Robust Estimation, Nonnormalities, and Generalized Exponential Distributions

Abstract

The detection and treatment of outliers in data has represented an important area of research. The general approach that has been adopted involves observing that outliers contribute to the “fatness” in the tails of the error distribution and using an appropriate model of the error term to take this into account. This article introduces a general dass of distributions to capture nonnormalities in the data based on the generalized exponential family. This family of distributions provides great flexibility in modeling not only Symmetrie fat-tailed distributions, but also skewed and possibly even multimodal distributions. The approach taken is to use subordinate distributions within the generalized exponential family to model the empirical distribution. This family represents a generalization of the unimodal exponential family, which contains as subordinates the normal, gamma, beta, Student t, and so forth. The parameters of the generalized exponential distribution can be estimated using maximum likelihood, weighted least squares, or minimum chi-squared methods, depending on the nature of the data. Special attention is given to analyzing the distributional properties of two subordinates of this family; namely, the generalized Student t and generalized lognormal distributions. The flexibility of these distributions is illustrated by applying them to two empirical problems and comparing the results with previous methods.