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Empirical distribution function under heteroscedasticity. (English) Zbl 1229.62050

Summary: Neglecting heteroscedasticity of error terms may imply the wrong identification of a regression model (see the appendix). Employment of the (heteroscedasticity resistent) H. White’s [Econometrica 48, 817–838 (1980; Zbl 0459.62051)] estimator of the covariance matrix of estimates of regression coefficients may lead to the correct decision about the significance of individual explanatory variables under heteroscedasticity. However, White’s estimator of the covariance matrix was established for least squares (LS)-regression analysis (in the case when error terms are normally distributed, LS- and maximum likelihood (ML)-analysis coincide and hence then White’s estimate of covariance matrix is available for ML-regression analysis). To establish White’s-type estimate for another estimator of regression coefficients requires Bahadur representation of the estimator in question, under heteroscedasticity of error terms. The derivation of Bahadur representation for other (robust) estimators requires some tools. As the key tool proved to be a tight approximation of the empirical distribution function (d.f.) of residuals by the theoretical d.f. of the error terms of the regression model. We need the approximation to be uniform in the argument of the d.f. as well as in the regression coefficients. The present paper offers this approximation for the situation when the error terms are heteroscedastic.

MSC:

62G08 Nonparametric regression and quantile regression
62G30 Order statistics; empirical distribution functions
62J02 General nonlinear regression

Citations:

Zbl 0459.62051
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