Point Estimation of the Location Parameter of a Skew-Normal Distribution: Some Fixed Sample and Asymptotic Results

Abstract

A three-parameter skew-normal distribution (SND), which is a nice generalization of the regular normal model, can accommodate both positively skewed and negatively skewed data. This work deals with estimation of the location parameter of a SND, which had eluded the attention of the researchers due to the complicated nature of the relevant sampling distributions. First, we consider the easier case of estimating the location when both the scale and shape parameters are known. The developed ideas are applied to the case of unknown scale and shape parameters. All the estimators have been studied in terms of bias and mean squared error for fixed sample sizes (through extensive simulation) as well as asymptotically. As a by-product of our investigation, we have provided an excellent approximate distribution of the sample average, how much information does it carry for the location parameter (in the presence of known scale and shape) and an approximation of the Fisher information number which is convenient to work with. This Fisher information is useful to study the asymptotic behavior of the maximum likelihood estimator (MLE). But when the scale and shape parameters are unknown, the asymptotic behavior of the estimators cannot be studied by using the Fisher information matrix always since it develops singularity at three values of the shape parameter two of which had not been reported before in the literature. All earlier works reported the singularity of the Fisher information matrix when the shape parameter is 0. But the existence of two extra singularity points, as observed in this study, is something new to best of our knowledge. Hopefully this phenomenon will draw the attention of other researchers to study SND further. This paper ends with an application of SND to model a stock market data set from Thailand.