We consider a class of generalized skew-elliptical distributions (GSE) which is useful for selection modeling and robustness analysis. We propose an approximation of this generalized skew-elliptical distributions (AGSE) by constructing an enumerable dense subset of skewing functions on a compact set. This approximation enables us to consider a flexible family of distributions which can capture skewness, heavy tails, and multimodality systematically. It is straightforward to simulate from this family using its stochastic representation. We present some illustrative examples on fiber-glass data, simulated data from a mixture of two normal distributions, and Swiss bills data. This family of distributions is used to relax the standard normal assumption of the random effects in linear random effect models. Computational issues is addressed using Monte Carlo type algorithms. Some advantages of using such family is discussed. If time permits, we will present a class of semiparametric estimators for the location and scale parameters of the central part of the GSE model. These estimators are shown to be consistent and asymptotically normal. We derive the locally efficient estimator that achieves the optimal efficiency if the model for the skewing function is correctly specified. The estimators we propose are consistent and asymptotically normal even if the model for the skewing function is misspecified and the efficiency on estimating the nuisance parameters does not influence the efficiency on estimating the parameters of interest. The semiparametric inference also naturally motivates the AGSE distribution described above.
Return to Biostatistics Working Group