Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM) Vol. 14, Issue 2 (December 2019), pp. 784 - 804 Alpha-Skew Generalized Normal Distribution and its Applications 1*Eisa Mahmoudi, 2Hamideh Jafari and 3RahmatSadat Meshkat Department of Statistics Yazd University Yazd, Iran [email protected]; [email protected]; [email protected] * Corresponding Author Received: July 8, 2018; Accepted: November 11, 2019 Abstract The main object of this paper is to introduce a new family of distributions, which is quite flexible to fit both unimodal and bimodal shapes. This new family is entitled alpha-skew generalized normal (ASGN), that skews the symmetric distributions, especially generalized normal distribution through this paper. Here, some properties of this new distribution including cumulative distribution function, survival function, hazard rate function and moments are derived. To estimate the model parameters, the maximum likelihood estimators and the asymptotic distribution of the estimators are discussed. The observed information matrix is derived. Finally, the flexibility of the new distribution, as well as its effectiveness in comparison with other distributions, are shown via an application. Keywords: Alpha-skew distribution; Generalized normal distribution; Maximum likelihood estimator; Skew-normal distribution MSC 2010 No.: 60E05, 62F10 1. Introduction Recently, the skew-symmetric distributions have received considerable amount of attention, for example, the alpha skew normal (ASN) distribution is studied by Elal-Olivero (2010). First, he introduced a new symmetric bimodal-normal distribution with the pdf 784 AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 785 f( y ) y2 ( y ), where is standard normal density function, and then defined alpha-skew-normal distribution as (1 y )2 1 f( y ; ) ( y ), y ¡¡, , 2 2 where is skew parameter. A random variable with ASN distribution is denoted by X~ ASN ( ). Also, Shams Harandi and Alamatsaz (2012) introduced the alpha-skew Laplace (ASL) distribution with the pdf (1 y )2 1 f( y ; ) e||y , y ¡¡, , 4(1 2 ) and investigated some of its properties and denoted this random variable with X~ ASL ( ) . In this paper, a more general case of these two mentioned distributions is presented, which is called as alpha-skew generalized normal (ASGN) distribution which is obtained by adding a skew parameter to the generalized normal distribution (GND) proposed first by Lee et al. (2013) with pdf 1 ||y f( y ; ) e , y ¡ , 0. 1 1 2 The motivations for considering this density are: 1. The alpha-skew generalized normal distribution with at most two modes is very flexible and includes five classes of important distributions: Normal, Laplace, Alpha-Skew- Normal, Alpha-Skew-Laplace and Bimodal-Generalized Normal as special cases. 2. It seems that the admissible intervals for the skewness and the kurtosis parameters are 0.685, 3.803 and 1.250, 3.325 , respectively, which are wider than those of the Azzalini’s the ASN distributions, 0.811, 0.811 and 1.300, 0.749 , respectively. 3. In many applied studies, data may be skew and bimodal with thinner or thicker tails than normal. We believe that the distribution ASGN can illustrate a better fit with respect to the fitted models for some this kind of real data sets. It is obvious that the new introduced models are not provided best fit in compared to all other models, but of course it is necessary to provide best fit in compared to its sub-models. The remainder of the paper is organized as follows: In Section 2, we define the ASGN distribution and outline some special cases of the distribution. Also, some properties of the distribution are 786 E. Mahmoudi et al. investigated in this section. cumulative distribution, survival and hazard rate functions are obtained in Section 3. we provide a general expansion for the moments of the ASGN distribution in Section 4. Skewness and kurtosis indices are presented in Section 5. Stochastic representation for this model is discussed in Section 6. We introduce the location-scale version of this distribution in Section 7. In Section 8, we focus on maximum likelihood estimation (MLE) and calculate the elements of the observed information matrix. A simulation study is performed in Section 9. Application of the ASGN distribution using a set of real data is given in Section 10. Finally, Section 11 concludes the paper. 2. Alpha-skew generalized normal distribution Definition 2.1. If a random variable Y has the following density function, 3 y ye2 f( y ; ) , y ¡ , 3 2 where 0 , then we say that follows the bimodal-generalized normal BGN distribution. Definition 2.2. If a random variable X has the following density function, 1 1 2 x (1x ) 1 f(;,) x e , x ¡ , (1) 2 2/ 31 22 where ¡ and 0 , then we say that follows the alpha-skew generalized normal distribution with parameters and , which is denoted by X~ ASGN ( , ) . If X~ ASGN ( , ) , the following properties are concluded directly from the definition: If 0 and 1, then X~ Laplas (0,1) . If 0 and 2 , then XN~ (0,1) . If , then X~ ASL ( ) . If , then X~ ASN ( ) . If , then X~ BGN ( ) . X~ ASGN ( ) . AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 787 Theorem 2.1. The alpha-skew generalized normal density function has at most two modes. Proof: Differentiating (1), we have 1 x 2 e2 ( x 1) 22 x 2 x 2 x x ) fx . 2 2/ 31 22 Due to , the above equation is not solved simply. So, it is sufficient to prove the especial cases. Putting 1 in the above equation, we get ex 22 x2 1 x 2 2 if x 0, 1 fx( ) (2) 2 41 x 22 e x 2 1 x 2 2 if x 0. It is obvious that each expression in the above has at most two roots. The roots of second 11 2 11 2 expression occur in x and x and the roots of first expression 1 2 11 2 11 2 occur in x* and x* . While 2 10, we have the following 1 2 cases * * a) for 1: it is easily observed that x2 0 , xx11, 0 and x2 0; * * b) for 1: it is observed that x2 0 , xx11,0 and x2 0 . Therefore, there are only three acceptable roots in each case. Hence, fx can have at most two modes. For 11 , fx () has not real root, but it is positive for x 0 and is negative for x 0 . In this case, is unimodal. Putting 2 , we obtain ()x 2 3 2 2 f( x )2 x 2 x 2 2 x 2 , 2 788 E. Mahmoudi et al. where fx () has at most three roots and also in this case, fx has at most two modes. For 2, the complicated mathematical computations are needed, but the claim can be proved by drawing some plots of density function for different values of . Plots of the alpha-skew generalized normal density function for selected parameter values are given in Figure 1. Figure 1. Plots of the alpha skew generalized normal density function for different values of and 3. The cumulative distribution, survival and hazard rate functions If X~ ASGN ( , ) , then cumulative distribution function of X is given as AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 789 2 2/3 (x ) 1/ 2 ( x ) 1 ( x ) , 2 , 2 , if x 0, 2 2/ 31 22 Fx( ; , ) 2 2/3 2 2/ 3x 1/ 2 x 1 x 1 2 , 2 , 2 , 4 , if x 0, 2 2/ 31 2 2 where the incomplete gamma is defined as (a , x ) ta1 exp( t ) dt . x The survival and hazard rate functions of random variable from the ASGN distribution is as following, if x 0 , 1 1 2 2/3 2 1/ 2 2 2/ 3 (x ) 2 2 , Sx() 2 2/ 31 22 1/ 2 (xx ) 1 ( ) 1 1 , , 1 , 2 2/ 31 2 1 1 2 2/3 2 1/ 2 2 2/ 3 (x ) 2 2 , Hx() 3 (x ) 2 ( x ) 1 ( x ) 1/ 1/ , 2 , 2 , 1 1/ 2 (xx ) 1 ( ) 11 2 , 2 , 2 1 2 , 3 (x ) 2 ( x ) 1 ( x ) 1/ 1/ , 2 , 2 , and if x 0 , X 790 E. Mahmoudi et al. 2 2/3x 1/ 2 x 1 x , 2 , 2 , Sx(), 2 2/ 31 22 2 2/3 2 2/ 3x 1/ 2 x 1 x 1 2 , 2 , 2 , 4 Hx() . 1/ 1/ 3 x 21xx , 2 , 2 , 4. Central moments and moment generating function If X~ ASGN ( , ) , then the r th central moments of X can be obtained as r/ 1/ r 1/ rr 32 r ( 1) 1 2 ( 1) 1 r EX() 2 2/ 31 24 rr/ r 1 2 ( 1) 1 , 2 2/ 31 24 and the moment generating function of can be written as nn tX t E() X MX ()() t E e n0 n! r/ 1/ r 1/ rr 32 r ( 1) 1 2 ( 1) 1 t n n0 n! 2 2/ 31 24 r1 r r 1 2 ( 1) 1 . 2 2/ 31 24 By applying mathematical calculations and separating even and odd expressions, we get AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 791 2nn 2 2 2n 1 2 n 3 2 t 2n 42 t 21n MtX (). n0 (2n )!2 2/ 31 24 5. The indices of skewness and kurtosis Suppose X~ ASGN ( , ) . Using first to fourth central moments, 1 and 2 can be obtained which are the indices of skewness and kurtosis of ASGN distribution, respectively. 2/ 3 2 EX(),1 2 2/ 31 2 2/ 2 2/ 53 2 2 EX(),2 2 2/ 31 2 4/ 5 2 3 EX(),3 2 2/ 31 2 4/ 2 2/ 7 5 2 4 EX( ) 4 .