The Annals of Mathematical Statistics

Maximum-Likelihood Estimation of Parameters Subject to Restraints

J. Aitchison and S. D. Silvey

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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.

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

First available in Project Euclid: 27 April 2007

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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.

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