Advances in Computational Mathematics and its Applications (ACMA) Vol. 1, No. 4, 2012, ISSN 2167-6356 Copyright © World Science Publisher, United States www.worldsciencepublisher.org
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Simulating skew normal distribution and improving the results Behrouz Fathi-Vajargah1, Solma Osouli2 1
Department of Statistics, University of Guilan, Rasht, Iran,
2
Department of Statistics, Islamic Azad University, North Branch, Tehran, Iran 1
[email protected] [email protected]
Abstract: In this paper we present a suitable way to simulate the skew normal distribution. We examine that the simulated data fit on the exact data and its performance will increase when we increase the number of simulated data. Keywords: Skew normal distribution; Moment generating function; Simulation.
transformation may not be found some cases, and then the skew normal distribution shows better performances. Skew normal distribution plays a main role on analyzing of non symmetric distributions, and it shows better flexibility on applicable problems. Azallini introduced standard Skew Normal (SN) random variable λ with skewness parameter λ ∈ and its density is performed by the following equation:
Introduction In most of cases on different phonemes we reach to skew normal distribution such that the exact normal distribution does not fit on our desired data. In this case, we may use some useful transformation on basic non normal data to make an opportunity to reach to the normal distribution. It is well-known that this ;
2
where ϕ . and Ф . are normal density function and normal distribution function, respectively. We show the above description by X ~ . Here, the parameter λ is called the skewness parameter of distribution, and shows the direction and the measure of skewness of distribution such that the λ 0 shows the skewness toward to right hand side and also, λ 0 shows the skewness of distribution toward to left hand side. Dalla Valle (2007), [1] considered a test for investigating the normality of population distribution. Then, Skewnormal ARMA models with nonlinear heteroscedastic predictors presented by Pourahmadi, M. (2007) [2]. Jamalizadeh, Behboodian and Balakrishnan (2008) introduced a two-parameter generalized skew-normal distribution [3]. Arellano-Valle and Azzalini (2008) described in their paper[4] that for statistical inference connected to the scalar skew-normal distribution, it is known that the so-called centred parametrization provides a more convenient parametrization than the one commonly employed for writing the density function. They extended the definition of the centred parametrization to the multivariate case, and study the corresponding information matrix. Azzalini and Genton
Ф
, ∈ (2008)[5]also claimed that the robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behavior of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew-t distribution is explored in more detail, and reasons are given to adopt this option as a sensible general-purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide-ranging examples with real data are provided in support of the claim. Azzalini (2008)[6] and also its joined developed work with Bacchieri [7] describes that in the context of clinical trials where one of several doses or treatments is selected in a phase II study to be examined further in a phase III study, they develop a formulation for the combination of the overall information obtained from such studies, which mimics the logic followed in actual drug development. The associated distribution theory is exact under the normality assumption. Extensions to more complex situations are sketched briefly. Azzalini, Genton and Scarpa (2010)[8], they develop estimating equations for the parameters of
Behrouz Fathi-Vajargah & Solma Osouli, ACMA, Vol. 1, No. 4, pp. 183-187, 2012
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pioneering work of Fernando de~Helguero who in 1908 put forward a formulation for the genesis of non-normal distributions via a selection mechanism which perturbs a normal distribution, hence employing an argument closely connected with the one now widely used in this context. Arguably, de~Helguero can then be considered the precursor of the current idea of skew-symmetric distributions.
the base density of a skew symmetric distribution. The method is based on an invariance property with respect to asymmetry. Various properties of this approach and the selection of a root are discussed. They also present several extensions of the methodology, namely to the regression setting, the multivariate case, and the skew-t distribution. The approach is illustrated on several simulations and a numerical example. Arellano-Valle and Azzalini (2011) [9], the skew-t family, in its univariate and multivariate versions, is a parametric family of probability distributions which is currently under intense investigation because of several appealing properties. The present paper addresses the question of the choice of its parameterization, and more generally of the selection of quantities of interest associated to this distribution. Azzalini (2012) [10], an active stream of literature has followed up the idea of skew-elliptical densities initiated by Azzalini and Capitanio (1999), their original formulation was based on a general lemma which is however of broader applicability than usually perceived. This note examines new directions of its use, and illustrates them with the construction of some probability distributions falling outside the family of the so-called skew-symmetric densities. Azzalini and Regoli (2012) [11], the family of skew-symmetric distributions is a wide set of probability density functions obtained by suitably combining a few components which can be quite freely selected provided some simple requirements are satisfied. Although intense recent work has produced several results for certain sub-families of this construction, much less is known in general terms. The present paper explores some questions within this framework, and provides conditions for the abovementioned components to, ensure that the final distribution enjoys specific properties. The current literature on so-called skew-symmetric distributions is closely linked to the idea of a selection mechanism operated by some latent variable. They illustrate the
In rare above articles they used simulation, they usually have collected data just to test whether these data follow the skew normal distribution or not. But in this paper, we first prove that how we can simulate the skew normal distribution then we simulate these data as much as we want. In fact, we take the empirical data (empirical data) which follow the skew normal distribution and then we make thousands copy of the empirical data via successful simulation method, introduce in this paper. Finally, we show that with increasing the number of simulated data we have more qualification in simulation. Most of researchers to realize these ideas leave this part and let them for future works and presentations. Some of simple properties of distribution have given in the following [1]: 1- For λ 0, presents the standard normal distribution, then the random variable – has . 2- When λ → ∞ , then ; tends to the half normal distribution. 3- If has distribution then ; is a convex function of . 4- If has distribution then ! has χ"! distribution. The above properties are immediately valid from the definition of skew normal distribution. The Moments of skew normal distribution can be evaluate by the following lemma.
Lemma: If # be random variable from 0,1 , then for every real values % and & we have: '(Ф %# ) & * Ф +&⁄, 1 ) %! .. Theorem1: The probability generating function of skew normal distribution is the following form: / 0 21 2 0 ! ⁄2 Ф 30 ⁄,1 ) λ! . where δ Proof: / 0
' 1 45
∞
6 1 47 2 8∞
Using changing variable we have:
6
∞
2
Ф
8∞ √2;
7
√1 ) ! Based on this probability generating function we obtain expectation, variance, skewness and elongation for skew normal distribution as follow: ' ?3 , @AB 1 ?3 ! K⁄! (' *! 1 C" D2 4 ; FG H I J @AB ! (' *! C! 2 ; 3 I J @AB where ? and δ are as below:
4
