated that a four-valued statistic is sufficient to resolve composite hypothesis testing problems which may be reduced to the form $H_0:p > p_0 \operatorname{vs.} H_1:p < p_0$ where $X_1, X_2, \cdots$ is a Bernoulli sequence with bias $p$.

#### Article information

**Source**

Ann. Math. Statist., Volume 40, Number 3 (1969), 828-835.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177697590

**Digital Object Identifier**

doi:10.1214/aoms/1177697590

**Mathematical Reviews number (MathSciNet)**

MR240906

**Zentralblatt MATH identifier**

0181.45502

**JSTOR**

links.jstor.org

#### Citation

Cover, Thomas M. Hypothesis Testing with Finite Statistics. Ann. Math. Statist. 40 (1969), no. 3, 828--835. doi:10.1214/aoms/1177697590. https://projecteuclid.org/euclid.aoms/1177697590