Nonparametric Transformations for Both Sides of a Regression Model

Abstract

SUMMARY One way to model heteroscedasticity and skewness of the error distribution in regression is to transform both sides of the regression equation. If it is possible to transform the regression equation to result in normally distributed errors, then we can obtain more efficient parameter estimates and valid prediction intervals. One problem with this approach is that the choice of transformation is usually restricted to the power or shifted power family. Often there is no scientific basis for this model and the limited flexibility of this parametric family may miss important features of the distribution. A more comprehensive approach is to estimate the transformation by using nonparametric methods based on maximizing a penalized likelihood function. This maximization problem leads naturally to a spline estimate for the log-derivative of the transformation. An algorithm for computing the estimate is given along with results on the existence and uniqueness of the estimate. This nonparametric method is illustrated by using a non-normally distributed fisheries data set.