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Consistency of \(M\)-estimates in general regression models. (English) Zbl 0872.62071

The authors prove the consistency of an \(M\)-estimator defined through a minimum property under two major assumptions: identifiability of the parameter and uniform convergence of the objective function. The regression model studied contains linear and nonlinear functions. A series of sufficient conditions for consistency is first provided for compact parameter spaces. For more general parameter spaces, conditions are given to ensure that the \(M\)-estimate eventually falls into a compact subspace.
In the discussions and applications, a finite variance of the error distribution was often assumed. \(M\)-estimators with a bounded score function (such as the least absolute deviation estimator) should be able to handle more general error distributions. An earlier treatment of consistency for general \(M\)-estimators can be found in a paper by P. J. Huber [Proc Fifth Berkeley Symp. Math. Stat. Probab. Berkeley 1965/66, Vol. I, 221-233 (1967)], but the present paper allows for non-stochastic designs in the regression setting.

MSC:

62J02 General nonlinear regression
62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)

Citations:

Zbl 0212.21504
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