Evaluating the CDF of the Skew Normal distribution

Posted: April 4, 2020 in Research papers, Uncategorized
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A paper I wrote together with Christine Amsler and Peter Schmidt (yes, I cannot resist to say, the Peter Schmidt of the KPSS time series stationarity test, and one of the founders of Stochastic Frontier Analysis), has just been approved for publication in a special issue of Empirical Economics that will be dedicated to efficiency and productivity analysis. The paper is

Amsler C, A Papadopoulos and P Schmidt (2020). “Evaluating the CDF of the Skew Normal distribution.” Forthcoming in Empirical Economics. Download the full paper incl. the supplementary file.

ABSTRACT. In this paper we consider various methods of 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.


The paper computes values of the Skew Normal distribution using 17 different mathematical formulas (approximations or exact), and/or algorithms and different software. with particular focus on the accuracy of computation of the Skew Normal CDF by the use of the Bivariate standard Normal CDF, since the latter is readily available, but also on what happens deep into the tails. There, the CDF values as so close to zero or unity that it would appear it wouldn’t matter for empirical studies, if one simply imposed a non-zero floor and a non-unity ceiling, and be ok. It is not ok. In Section 7 of the paper we show by a simulated example, that using the Bivariate standard Normal CDF only (with or without floor/ceiling) may lead to failed estimation, while inserting an approximate expression in its place for the left tail solves the problem. This is a result we did not anticipate: it says that approximate mathematical expressions may perform better than exact formulas due to computational limitations related to the latter.


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