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On the asymptotics of constrained \(M\)-estimation. (English) Zbl 0829.62029

Summary: 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 [H. Chernoff, Ann. Math. Stat. 25, 573–578 (1954; Zbl 0056.37102)] of the parameter set. A \(\sqrt {n}\)-consistent sequence of local optimizers converges under Clarke regularity [F. H. Clarke, Optimization and nonsmooth analysis. New York: John Wiley (1983; Zbl 0582.49001)] 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.

MSC:

62F12 Asymptotic properties of parametric estimators
62F05 Asymptotic properties of parametric tests
62F30 Parametric inference under constraints
49J55 Existence of optimal solutions to problems involving randomness
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