# A Method to Simulate the Skew Normal Distribution

Dariush Ghorbanzadeh, Luan Jaupi, Philippe Durand

► To cite this version:

Dariush Ghorbanzadeh, Luan Jaupi, Philippe Durand. A Method to Simulate the Skew Normal Distribution. Applied Mathematics, 2014, 5 (13), pp.2073-2076. 10.4236/am.2014.513201. hal-02467997

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# A Method to Simulate the Skew Normal Distribution

Dariush Ghorbanzadeh, Luan Jaupi, Philippe Durand

CANAM-Département IMATH, Paris, France

Email: dariush.ghorbanzadeh@cnam.fr, jaupi@cnam.fr, philippe.durand@cnam.fr

Received 2 May 2014; revised 10 June 2014; accepted 17 June 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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Open Access

## Abstract

A new method is developed to simulate the skew normal distribution. The result is interesting from a practical as well as a theoretical viewpoint. The new method is simple to program and is more efficient than the standard method of simulation by acceptance-rejection method.

## Keywords

Normal Distribution, Skew Normal Distribution

## 1. Introduction

In this paper, we denote by $SN(\theta)$ the skew normal distribution of parameter $\theta$ and density :

$$f(x) = 2\varphi(x)\Phi(\theta x) \quad (1)$$

where $\varphi$ and $\Phi$ denote the standard normal $N(0,1)$ probability density function and cumulative distribution function, respectively.

The skew normal distribution, due to its mathematical tractability and inclusion of the standard normal distribution, has attracted a lot of attention in the literature. Azzalini [1], Azzalini [2], Chiogna [3], Genton & Liu [4] and Henze [5] discussed basic mathematical and probabilistic properties of the skew normal family. The multivariate skew normal distribution is studied by Azzalini & Capitanio [6] and Azzalini & Dalla Valle [7]. For additional references and a review on related literature, see Azzalini & Capitanio [8] and Pewsey [9] for a collection of papers on the subject. Henze [5], in his paper showed that if $U_1$ and $U_2$ are identically and independently distributed $N(0,1)$ random variables, then

$$\frac{\theta|U_1|+U_2}{\sqrt{1+\theta^2}}$$ has the skew normal distribution.

For the simulation of the skew normal distribution, we propose a combinations of maximum and minimum of
the independent and identically distributed $N(0,1)$ random variables.

## 2. Method

Let $U_1$ and $U_2$ two independent and identically distributed $N(0,1)$ random variables and $U = \max(U_1, U_2)$ and $V = \min(U_1, U_2)$. For simulation of the random variable $X \sim SN(\theta)$, we take the combination of $U$ and $V$. First note that:

* if $\theta = 0$, the density (1) becomes: $f(x) = \varphi(x)$, simply simulate $X \sim N(0,1)$.

* if $\theta = -1$, the density (1) becomes: $f(x) = 2\varphi(x)(1-\Phi(x))$, we take $X = V$.

* if $\theta = 1$, the density (1) becomes: $f(x) = 2\varphi(x)\Phi(x)$, we take $X = U$.

For $\theta \notin \{-1,0,1\}$, note:

$$ \lambda_1 = \frac{1+\theta}{\sqrt{2(1+\theta^2)}}, \quad \lambda_2 = \frac{1-\theta}{\sqrt{2(1+\theta^2)}} \qquad (2) $$

We note that: $\lambda_1^2 + \lambda_2^2 = 1$. For simulation of the random variable $X \sim SN(\theta)$, we take the combination of $U$ and $V$ in the form:

$$ X = \lambda_1 U + \lambda_2 V \qquad (3) $$

**Proposition** The random variable $X$ defined in the Equation (3) has the skew normal distribution $SN(\theta)$.

*Proof.* The pair $(U,V)$ has density: $f_{U,V}(u,v) = 2\varphi(u)\varphi(v)\mathbb{I}_{\{v \le u\}}(u,v)$, where $\mathbb{I}$ is the indicator function.

Consider the transformation: $x = \lambda_1 u + \lambda_2 v$, $y = \lambda_1 u$. The inverse transform is defined by: $u = \frac{y}{\lambda_1}$, $v = \frac{x-y}{\lambda_2}$

and the corresponding Jacobian is: $J = \frac{1}{\lambda_1 \lambda_2}$. $X$ density is defined by:

$$ f(x) = \frac{2}{|\lambda_1 \lambda_2|} \int_{\Delta} \varphi\left(\frac{y}{\lambda_1}\right) \varphi\left(\frac{x-y}{\lambda_2}\right) dy \qquad (4) $$

where $\Delta = \left\{ \frac{x-y}{\lambda_2} \le \frac{y}{\lambda_1} \right\}$. Taking into account $\lambda_1^2 + \lambda_2^2 = 1$, we can write:

$$ \varphi\left(\frac{y}{\lambda_1}\right) \varphi\left(\frac{x-y}{\lambda_2}\right) = \varphi(x) \varphi\left(\frac{y-\lambda_1^2 x}{|\lambda_1 \lambda_2|}\right) \qquad (5) $$

Equation (4) becomes,

$$ f(x) = \frac{2\varphi(x)}{|\lambda_1 \lambda_2|} \int_{\Delta} \varphi\left(\frac{y-\lambda_1^2 x}{|\lambda_1 \lambda_2|}\right) dy \qquad (6) $$

For the domain $\Delta$, we have the following three cases:

Case 1: $\theta \in (-1,0) \cup (0,1)$, we have: $|\lambda_1 \lambda_2| = \frac{1-\theta^2}{2(1+\theta^2)}$ and $\Delta = \left\{ y \ge \frac{\lambda_1 x}{\lambda_1 + \lambda_2} \right\}$

Case 2: $\theta < -1$, we have: $|\lambda_1 \lambda_2| = \frac{\theta^2 - 1}{2(1+\theta^2)}$ and $\Delta = \left\{ y \ge \frac{\lambda_1 x}{\lambda_1 + \lambda_2} \right\}$

Case 3: $\theta > 1$, we have: $|\lambda_1 \lambda_2| = \frac{\theta^2 - 1}{2(1+\theta^2)}$ and $\Delta = \left\{ y \le \frac{\lambda_1 x}{\lambda_1 + \lambda_2} \right\}$

Using Equation (6) and the three cases above, we get the result.

## 3. Simulation Results

We simulated a sample of size 500,000 for the values $\theta = -7$ and $\theta = 13$. The following Figure 1, Figure 2
and Table 1, Table 2 show the results obtained by our method and the method Henze [5].

The results of Table 1 and Table 2 show that our method provides results close to the theoretical values and secondly, we obtain results similar to those obtained by Henze [5] results.

## 4. Conclusion

In this article we propose a very simple method to simulate skew normal family distribution. The obtained

<table><caption>Table 1. Simulation results for a sample size of 500,000 and $\theta = -7$.</caption><thead><tr><th></th><th>Theoretical value</th><th>Simulated value<sup>a</sup></th><th>Simulated value<sup>b</sup></th></tr></thead><tbody><tr><td>Mean</td><td>-0.789865</td><td>-0.789651</td><td>-0.790496</td></tr><tr><td>Variance</td><td>0.376113</td><td>0.375669</td><td>0.376671</td></tr><tr><td>Skewness</td><td>-0.916950</td><td>-0.915649</td><td>-0.924019</td></tr><tr><td>Kurtosis</td><td>0.779197</td><td>0.768095</td><td>0.796781</td></tr></tbody></table>

<table><caption>Table 2. Simulation results for a sample size of 500,000 and $\theta = 13$.</caption><thead><tr><th></th><th>Theoretical value</th><th>Simulated value<sup>a</sup></th><th>Simulated value<sup>b</sup></th></tr></thead><tbody><tr><td>Mean</td><td>0.795534</td><td>0.795174</td><td>0.795930</td></tr><tr><td>Variance</td><td>0.367125</td><td>0.366461</td><td>0.367464</td></tr><tr><td>Skewness</td><td>0.971447</td><td>0.969756</td><td>0.970813</td></tr><tr><td>Kurtosis</td><td>0.841547</td><td>0.836937</td><td>0.831219</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Figure 1. Histogram of simulations for a sample size of 500,000 and $\theta = -7$. (a) histogram of simulations using our method; (b) histogram of simulations using the method of Henze [5].</figcaption></figure>

<footnote><sup>a</sup>by our method; <sup>b</sup>by the method of Henze [5].</footnote>

<footnote><sup>a</sup>by our method; <sup>b</sup>by the method of Henze [5].</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 2. Histogram of simulations for a sample size of 500,000 and $\theta = 13$. (a) histogram of simulations using our method; (b) histogram of simulations using the method of Henze [5].</figcaption></figure>

results are very close to theoretical values and the method is more efficient than the standard one. The method is simple to program and exploit for practical applications.

References

[1] Azzalini, A. (1985) A Class of Distributions Which Includes the Normal Ones. *Scandinavian Journal of Statistics*, **12**, 171-178.

[2] Azzalini, A. (1986) Further Results on the Class of Distributions Which Includes the Normal Ones. *Statistica*, **46**, 199-208.

[3] Chiogna, M. (1998) Some Results on the Scalar Skew-Normal Distribution. *Journal of the Italian Statistical Society*, **1**, 1-13. http://dx.doi.org/10.1007/BF03178918

[4] Genton, M.G., He, L. and Liu, X. (2001) Moments of Skew Normal Random Vectors and Their Quadratic Forms. *Statistics & Probability Letters*, **51**, 319-325. http://dx.doi.org/10.1016/S0167-7152(00)00164-4

[5] Henze, N. (1986) A Probabilistic Representation of the Skew-Normal Distribution. *Scandinavian Journal of Statistics*, **13**, 271-275.

[6] Azzalini, A. and Capitanio, A. (1999) Statistical Applications of the Multivariate Skew-Normal Distributions. *Journal of the Royal Statistical Society, Series B*, **61**, 579-602. http://dx.doi.org/10.1111/1467-9868.00194

[7] Azzalini, A. and Dalla Valle, A. (1996) The Multivariate Skew-Normal Distribution. *Biometrika*, **83**, 715-726. http://dx.doi.org/10.1093/biomet/83.4.715

[8] Azzalini, A. and Capitanio, A. (2003) Distributions Generated by Perturbation of Symmetry with Emphasis on a Multivariate Skew t Distribution. *Journal of the Royal Statistical Society, Series B*, **65**, 367-389. http://dx.doi.org/10.1111/1467-9868.00391

[9] Pewsey, A. (2000) Problems of Inference for Azzalini's Skew-Normal Distribution. *Journal of Applied Statistics*, **27**, 859-870. http://dx.doi.org/10.1080/02664760050120542
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