Evaluating the cdf of the Skew Normal distribution
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Evaluating the cdf of the Skew Normal distribution Christine Amsler1 · Alecos Papadopoulos2
· Peter Schmidt1
Received: 21 November 2019 / Accepted: 4 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Skew Normal distribution · Bivariate Normal distribution · Stochastic frontier · Simulation · Computational software JEL Classification C46 · C87 · C88 · C13
1 Introduction This article deals with the evaluation of the cumulative distribution function of the Skew Normal distribution (hereafter SN-cdf). This is the distribution of the
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00181020-01868-6) contains supplementary material, which is available to authorized users.
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Alecos Papadopoulos [email protected]
1
Michigan State University, East Lansing, USA
2
Athens University of Economics and Business, Athens, Greece
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C. Amsler et al.
Normal/Half-Normal composed error in stochastic frontier (SF) models, and that is our main interest. However, the Skew Normal distribution also arises in a number of other contexts. For example, the maximum and the minimum of two correlated bivariate standard Normal random variables follow Skew Normal distributions (Loperfido 2002). As another example, if X , Y are correlated bivariate Normal variables, then the distribution of X |Y > 0 is Skew Normal (Azzalini and Capitanio 2014, p. 28). As a third example, in the Normal/Half-Normal specification of the two-tier stochastic frontier model, the composed error is made up of three components and its density includes the difference of two Skew Normal cdfs (Papadopoulos 2015). As a final more general example, the Skew Normal distribution is a useful way to model a moderately skewed regression error (which is often the case), and if one wants to account for sample selection bias, the SN-cdf will appear in the likelihood. Our work is an extension of Amsler et al. (2019)—hereafter AST—who a
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