Asymptotic Relative Efficiency and Exact Variance Stabilizing Transformation for the Generalized Gaussian Distribution

• Activity Number: 188
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Computing
• Author(s): Kai-Sheng Song*+
• Companies: University of North Texas
• Keywords: convexity ; generalized Gaussian distribution ; inverse hyperbolic cotangent ; maximum likelihood ; variance stabilizing transformation

Abstract

It is demonstrated that the sampling distributions of the maximum likelihood (ML)estimator and its Studentized statistic for the generalized Gaussian distribution do not pass the most powerful normality tests even for fairly large sample sizes. This disagreement with what the standard large sample ML theory predicts and the computational burden of having to deal with its associated polygamma functions motivate the consideration of a competing convexity-based estimator. It is shown that the competing estimator is almost as efficient as the ML estimator and its asymptotic relative efficiency to the ML estimator is equal to 1 in the limit. More important, its asymptotic distribution admits an exact variance stabilizing transformation, whereas the asymptotic variance function of the ML estimator does not have a closed form variance stabilizing transformation. The exact transformation is a composition of the inverse hyperbolic cotangent and square root functions. Besides stabilizing the variance, the transformation is remarkably effective for symmetrizing and normalizing the sampling distribution of the the estimator and hence improving the standard normal approximation.