Adapting for heteroscedasticity in linear models. (English) Zbl 0571.62058

The heteroscedastic linear model, \(Y_{ij}=x^ T_ i\beta +\sigma_ i\epsilon_{ij}\), \(i=1,2,...,n\), \(j=1,2,...,m_ i\), is considered. The author and D. Ruppert proved [ibid. 10, 429-441 (1982; Zbl 0497.62034)] that in the parametric cases \(\sigma^ 2_ i=H(x_ i,\theta)\) or \(\sigma^ 2_ i=H(x^ T_ i\beta,\theta)\) with H known, the weighted least squares estimates based on the true weights and the estimated weights have the same asymptotic distributions. This says that there is no cost asymptotically for not knowing \(\theta\). W. A. Fuller and J. N. K. Rao [ibid. 6, 1149-1158 (1978; Zbl 0388.62064)] considered the fully nonparametric cases, i.e., \(\sigma^ 2_ i=H(x_ i)\) or \(\sigma^ 2_ i=H(x^ T_ i\beta)\) with nothing known about H, except \(H>0\). In such cases they found that there is an asymptotic cost for not knowing H.
In this paper, simple linear regression is discussed with \(\sigma^ 2_ i=H(c_ i)\), where the \(c_ i\) are the design points. Kernel estimates of H are used to provide weights for a weighted least squares estimate, and it is shown, under suitable regularity conditions including the smoothness of H, that this weighted least squares estimate has the same asymptotic distribution as the one based on the true weights.
A similar result is obtained for the heteroscedastic linear model above if \(\sigma^ 2_ i=H(x^ T_ i\beta)\). A Monte Carlo study was conducted for the simple linear regression model and the proposed estimate seems to work well, especially for larger sample sizes.


62J05 Linear regression; mixed models
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
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