Variance Function Estimation

We develops a general theory for variance function estimation in regression. Most methods in common use are included in our development. The general qualitative conclusions are these. First, most variance function estimation procedures can be looked upon as regressions with responses' being transformations of absolute residuals from a preliminary fit or sample standard deviations from replicates at a design point. Our conclusion is that the former is typically more efficient, but not uniformly so. Secondly, for variance function estimates based on transformations of absolute residuals, we show that efficiency is a monotone function of the efficiency of the fit from which the residuals are formed, at least for symmetric errors. Our conclusion Is that one should iterate so that the residuals are based on generalized least squares. Finally, robustness issues are of even more importance here than in estimation of a regression function for the mean. The loss of efficiency of the standard method away from the normal distribution is much more rapid than in the regression problem