An Empirical Characteristic Function Approach to Selecting a Transformation to Normality

Abstract

In this paper, we study the problem of transforming to normality. We propose to estimate the transformationparameter by minimizing a weighted squared distance between the empirical characteristic function oftransformed data and the characteristic function of the normal distribution. Our approach also allows for othersymmetric target characteristic functions. Asymptotics are established for a random sample selected from anunknown distribution. The proofs show that the weight function t^{-2} needs to be modified to have thinner tails. We also propose the method to compute the influence function for M-equation taking the form of U-statistics. The influence function calculations and a small Monte Carlo simulation show that our estimates are less sensitiveto a few outliers than the maximum likelihood estimates.