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Empirical Economics

Erschienen in:

15.05.2020

Evaluating the cdf of the Skew Normal distribution

verfasst von: Christine Amsler, Alecos Papadopoulos, Peter Schmidt

Erschienen in: Empirical Economics | Ausgabe 6/2021

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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.

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Ashour and Abdel-hameed (2010) presented also a closed-form approximation to the Skew Normal density and cdf, but they were functions with branches, while the partition of the support depended on the value of \(\lambda \). They considered only the [− 4,4] interval.

This is the cdf of the maximum of two i.i.d. standard Normal variables and can also be obtained as the limiting case of the results in Loperfido (2002) mentioned earlier.

Azzalini (1985), p. 174. This paper by Adelchi Azzalini has become by all accounts a classic of the statistical literature, and so, the Mark Twain remark on classic works applies.

Gupta and Chen (2001) used it also as a validation tool for their SN-cdf tables. They applied Simpson’s rule to compute the integral and examined values in the \([-\,4,4]\) interval.

The fact that one of the two arguments is fixed at zero helps the affirmative argument in our case.

MATLAB online documentation, https://www.mathworks.com/help/stats/mvncdf.html. Website accessed July 6th, 2019.

Some authors define the “signal-to-noise” ratio to be the ratio of the variance of the inefficiency component to the variance of the composed error. This is a useful metric, but it is not a signal-to-noise ratio, as the concept is used in most scientific fields.

Casio-Keisan is an online general computing tool, https://keisan.casio.com. It uses a proprietary method of “high-precision” arithmetic, which allows for computational accuracy far beyond that of the floating-point, double-precision arithmetic widely implemented in most available hardware and software. The Mathematica software also uses a high-precision proprietary method.

Using the negatively skewed distribution, we can obtain nonzero values for \(Q =-\,38\), that are of order \(10^{-316}\), where we hit the limits of double-precision arithmetic.

“Gnu Regression, Econometrics and Time-series Library, http://gretl.sourceforge.net/.

For the 132 values per computing algorithm that we computed in our tables, the two algorithms differed only in 5 left tail cases for \(\lambda \ge 2\), for which gretl computed exactly zero, while MATLAB gave a nonzero value. On the other hand, gretl is more tidy in that, where MATLAB gives negative values, gretl gives zeros.

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Titel: Evaluating the cdf of the Skew Normal distribution
verfasst von: Christine Amsler
Alecos Papadopoulos
Peter Schmidt
Publikationsdatum: 15.05.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 6/2021
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-020-01868-6

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