The Annals of Statistics

On the Asymptotics of Constrained $M$-Estimation

Charles J. Geyer

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Limit theorems for an $M$-estimate constrained to lie in a closed subset of $\mathbb{R}^d$ are given under two different sets of regularity conditions. A consistent sequence of global optimizers converges under Chernoff regularity of the parameter set. A $\sqrt n$-consistent sequence of local optimizers converges under Clarke regularity of the parameter set. In either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value. Limit theorems for the optimal value are also obtained, agreeing with Chernoff's result in the case of maximum likelihood with global optimizers.

Article information

Ann. Statist. Volume 22, Number 4 (1994), 1993-2010.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F12: Asymptotic properties of estimators
Secondary: 49J55: Problems involving randomness [See also 93E20] 60F05: Central limit and other weak theorems

Central limit theorem maximum likelihood $M$-estimation constraint tangent cone Chernoff regularity Clarke regularity


Geyer, Charles J. On the Asymptotics of Constrained $M$-Estimation. Ann. Statist. 22 (1994), no. 4, 1993--2010. doi:10.1214/aos/1176325768.

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