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Computational Methods Applied to a Skewed Generalised Normal Family

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Computational Methods Applied to a Skewed Generalised Normal Family Computational methods applied to a skewed generalised normal family A. Bekker,J.T.Ferreira1,M.Arashi,&B.W.Rowland Department of Statistics, University of Pretoria, Pretoria, South Africa Department of Statistics, School of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran ABSTRACT Some characteristics of the normal distribution may not be ideal to model in many applications. We develop a skew generalised normal ( ) distribution by applying a skewing method to a generalised normal distribution, and study some meaningfulSGN statistical characteristics. Computational methods to approximate, and a well-constructed efficient computational approach to estimate these characteristics, are presented. A stochastic representation of this distribution is derived and numerically implemented. The skewing method is extended to the elliptical class resulting in a more general framework for skewing symmetric distributions. The proposed distribution is applied in a fitting context and to approximate particular binomial distributions. Keywords and phrases: Approximating binomial distribution; Skew-normal; Skew generalised normal; Stochastic representation. 2000 Mathematics Subject Classification: 62F03; 62F10; 62E99. 1Introduction Azzalini (Azzalini 1985) introduced the skew-normal ( ) distribution, which includes the standard normal dis- tribution as a special case, and studied the basic mathemSNatical properties thereof. The skewing methodology that is used to skew existing symmetric probability density functions (PDFs) is stated: Proposition 1 (Azzalini 1985) Denote by 0 ( ) aprobabilitydensityfunction(PDF)onR by 0 ( ) acontinuous · · cumulative distribution function (CDF) on R,andby ( ) a real-valued function on R such that 0 ( )=0 (), · − ( )= () and 0 ( )=1 0 () for all R R Then − − − − ∈ ∈ ()=20 () 0 () (1) { } is a PDF on R Note that 0 is termed the symmetric base PDF, 20 () is termed the skewing mechanism and is termed the skewed version of the symmetric base PDF. { } Azzalini’s innovation of coupling the symmetric component with a skewing mechanism generates distributions with flexible tail behaviour. This methodology provides a platform for considering other candidates rather than the normal distribution as the symmetric component. (Azzalini 2013) remarked that the distribution has short tails making it unsuitable for use when there is a need for a model to have heavier tailsSN than the normal distribution. One method to solve this problem, as suggested in (Azzalini 2013), is to use a symmetric base PDF 0, (see Proposition 1) with heavier tails than the normal distribution. Following this motivation, this paper focuses on the use of Subbotin’s generalised normal distribution (Subbotin 1923) as 0, which provides enough flexibility to allow for tails heavier than that of the normal distribution. This idea has been briefly discussed in (Salinas, Arellano-Valle, and Gómez 2007; Azzalini 2013) and is used as a point of departure. The purpose of this paper is to enrich and develop theory in this area and provide novel methods of estimating the characteristics of this distribution. Literature on Azzalini’s model and related models are widely presented in literature (Azzalini and Regoli 2011; Arellano-Valle, Del Pino, and San Martın 2002; Bahrami and Qasemi 2015; Arnold, Beaver, Azzalini, Balakrishnan, Bhaumik, Dey, Cuadras, and Sarabia 2002; Pourahmadi 2007; Yadegari, Gerami, and Khaledi 2008). In Section 2 the generalised normal distribution (Subbotin 1923) is given and a stochastic representation is presented. Thereafter, the use of the generalised normal ( ) distribution as a symmetric base (see Proposition 1) gives rise to a skew generalised normal distribution termedGN the distribution. Methods to calculate basic SGN 1 [email protected] 1 statistical properties of the and a corresponding stochastic representation is presented in Section 3. In Section 4 a generalised skewing mechanismSGN is coupled with the generalised normal distribution. In Section 5 the is firstly applied in a distribution fitting context. Thereafter, the distribution is considered as a candidateSGN to approximate the binomial distribution. Final remarks are madeSGN in Section 6. 2 Skew generalised normal distribution In this section the distribution is revisited and a sampling scheme to generate random variates from this distribution is proposed.GN Thereafter, the distribution is focussed on. Three methods of evaluating the characteristics (i.e. expected value, variance,SGN skewness and kurtosis) of this distribution are presented. Finally, a stochastic representation of the distribution is developed. SGN Definition 2 (Subbotin 1923) A random variable has the generalised normal distribution with location parameter and scale parameter if its PDF is given by − ∗ (; )= − (2) 2Γ 1 | | R ( ) ∈ where Γ ( ) denotes the gamma function, R and R+.Thisisdenotedby ( ). · ∈ ∈ ∼ GN Following a similar approach in (Azzalini 2013), a stochastic representation of the ( ) distribution is proposed. This provides a method to generate random numbers from ( )GN. ∼ GN Theorem 3 If ( ) then, ∼ GN 1 1 + with probability 2 = 1 with probability 1 ( − 2 where 1 1 . ∼ ³ ´ 1 Proof. Let = − .Theproperty,P [ 0] = P [ 0] = , follows from the symmetry of − − 2 ( ) It follows¯ ¯ from (Bain and Engelhardt 1992) that ∼ GN ¯ ¯ ¯ ¯ 2 1 1 1 1 1 ()= − () − () = − − 1 =1 ¯ ¯ Γ X ¡ ¢ ¯ ¯ ¯ ¯ ³ ´ 1¯ ¯ for R+, and the result follows that 1 . ∈ ∼ Since software that can generate gamma distributed³ random´ numbers is readily available, Theorem 3 provides a representation to easily generate random numbers from a generalised normal distribution. Using the same notation defined in Proposition 1, the case where 0 = ∗0 = Φ (with ∗ denoting the PDF defined in (2), Φ ( ) representing the standard normal CDF) and where ()=√2 for R yields the following definition of the · distribution. ∈ SGN Definition 4 A random variable has the distribution with location parameter and scale parameter if its PDF is given by SGN 2 (; )= ∗ − ; Φ √2 − R (3) ∈ where R, R+ and R.Thisisdenotedby ( ). ∈ ∈ ∈ ¡ ∼ SGN¢ ¡ ¡ ¢¢ When =0, = √2 and =2the distribution with PDF as in (3), the PDF collapses to that of Azzalini’s (Azzalini 1985). SGN SN 3 Statistical Properties In this section, three methods are proposed to evaluate some statistical characteristics (expected value, standard deviation, skewness and kurtosis) of the distribution. Four different parameter structures of the are considered as candidates and the results areSGN tabulated. Finally, a stochastic representation of the distributionSGN is proposed. SGN 2 3.1 Method 1 The acceptance-rejection (AR) method (Lange 2010) is used to generate random skew generalised normal variates from ( ) distribution with PDF given by (3). Fig. 1 shows the results of the AR algorithm for the first parameterSGN structure considered. The green and red points indicate the random numbers that are respectively accepted and rejected as candidates from the ( ) distributionwithPDFgivenby(3). SGN Figure 1. Histogram of the realised random variates drawn from ( ) with theoretical PDF given by (3) overlaid for the first parameter structure i.e.SGN( =0=4 =2=2) Estimates of the characteristics of the ( ) distribution can be obtained by performing calculations on the realised random variates. The drawbackSGN of this method is that the AR algorithm may perform poorly or fail outright for certain parameter structures. 3.2 Method 2 This method uses the stochastic representation of the distribution as derived in Section 2. GN Theorem 5 If ( ) with PDF as defined in (3) then ∼ SGN E [ ]=E 2 Φ √2 ∗ − ∗ ∗ ∙ µ µ ¶¶¸ where ( ) has PDF as in (2). ∗ ∼ GN Proof. By definition − E [ ]= − Φ √2 − 1 | | ZR Γ µ µ ¶¶ ³ ´ = E 2 Φ √2 ∗ − ∗ ∗ ∙ µ µ ¶¶¸ where ( ) has PDF as in (2). Since Theorem∗ ∼ GN 3 can be used to generate random variates, Theorem 5 provides a method to calculate the moments of ( ) with PDF given byGN (3), which are then used to calculate the characteristics of the respective distribution.∼ SGN 3 3.3 Method 3 A novel expression for the moments of ( ) with PDF given by (3), is presented. ∼ SGN Theorem 6 If ( ) with PDF given by (3) then ∼ SGN []= (4) E − E =0 µ ¶ X £ ¤ where Γ( +1 ) for even Γ( 1 ) = E ⎧ Γ( +1 ) 1 2 Φ √2 1 for odd ⎪ Γ 1 E £ ¤ ⎨ ( ) − n h ³ ´i o where +1 1 . ⎩⎪ ∼ ³ ´ Proof. Consider a random variable (0 1) with PDF as defined in (3) then ∼ SGN ∞ ∞ E = − Φ √2 +( 1) − Φ √2 0 1 − 0 1 − Z Γ Z Γ £ ¤ ³ ´ ³ ´ = 1 + 2 ³ ´ ³ ´ Let = then it follows that +1 Γ 1 = E Φ √2 Γ³ 1 ´ h ³ ´i Similarly, ³ ´ +1 ( 1) Γ 1 − 2 = E 1 Φ √2 Γ 1³ ´ − h ³ ´i Therefore result (??) follows. To extend this³ approach´ for a random variable = + ,sothat ( ) with PDF given by (3), it follows from the binomial theorem that ∼ SGN []= E − E =0 µ ¶ X £ ¤ 3.4 Simulation results Four different parameter structures of the ( ) distribution are considered. The three methods presented are used to estimate the characteristics ofSGN the distribution for a certain parameter structure. The time taken by each of the methods to obtain a result is also recorded. The results of the simulation are summarised in Table 1. Remark 7 It is clear that Method 2 performs comparatively poorly. Method 1 may outperform Method 3, however, the latter will work for any valid parameter structure which is not the case with Method 1. 3.5 Stochastic representation
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