Skew-Normal Distribution with a Cauchy Skewing Function

The skew-normal distribution using the Cauchy CDF “skew-normal-Cauchy or SNC(λ)” is introduced to relieve some problems associated with the well-known skew-normal distribution. Two versions of maximum likelihood estimation methods are used. The first one is the maximum likelihood estimation with a penalty function used on its bias which was developed by Firth. 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. Also some features of SNC(λ) are compared to those in the well-known skew-normal distribution. Two versions of goodness of fit test for skew-normal using Cauchy SNC(λ) are conducted. The first test is Pearson’s Chi-squared test and the second is the Anderson-Darling (A-D) test. Finally probability tables for several values of (λ) were produced to conduct the Anderson-Darling (A-D) test.