## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 4 (1973), 633-643.

### Open-Ended Tests for Koopman-Darmois Families

#### Abstract

The generalized likelihood ratio is used to define a stopping rule for rejecting the null hypothesis $\theta = \theta_0$ in favor of $\theta > \theta_0$. Subject to a bound $\alpha$ on the probability of ever stopping in case $\theta = \theta_0$, the expected sample sizes for $\theta > \theta_0$ are minimized within a multiple of $\log \log \alpha^{-1}$, the multiple depending on $\theta$. An heuristic bound on the error probability of a likelihood ratio procedure is derived and verified in the case of a normal mean by consideration of a Wiener process. Useful lower bounds on the small-sample efficiency in the normal case are thereby obtained.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 4 (1973), 633-643.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342459

**Digital Object Identifier**

doi:10.1214/aos/1176342459

**Mathematical Reviews number (MathSciNet)**

MR426318

**Zentralblatt MATH identifier**

0282.62072

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

**Keywords**

Likelihood ratio sequential probability ratio test open-ended test asymptotic efficiency

#### Citation

Lorden, Gary. Open-Ended Tests for Koopman-Darmois Families. Ann. Statist. 1 (1973), no. 4, 633--643. doi:10.1214/aos/1176342459. https://projecteuclid.org/euclid.aos/1176342459