On the Asymptotics of Constrained $M$-Estimation

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.