## The Annals of Statistics

### On the Asymptotics of Constrained $M$-Estimation

Charles J. Geyer

#### Abstract

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

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

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176325768

Digital Object Identifier
doi:10.1214/aos/1176325768

Mathematical Reviews number (MathSciNet)
MR1329179

Zentralblatt MATH identifier
0829.62029

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