Effects of Variance Function Estimation on Prediction and Calibration. An Example

摘要

This reprint considers a heteroscedastic regression model. In the linear regression model with a reasonably sized data set, since unweighted least squares is consistent its fitted values rarely differ much from the fitted values from a generalized lease squares fit. Consequently, the usual practice is to treat the estimation of the variance function g(chi, beta, theta) fairly cavalierly, if at all. The narrow focus on estimating the mean is misplaced, as Schwartz later notes. Box & Meyer (1985) state that one distinctive feature of Japanese quality control improvement techniques is the use of statistical experimental design to study the effect of a number of factor on variance as well as the mean. Other times the variance function essentially determines the quantity of interest. It is perhaps trite to state that how well one estimates the variance function has a large effect on how well one can do prediction and calibration. It is, however, a point that is rarely taken into account in practice, as any review of the techniques in the assay literature will show. We construct an asymptotic theory outlined in section 3, where we show that the difference in the length of a prediction interval between theta known and unknown is asymptotically distributed with variance a monotone function of how well one estimates theta. (kr)