Parametric generalized Gaussian density estimation

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Abstract
The primary objective of this paper is to compare the large‐sample as well as the small‐sample properties of different methods for estimating the parameters of a three‐parameter generalized Gaussian distribution. Three estimators, namely, the moment method (MM), the maximum‐likelihood (ML), and the moment/Newton‐step (MNS) estimators, are considered. The applicability of general asymptotic optimality results of the efficient ML and MNS estimation techniques is studied in the generalized Gaussian context. The asymptotic normal distributions of the estimators are obtained. The asymptotic relative superiority of the ML estimator or its variant, the MNS estimator, over the moment method is studied in terms of asymptotic relative efficiency. Based on this study, it is concluded that deviations from normality in the underlying distribution of the data necessitate the use of the efficient ML or MNS methods. In the small‐sample case, a detailed comparative study of the estimators is made possible by extensive Monte Carlo simulations. From this study, it is concluded that the maximum‐likelihood method is found to be significantly superior for heavy‐tailed distributions. In a region of the parameter space corresponding to the vicinity of the Gaussian distribution, the moment method compares well with the other methods. Further, the MNS estimator is shown to perform best for light‐tailed distributions. The simulation results are shown to lend support to analytically derived asymptotic results for each of the methods.

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