Evaluating the cdf of the Skew Normal distribution

Abstract

In this paper, we consider various methods for evaluating the cdf of the Skew Normal distribution. This distribution arises in the stochastic frontier model because it is the distribution of the composed error, which is the sum (or difference) of a Normal and a Half-Normal random variable. The cdf must be evaluated in models in which the composed error is linked to other errors using a Copula, in some methods of goodness of fit testing, or in the likelihood of models with sample selection bias. We investigate the accuracy of the evaluation of the cdf using expressions based on the bivariate Normal distribution, and also using simulation methods and some approximations. We find that the expressions based on the bivariate Normal distribution are quite accurate in the central portion of the distribution, and we propose several new approximations that are accurate in the extreme tails. By a simulated example, we show that the use of approximations instead of the theoretical exact expressions may be critical in obtaining meaningful and valid estimation results.