Consider the regression model $Y_i = g(t_i) + \varepsilon_i, 1 \leq i \leq n$, with nonrandom design variables $(t_i)$ and measurements $(Y_i)$ for the unknown regression function $g(\cdot)$. We assume that the data are heteroscedastic, i.e., $E(\varepsilon^2_i) = \sigma^2_i \not\equiv \operatorname{const.}$ and investigate how to estimate $\sigma^2_i$. If $\sigma^2_i = \sigma^2(t_i)$ with a smooth function $\sigma^2(\cdot)$, initial estimators $\tilde{\sigma}^2_i$ can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent. These estimates can be used to improve the estimation of the regression function $g$ itself in parametric and nonparametric models. Further applications are suggested.
Publié le : 1987-06-14
Classification:
Local variance,
kernel estimators,
rates of uniform convergence,
nonparametric regression,
parametric regression,
bandwidth variation in kernel estimators,
convergence of weighted averages of $m$-dependent random variables,
weighted least squares,
62G05,
62J02
@article{1176350364,
author = {Muller, Hans-Georg and Stadtmuller, Ulrich},
title = {Estimation of Heteroscedasticity in Regression Analysis},
journal = {Ann. Statist.},
volume = {15},
number = {1},
year = {1987},
pages = { 610-625},
language = {en},
url = {http://dml.mathdoc.fr/item/1176350364}
}
Muller, Hans-Georg; Stadtmuller, Ulrich. Estimation of Heteroscedasticity in Regression Analysis. Ann. Statist., Tome 15 (1987) no. 1, pp. 610-625. http://gdmltest.u-ga.fr/item/1176350364/