Efficient estimation of the regression parameter in a heteroscedastic regression model where heteroscedasticity is modeled as a function of the mean response

摘要

A heteroscedastic linear regression model is studied in which the heteroscedasticity is modeled as a function of the mean response. Efficient estimators of the regression parameter are characterized and constructed. The efficiency concept used is the notion of a least dispersed regular estimator. Such estimates are characterized via the convolution theorem as possessing a certain influence function, the so-called efficient influence function. An explicit form of the efficient influence function is given. A construction of an efficient estimate is then provided, based on a modification of the work of Schick (1994) to allow for random weights and local sequences. For this estimation, nonparametric estimators of the scale function are derived, using locally weighted least squares techniques.