## The Annals of Mathematical Statistics

### Characterizations of Complete Classes of Tests of Some Multiparametric Hypotheses, with Applications to Likelihood Ratio Tests

Allan Birnbaum

#### Abstract

For the problem of testing a simple hypothesis on a density function of the form $f_\theta(e) = \exp \{\psi_0(\theta) + \sum^k_1 \psi_i(\theta)t_i(e) + t_0(e)\}$, explicit characterizations are given of a minimal essentially complete class of tests, the minimal complete class, and the closure of the class of Bayes' solutions, under certain assumptions. Applications are made to discrete distributions of the above form and to some problems of testing composite hypotheses. The likelihood ratio tests of these hypotheses are characterized and shown to be admissible under certain assumptions.

#### Article information

Source
Ann. Math. Statist., Volume 26, Number 1 (1955), 21-36.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177728590

Digital Object Identifier
doi:10.1214/aoms/1177728590

Mathematical Reviews number (MathSciNet)
MR67438

Zentralblatt MATH identifier
0064.13802

JSTOR