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The Alpha-Beta-Gamma Skew Normal Distribution and Its Application Open Journal of Statistics, 2020, 10, 1057-1071 https://www.scirp.org/journal/ojs ISSN Online: 2161-7198 ISSN Print: 2161-718X The Alpha-Beta-Gamma Skew Normal Distribution and Its Application Zhengyuan Wei, Tiankui Peng*, Xiaoya Zhou College of Science, Chongqing University of Technology, Chongqing, China How to cite this paper: Wei, Z.Y., Peng, Abstract T.K. and Zhou, X.Y. (2020) The Alpha- Beta-Gamma Skew Normal Distribution In this paper, a new class of skew multimodal distributions with more flexible and Its Application. Open Journal of Statis- than alpha skew normal distribution and alpha-beta skew normal distribution tics, 10, 1057-1071. is proposed, which makes some important distributions become its special https://doi.org/10.4236/ojs.2020.106060 cases. The statistical properties of the new distribution are studied in detail, Received: November 20, 2020 its moment generating function, skewness coefficient, kurtosis coefficient, Accepted: December 27, 2020 Fisher information matrix, maximum likelihood estimators are derived. Published: December 30, 2020 Moreover, a random simulation study is carried out for test the performance Copyright © 2020 by author(s) and of the estimators, the simulation results show that with the increase of sample Scientific Research Publishing Inc. size, the mean value of maximum likelihood estimators tends to the true val- This work is licensed under the Creative ue. The new distribution family provides a better fit compared with other Commons Attribution International known skew distributions through the analysis of a real data set. License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Keywords Open Access Skew Distribution, Multimodal Distribution, Alpha-Beta-Gamma Skew Normal Distribution, Fisher Information Matrix 1. Introduction Over the years, the family of skew-symmetric distributions has received consi- derable attention by many researchers. Azzalini [1] first proposed the skew normal distribution (SN) with the density T 2 fz( ) =2Φ∈(αφ z) ( z) ,,( z α) , (1) where φ (⋅) and Φ⋅( ) are probability density function (pdf) and cumulative distribution function (cdf) of standard normal distribution, respectively. Bala- krishnan [2] and Gupta [3] researched a generalization of the skew normal dis- tribution (GSN) and further discussed its properties, Yadegaria et al. [4] intro- duced a generalization of the Balakrishnan skew normal distribution which in- DOI: 10.4236/ojs.2020.106060 Dec. 30, 2020 1057 Open Journal of Statistics Z. Y. Wei et al. cludes GSN as special case, an epsilon skew normal distribution and a new class of skew-Cauchy distribution is studied in [5] and [6], respectively. Another skew normal distribution named power normal distribution is proposed by [7] which solved the estimation of the skewness parameter of SN when the sample size is not very large. Recently, Elal-Olivero [8] proposed a new form of skew distribution with bi- modal behavior named alpha skew normal distribution (ASN) with the density 2 (11−+α z) T 2 fz( ;α) = φα( z),,( z ) ∈ , (2) α 2 + 2 where α controls the skewness and kurtosis, Equation (2) has at most two modes. Alpha-skew-Laplace distribution (ASL) and a generalization of alpha skew normal distribution (GASN) are studied in [9] and [10], respectively. Moreover, Chakraborty et al. [11] proposed the Balakrishnan alpha skew normal distribution and discussed a generalized bimodal normal distribution denoted by BN(n) with the density zn fz( ) = φ ( z),, z∈ R (3) M where n is positive even integers and M is normalizing constant. Furthermore, Shafiei et al. [12] proposed a new class of skew normal distributions called al- pha-beta skew normal distribution (ABSN) that is more flexible than SN and ASN, with the density 2 11−−αβzz3 + ( ) T 3 fz( ;,αβ) = φ( z),( z ,, αβ) ∈ , (4) 6αβ+ 15 β22 ++ α 2 where α , β control the skewness and kurtosis, Equation (4) has at most four modes. The motivations for considering this new family of distributions are as follows: Firstly, the new distribution family contains some classical distributions, such as normal distribution, ASN, ABSN, etc. Secondly, the admissible intervals for the skewness and the kurtosis parameters are wider than ASN and ABSN. Lastly, the new distribution family is more flexible than some other known skew distribu- tions through the analysis of a real data set. Consequently, the reminder of the paper is organized as follows: In Section 2, we propose the new class of skew normal distributions and discussed its properties. In Section 3, we provide the maximum likelihood estimates (MLEs) of the parameters, the performance of the estimates are verified by random simulation. The real application of the new distribution family is considered in Section 4. Section 5 presents some conclu- sions. Lastly, some proofs and elements of Fisher information (FI) matrix are given in the appendix. 2. Alpha-Beta-Gamma Skew Normal Distribution In this section, we define a class of skew distributions that allow the fitting of multimodal data sets. DOI: 10.4236/ojs.2020.106060 1058 Open Journal of Statistics Z. Y. Wei et al. Definition 1 A random variable X follows alpha-beta-gamma skew normal distribution, denoted by ABGSN (αβγ, , ) , if it has the pdf 2 11−−αβxx35 − γ x + ( ) T 4 fx( ;,,αβγ) = φ( x),( x ,,, αβγ) ∈ , (5) C where C =945γ2 + 210 γβ + 30 αγ + 6 αβ + 15 β22 ++ α 2 . Figure 1 and Figure 2 show the pdf of the ABGSN (αβγ, , ) for different parameter values. As can be seen that the proposed density is very general, the parameters α , β as well as γ have substantial impact on the skewness and the number of extreme points of the distribution. Remark 1 Derivative of pdf (5) with respect to x is given by ∂fx( ;,,αβγ) ∂x φ ( x) = −γ2x 11 +(10 γ 2 − 2 βγ) x9 +( 16 βγ −− β2 2 αγ) xx7 + 2 γ 6 C (6) +(12αγ +− 6 β2 2 αβ) xx 5 +−( 2 β 10 γ) 4 +( 8 αβ − α 23) x αβ22 α α +−(2 6) xx +( 2 −− 2) 2, Figure 1. Alpha-beta-gamma skew normal pdf. ABGSN(20,− 10,0.8) has six modes (black solid line), ABGSN(4.5,−− 8, 10) has two modes (red dashed line). Figure 2. Alpha-beta-gamma skew normal pdf. ABGSN(0.4,− 0.3,0.1) has three modes (black solid line), ABGSN(−−0.4,2, 0.2) has five modes (red dashed line). DOI: 10.4236/ojs.2020.106060 1059 Open Journal of Statistics Z. Y. Wei et al. Equation (6) has at most eleven zeros, therefore the pdf up to six modes, see Figure 1 and Figure 2. T Proposition 1 The cdf of X ~ ABGSN(αβγ , , ) is FxX ( ;,,αβγ) =−− γ29 βγ + γ2 7 −+ β 2 αγ + γ2 + βγ5 + γ 4 x(2 9 ) x( 2 63 14) xx 2 −(5β2 + 10 αγ + 315 γ 2 + 2 αβ + 70 βγ) xx 32 ++( 2 β 8 γ ) −C −2 x + 2α + 4 β + 16 γ φ xC+Φ x,,,,. xαβγ ∈ 4 ( ) ( ) ( ) ( ) Remark 2 Some properties of the ABGSN (αβγ, , ) are as follows 1) ABGSN (0,0,0) , ABGSN (αβ,0, ) and ABGSN (α,0,0) degenerate to N(0,1) , ABSN (αβ, ) and ASN (α ) , respectively. 2) If α = 0 , pdf (5) reduces to 2 11−−βγxx35 + ( ) T 3 fx( ;,βγ) = φ( x),( x ,, βγ) ∈ , (7) 945γ22+ 210 γβ ++ 15 β 2 in sequel, we name Equation (7) the beta-gamma skew normal distribution, de- noted by BGSN (βγ, ) . 3) If β = 0 , pdf (5) reduces to 2 11−−αγxx5 + ( ) T 3 fx( ;,αγ) = φ( x),( x ,, αγ) ∈ , (8) 945γ22+ 30 αγ ++ α 2 in sequel, we name Equation (8) the alpha-gamma skew normal distribution, denoted by AGSN (αγ, ) . 4) If αγ= = 0 , pdf (5) reduces to 2 11−+β x3 ( ) T 2 fx( ;β) = φβ( x),,( x ) ∈ , (9) 15β 2 + 2 Equation (9) is known as the beta skew normal distribution and denoted by BSN (β ) , see Shafiei [12]. 5) If αβ= = 0 , pdf (5) reduces to 2 11−+γ x5 ( ) T 2 fx( ;γ) = φγ( x),,( x ) ∈ , (10) 945γ 2 + 2 in sequel, we name Equation (10) the gamma skew normal distribution, denoted by GSN(γ ) . 6) If X ~ABGSN (αβγ ,,) , when α → ±∞ , β → ±∞ and γ → ±∞ , then X follows BN(2), BN(6) and BN(10), respectively. 7) If X ~ABGSN (αβγ ,,) , then −X ~ABGSN( −−−αβγ ,,) . Proposition 2 Let X ~ABGSN (αβγ ,,) , then for k ∈ , we have 1 2k = +α 22 ++ αγ β X22 hh2kk 22++( ) h26k C ++γ2h2 αβ hh + 2, βγ 2k+ 10 2 kk ++ 4 2 8 DOI: 10.4236/ojs.2020.106060 1060 Open Journal of Statistics Z. Y. Wei et al. 21k − 1 X=−++(22αβ hh2kk 22++ 2 γ h 24 k) , C where h2ki+ =××13 ×( 2ki +− 1) , i = 0, 2, 4,6, . Remark 3 In particularly, we have 1 [ X ] =−++(2αβ 6 30 γ) , C 1 2 22 Var[ X ] =(2 ++ 3α 210 αγ + 105 β + 10395 γ + 30 αβ + 1890 βγ ) C −αβ ++ γ2 (2 6 30) C , We use R software to optimizing [ X ] and Var[ X ] with respect to αβγ,,, we can get the bounds [ X ]∈−( 0.71,0.71) , Var[ X ]∈(0.81,14.07) . Remark 4 Let ξ1 and ξ2 stand for its skewness and kurtosis coefficients, then we have 3 23 X−+32 [ XX] [ X] ξ = 1 32 22 (XX− [ ]) =−++ α β γ3 + α3 + β 3 + γ 32 + αγ 16( 3 15) 12C ( 120 61425 95 2 22 2 2 +++14α β 75 αβ 2835 β γ + 5775 αγ + 23730 βγ 22 +1200αβγ −− 2 β 20 γ)( 2 + 3 α + 210 αγ + 105 β 32 +γ2 + αβ + βγ − α ++ β γ 2 10395 30 1890)C ( 2 6 30) , 4 32 24 X−+46 [ XX] [ XX] − 3 [ X] ξ = − 2 2 3 22 (XX− [ ]) =−+− α + β + γ4 +32 α + β 2 + γ 2 + αβ 3 48( 3 15) C ( 15 945 135135 210 +1890αγ + 20790 βγ ++ 6) 24( α + 3 β + 15 γ)C ( α33 + 165 β +89775γ322 + 125 α γ ++ 17 α β 105 αβ 2 + 4095 β 2 γ + 9555 αγ 2 +βγ2 + αβγ −− α β− γ ++ α2 αγ 35385 1830 2 14 110)( 2 3 210 2 +β22 + γ + αβ + βγ − α ++ β γ 2 105 10395 30 1890)C ( 2 6 30) , calculated by R software with the package DEoptim we can get the bounds ξξ12∈−( 1.29,1.29) ,∈−( 1.88,5.60) .
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