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December, 1994 On the Asymptotics of Constrained $M$-Estimation
Charles J. Geyer
Ann. Statist. 22(4): 1993-2010 (December, 1994). DOI: 10.1214/aos/1176325768


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.


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Charles J. Geyer. "On the Asymptotics of Constrained $M$-Estimation." Ann. Statist. 22 (4) 1993 - 2010, December, 1994.


Published: December, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0829.62029
MathSciNet: MR1329179
Digital Object Identifier: 10.1214/aos/1176325768

Primary: 62F12
Secondary: 49J55 , 60F05

Keywords: $M$-estimation , central limit theorem , Chernoff regularity , Clarke regularity , constraint , maximum likelihood , tangent cone

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 4 • December, 1994
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