PARAMETER ESTIMATION AND TEST OF FIT FOR SKEW NORMAL DISTRIBUTION

Abstract

The celebrated Gaussian (Normal) distribution has been known for centuries. Its popularity has been driven by its analytical simplicity and the associated Central Limit Theorem. The multivariate extension is straightforward because the marginals and conditionals are both normal, a property rarely found in most of the other multivariate distributions. Yet there have been doubts, reservations, and criticisms about the unqualified use of normality. There are numerous situations when the assumption of normality is not validated by the data. In fact Geary (1947) remarked, Normality is a myth; there never was and never will be a normal distribution." As an alternative, many near normal distributions have been proposed. Some families of such near normal distributions, which include the normal distribution and to some extent share its desirable properties, have played a crucial role in data analysis.

It has been observed in vanous practical applications that data do not conform to the normal distribution, which is symmetric with no skewness. The skew normal distribution proposed by Azzalini (1 985) is appropriate for the analysis of data which is unimodal but exhibits some skewness. The skew normal distribution includes the normal distribution as a special case where the skewness parameter is zero. The skew normal family of probability distributions is a fairly recent family of distribution that attracted wide attention in the literature due to its strict inclusion of the normal distribution, its mathematical tractability and because it reproduces some properties of the normal distribution. Since the normal distribution is still the most commonly used distribution both in statistical theory and applications, a family of distributions that possesses the same properties have a great potential impact in theoretical and applied probability and statistics. However, this potential impact, there are still relatively few statisticians who use this family in their theoretical and applied works. The main reason of this potential impact that research on characterization and statistical inference for this family is still in its early stage.