Heteroscedastic regression models with non-normally distributed errors
Although regression estimates are quite robust to slight departure from normality, symmetric prediction intervals assuming normality can be highly unsatisfactory and problematic if the residuals have a skewed distribution. For data with distributions outside the class covered by the Generalized Linear Model, a common way to handle nonnormality is to transform the response variable. Unfortunately, transforming the response variable often destroys the theoretical or empirical functional relationship connecting the mean of the response variable to the explanatory variables established on the original scale. Further complication arises if a single transformation cannot both stabilize variance and attain normality. Furthermore, practitioners also find the interpretation of highly transformed data not obvious and often prefer an analysis on the original scale. The present paper presents an alternative approach for handling simultaneously heteroscedasticity and non-normality without resorting to data transformation. Unlike classical approaches, the proposed modeling allows practitioners to formulate the mean and variance relationships directly on the original scale, making data interpretation considerably easier. The modeled variance relationship and form of non-normality in the proposed approach can be easily examined through a certain function of the standardized residuals. The proposed method is seen to remain consistent for estimating the regression parameters even if the variance function is misspecified. The method along with some model checking techniques is illustrated with a real example.
heteroscedasticity non-normality data transformation linear regression