The Annals of Statistics

On the asymptotics of constrained local $M$-estimators

Alexander Shapiro

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We discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and,more generally, $M$-estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set $S$.We give a counterexample showing that regularity of $S$ in the sense of Clarke is not sufficient for asymptotic equivalence of $\sqrt{n}$-consistent locally optimal $M$-estimators.We argue further that stronger properties, such as so-called near convexity or prox-regularity of $S$ are required in order to ensure that any two $\sqrt{n}$-consistent locally optimal $M$-estimators have the same asymptotics.

Article information

Ann. Statist. Volume 28, Number 3 (2000), 948-960.

First available: 12 March 2002

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators

Maximum likelihood contrained $M$-estimation asymptotic distribution tangent cones Clarke regularity prox-regularity metric projection


Shapiro, Alexander. On the asymptotics of constrained local $M$-estimators. The Annals of Statistics 28 (2000), no. 3, 948--960. doi:10.1214/aos/1015952006.

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