## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 29, Number 3 (1958), 813-828.

### Maximum-Likelihood Estimation of Parameters Subject to Restraints

#### Abstract

The estimation of a parameter lying in a subset of a set of possible parameters is considered. This subset is the null space of a well-behaved function and the estimator considered lies in the subset and is a solution of likelihood equations containing a Lagrangian multiplier. It is proved that, under certain conditions analogous to those of Cramer, these equations have a solution which gives a local maximum of the likelihood function. The asymptotic distribution of this `restricted maximum likelihood estimator' and an iterative method of solving the equations are discussed. Finally a test is introduced of the hypothesis that the true parameter does lie in the subset; this test, which is of wide applicability, makes use of the distribution of the random Lagrangian multiplier appearing in the likelihood equations.

#### Article information

**Source**

Ann. Math. Statist., Volume 29, Number 3 (1958), 813-828.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706538

**Digital Object Identifier**

doi:10.1214/aoms/1177706538

**Mathematical Reviews number (MathSciNet)**

MR94873

**Zentralblatt MATH identifier**

0092.36704

**JSTOR**

links.jstor.org

#### Citation

Aitchison, J.; Silvey, S. D. Maximum-Likelihood Estimation of Parameters Subject to Restraints. Ann. Math. Statist. 29 (1958), no. 3, 813--828. doi:10.1214/aoms/1177706538. https://projecteuclid.org/euclid.aoms/1177706538