## The Annals of Statistics

### On the asymptotics of constrained local $M$-estimators

Alexander Shapiro

#### Abstract

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

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

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1015952006

Digital Object Identifier
doi:10.1214/aos/1015952006

Mathematical Reviews number (MathSciNet)
MR1792795

Zentralblatt MATH identifier
01828970

Subjects
Primary: 62F12: Asymptotic properties of estimators

#### Citation

Shapiro, Alexander. On the asymptotics of constrained local $M$-estimators. Ann. Statist. 28 (2000), no. 3, 948--960. doi:10.1214/aos/1015952006. http://projecteuclid.org/euclid.aos/1015952006.

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