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Estimation of Heteroscedasticity in Regression Analysis

Hans-Georg Muller and Ulrich Stadtmuller
The Annals of Statistics
Vol. 15, No. 2 (Jun., 1987), pp. 610-625
Stable URL: http://www.jstor.org/stable/2241329
Page Count: 16
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Estimation of Heteroscedasticity in Regression Analysis
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Abstract

Consider the regression model Yi = g(ti) + εi, 1 ≤ i ≤ n, with nonrandom design variables (ti) and measurements (Yi) for the unknown regression function g(·). 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 σ2 i. If σ2 i = σ2(ti) with a smooth function σ2(·), initial estimators σ̃2 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.

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