Unbiased Fitting versus Functional Inversion in Square-Root Models

Abstract

The question of fitting appropriate theoretical values for the response variable from a regression model is a classical one. When a model is constructed by making a transformation on the dependent variable to achieve stable variance and normality, the fitting of theoretical values for the ‘original’ dependent variable is generally obtained by a simple functional inversion of the transformation. However, these theoretical values are not unbiased and nothing is known about their properties. A functional inversion of the transformation appears functionally proper, but it is statistically inappropriate. In this paper, we address the question of obtaining unbiased theoretical value of an observation on the original (untransformed) dependent variable that has been subjected to the square-root transformation for model building. A comparison of the ‘functional inversion fitting’ approach with the proposed ‘statistical fitting’ (unbiased fitting) approach is carried out under a squared-error risk function and the latter is shown to be the preferred one with reduced mean-square risk. Extension to fitting of theoretical values simultaneously for multiple observations under a matrix mean-square risk is also presented.