Skew-normal distribution with a Cauchy skewing function

The skew-normal distribution using the Cauchy CDF is introduced to partially relieve some problems associated with the well-known skew-normal distribution proposed by Azzalini (1985). In addition to the shape parameter estimation problem, the skew-normal distribution possesses a practical range limitation of the skewing parameters and a quick departure to a half normal. To the best of our knowledge, these problems have never been addressed in the literature. Due to the heavy tails of Cauchy density the skew-normal distribution, with the Cauchy CDF as a skewing function, exhibits better behavior in terms of the practical range and the departure to a half normal compared to Azzalini's skew normal model. This class of distributions will hereafter be referred to as skew-normal-Cauchy or SNC(lambda).; To overcome the estimation problem two versions of maximum likelihood estimation methods will be used. The first one is the maximum likelihood estimation with a penalty function used in the bias reduction of the maximum likelihood estimation approach, developed by Firth (1993). The second one is the maximum likelihood estimation with a new penalty function that is suggested in this work. The latter method is based on an approximate penalty function which mimics the exact function suggested by Firth (1993). Furthermore, Bayesian estimation with the Uniform and Jeffreys' priors, will be used as an alternative way to estimate the shape parameters. Also, some features of SNC(lambda) will be compared to those derived by Azzalini (1985).; Finally, two versions of goodness of fit test for skew-normal using Cauchy SNC(lambda), will be conducted. The first test is Pearson's Chi-squared (chi2) test, and the second is the Anderson-Darling (A-D) test. The A-D test is a modification of the Kolmogorov-Smirnov's (K-S) test and gives more weight to the tails than does the K-S test. Carrying out these tests will require probability tables for SNC distributions with different values of the skew factor. The purpose is to examine whether the skew model under investigation is appropriate for a set of data.