## M-estimation in linear models under nonstandard conditions.(English)Zbl 1038.62026

Summary: The limiting distribution of M-estimators of the regression parameter in linear models is derived under nonstandard conditions, allowing, e.g., for discontinuities in density functions. Unlike usual regularity assumptions, our conditions are satisfied, for instance, in the case of regression quantiles, hence also in the context of $$L_1$$ estimation; our results thus extend those of K. Knight [Ann. Stat. 26, 755–770 (1998; Zbl 0929.62021)]. The resulting asymptotic distributions, in general, are not Gaussian. Therefore, the limiting bootstrap distributions of these estimators are also investigated. It is shown that bootstrap approximations are correct to the first order only when limiting distributions are Gaussian, or along specific sequences $$m_n$$ of bootstrap sample sizes. Numerical examples are given to illustrate these asymptotic results.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62F40 Bootstrap, jackknife and other resampling methods 62E20 Asymptotic distribution theory in statistics 62J05 Linear regression; mixed models

### Keywords:

Bootstrap; Limiting distribution; Linear model; M-estimators

Zbl 0929.62021

bootstrap
Full Text:

### References:

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