Transformations to Symmetry and Homoscedasticity

Abstract

This article considers transformations in regression to eliminate skewness and heteroscedasticity of the response. We work with the transform-both-sides model where the relationship between the median response and the independent variables has been identified, at least tentatively. To preserve this relationship, the response and the regression model are transformed in the same way. Extending the work of others for the location parameter case, we propose an estimator that eliminates skewness. We also develop an estimator to eliminate heteroscedasticity and an estimator that attempts to induce both symmetry and homoscedasticity. Both and appear new. By comparing and we develop a test of the null hypothesis that there exists a transformation to both symmetry and homoscedasticity. We study the question When does the estimator of λ behave (in terms of asymptotic variance) as if the regression parameter β were known (and vice versa)? The results are of use for telling when the optimal estimator of λ does not depend upon the regression model. In addition, we present two examples and discuss computation of the estimators. The transformation to symmetry is defined by setting the usual third-moment skewness coefficient of the residuals equal to 0, and the variance-stabilizing transformation is defined by setting the correlation between the squared residuals and the logarithms of fitted values equal to 0. Under normality, both of these estimators achieve minimum asymptotic variance within certain natural classes of estimators. The estimator of a transformation to both symmetry and constant variance uses a weighted average of the equations separately defining transformations to symmetry and homoscedasticity, and the optimal weights can be estimated. Robust modifications of the estimators are considered. One example studies efficiency in a class of robust estimators of transformations to symmetry. A second example, using real data, compares standard errors from asymptotic theory with bootstrap standard errors. The transformation parameter is nearly orthogonal to the regression parameters. For this reason we concentrate on estimation of the transformation parameter. A good estimate of the transformation parameter is especially important when constructing prediction intervals for future responses. For this reason we concentrate on estimation of the transformation parameter. A good estimate of the transformation parameter is especially important when constructing prediction intervals for future responses.

Key Words:

• M estimation
• Power transformations
• Robustness
• Skewness
• Test for heteroscedasticity
• Transform-both-sides model