Estimating Weights in Heteroscedastic Regression Models by Applying Least Squares to Squared or Absolute Residuals

摘要

This document considers a nonlinear regression model for which the variances depend on a parametric function of known variables. The authors focus on estimating the variance function, after what it is typical to estimate the mean function by weighted least squares. Most often, squared residuals from an unweighted least squares fit are compared to their expectations and used to estimate the variance function. If properly weighted such methods are asymptotically equivalent to normal-theory maximum likelihood. Instead, one could use the deviations of the absolute residuals from their expectations. Constructed is such an estimator of the variance function based on absolute residuals whose asymptotic efficiency relative to maximum likelihood is precisely the same for symmetric errors as the asymptotic efficiency in the one-sample problem of the mean absolute deviation relative to the sample variance. The estimators are computable using nonlinear least squares software. The results hold with minimal distributional assumptions. (Author)