The Generalized-Alpha-Beta-Skew-Normal Distribution: Properties and Applications
Sricharan Shah1, Subrata Chakraborty1, Partha Jyoti Hazarika1 and M. MasoomAli2 1Department of Statistics, Dibrugarh University, Dibrugarh, Assam 786004 India 2Department of Mathematical Sciences, Ball State University, Muncie, IN 47306 USA
Abstract In this paper we have introduced a generalized version of alpha beta skew normal distribution in the same line of Sharafi et al. (2017) and investigated some of its basic properties. The extensions of the proposed distribution have also been studied. The appropriateness of the proposed distribution has been tested by comparing the values of Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) with the values of some other known related distributions for better model selection. Likelihood ratio testhas been used for discriminating between nested models.
Keywords: Skew Distribution, Alpha-Skew Distribution, Bimodal Distribution, AIC, BIC. Math Subject Classification: 60E05, 62H10, 62H12
1. Introduction In many real life situations the data exhibit many modes as well as asymmetry, e.g., in the fields of demography, insurance, medical sciences, physics, etc. (for details see Hammel et al. (1983), Dimitrov et al. (1997), Sinha (2012), Chakraborty and Hazarika (2011), Chakraborty et al. (2015), among others). Azzalini (1985) first introduced the skew normal distribution and the density function of this distribution, denoted by SN () , is given by
fZ (z;) 2(z)(z); z , (1) where, (.) is the density function of standard normal distribution , (.) is the cumulative distribution function (cdf) of standard normal distribution and is the asymmetry parameter. The generalization of the skew normal distribution was proposed by Balakrishnan (2002) as a discussant in Arnold and Beaver (2002) and studied its properties and the density function is given by
n f Z (z;, n) (z)[(z)] Cn () (2)
1
n where, n , is a positive integer and Cn () E[ (U)], U ~ N(0,1) . The general formula for the construction of skew-symmetric distributions was proposed by Huang and Chen (2007) starting from a symmetric (about 0) pdf h(.) by introducing the concept of skew functionG(.) , a Lebesgue measurable function such that, 0 G(z) 1 and
G(z) G(z) 1, z R , almost everywhere. The density function of the same is given by
f (z) 2h(z)G(z) ; z R . (3) Elal-Olivero (2010) introduced a new form of skew distribution which has both unimodal as well as bimodal behavior and is known as alpha skew normal distribution, denoted by ASN() and its density function is given by
(1z)2 1 f (z;) (z); z R . (4) 2 2 Along the same line of the eqn. (4), Venegas et al. (2016) studied the logarithmic form of alpha-skew normal distribution and used for modeling chemical data. Sharafi et al. (2017) introduced a generalization of ASN() distribution. Shafiei et al. (2016) introduced a new family of skew distributions with more flexibility than the Azzalini (1985) and the Elal- Olivero (2010) distributions and is defined as follows. A random variable Z is said to be an alpha-beta skew normal distribution, denoted by ABSN(,) if its density function is given by
(1 z z 3 )2 1 f (z;, ) (z); z,, R . (5) 2 2 2 15 6 The reasons for considering this new distribution are as follows: First, this distribution includes four important classes of distribution which are normal, skew normal, alpha-beta skew normal, and the generalization of ASN() distribution. Second, this distribution is flexible enough to support both unimodality and bimodality (or multimodality) behaviors and has at most four modes. Third, for some real life data, this new proposed distribution provides better fitting as compared to the other distributions considered. The article is summarized as follows: In Section 2 we define the proposed distribution and some of its basic properties are investigated. In Section 3 we study some of its important distributional properties. Some extension of this distribution is discussed in Section 4. A stochastic representation and simulation procedure is given in Section 5. The parameter estimation and the real life applications of the proposed distribution are provided in Section 6. Lastly, conclusions are given in Section 7.
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2. The Generalized Alpha-Beta Skew Normal Distribution In this section we introduce a new generalized form of alpha-beta skew normal distribution and investigate some of its basic properties. Definition 1: If a random variable Z has a density function [(1 z z 3 )2 1] f (z;,,) (z)( z); z R (6) C(,,)
3 22 2 15 2 where C(, ,) 1 3 b b , then it is said to be generalized 1 2 2 2 alpha-beta skew normal distribution with skewness parameters (,,) R . We denote it as
GABSN(, ,) .The normalizing constant C(, ,) is calculated as follows:
C( , ,) [(1 z z 3 )2 1](z) ( z) dz
(2 2 z 2 z 2 2 z 3 2 z 4 2 z 6 )(z)( z) dz
2 2 (z;) dz E(Z ) E(Z 2 ) E(Z 3 ) E(Z 4 ) E(Z 6 ) 2 2
2 2 2 15 2 2 3 2 1 3 2 2 2 1 2 1 2 1
3 22 2 15 2 1 3 b b , 1 2 2 2
2 where b , and (z;) is the density function of Z ~ SN () . 1 2 2.1. Properties of GABSN(, ,) :
If 0 , then we get the generalized ASN() distribution of Sharafi et al. (2017) given by [(1 z)2 1] f (z;,) (z)( z) 2 1 b 2
[(1 z 3 )2 1] If 0 , then we get f (z;,) (z)( z) 3 22 15 2 1 b 1 2 2 This is known as generalized beta skew normalGBSN(,) distribution.
If 0 , then we get the ABSN(,) distribution of Shafiei et al. (2016)given by
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[(1 z z3 )2 1] f (z;, ) (z) 2 2 6 15 2
If 0 , then we get the SN () distribution of Azzalini (1985) and is given by f (z;) 2(z)( z)
If 0 , then we get the standard normal distribution and is given by f (z) (z)
If , then f (z;) 2z2(z)( z), where, 2 z 2(z) is the pdf of BN(2) (see
Hazarika et al. (2019)). Therefore, as , GABSN(,,) GBN(2)
If , then GABSN(,,) GBN(6) . the pdf of GBN (6) is
2 z 6 f (z;) (z)( z) 15
For fixed and , if then
[(1 z z 3 )2 1] f (z;,) (z) I(z 0) 2 15 2 1 b 2 b 3 2 2 and if then
[(1 z z 3 )2 1] f (z;,) (z) I(z 0) 2 15 2 1 b 2 b 3 2 2 If Z ~ GABSN(, , ) , then Z ~ GABSN(, ,) . 2.2. Plots of the density function The density function of GABSN(, ,) distribution for different choices of the parameters, and are plotted in Figure 1.
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Figure1: Plots of the density function ofGABSN(, ,)
2.3. Mode ofGABSN(, ,) :
Theorem 1: The GABSN(, ,) distribution has at most four modes.
Proof: Let Z ~ GABSN(, , ) distribution then by eqn. (5) and eqn. (6) we have,
2 2 15 2 6 f (z;,,) f (z;, )( z) .(7) C(,,) Then by differentiating we get,
2 2 15 2 6 f '(z;,,) f '(z;,)( z) f (z;, )( z) . (8) C(,,) To prove that the eqn. (7) has at most four modes, we have to show that the eqn. (8) has one or seven roots. For this, we apply a graphical approach, and so we have
f '(z; , , ) F1 (z) F2 (z) where 2 2 15 2 6 F (z) f '(z;, )( z) 1 C(, ,)
2 2 15 2 6 F2 (z) f (z;, )( z) . C(,,)
Then, set f '(z;,,) 0 , and hence F1 (z) F2 (z) . (9)
Since the eqn. (6) vanishes for (-5, 5), we draw the curves of C1: y F1(z)for
3.72, 1.08, 0.3 andC2: y F2 (z) for 1.53, 0.1, 3 in Figure 2.
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(a) (b) Figure 2: (a) The Plot of C1 and C2 for 3.72, 1.08, 0.3 . (b) The Plot of C1 and C2 for 1.53, 0.1, 3
From the figure 2, it is shown that the two curves have atleast one and at most seven intersection points and the values of z of these points are the roots of eqn. (9). Since lim f (z;,,) 0 , then if eqn.(9) has one root it should be the mode of eqn. (7) z and if eqn.(9) has seven roots,then eqn. (7) should have four modes and hence the GABSN(, ,) distribution has at least one and at most four modes.
3. Distributional Properties 3.1. Cumulative Distribution Function Theorem 1: The cdf of GABSN(, ,) distribution is given by
1 2 2 2 2 2 2 2 2 2 4 2 2 FZ (z) 5 (b ( (1 ) 2 (1 )(5 z (3 z ) ) (2z(1 ) z (1 ) 2(1 2 ) 2 1 z 2 (1 2 )(9 52 ) (33 402 154 )))(z 1 2 ) (1 2 ) 2 ( 2( 2 ) Erf (z 1 2 / 2) (2( 2 z 2(1 z(3 2 ) ) (2(2 2 ) z(15 5z 2 z 4 ) )) 1 2 (z) (z) 2b (z 1 2 ) 3(1 2 5 2 ) 1 2 (z;)))) (10)
z (1 t t 3 )2 1 Proof: F (z) P(Z z) (t)(t)dt Z C(, ,)
1 z (2 2 t 2t 2 2 t3 2 t 4 2t 6 )(t)(t)dt C(, ,)
z z z z 1 2 2 3 2(t)(t)dt 2 t(t)(t)dt t (t)(t)dt 2 t (t)(t)dt C(,,) z z 4 2 6 (11) 2 t (t)(t)dt t (t)(t)dt
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Since, we have
z 2(t)(t)dt (z;)
z Erf (z 1 2 ) / 2 t(t) (t)dt (z) (z) 2 2 2 1
z b z 1 2 t 2(t)(t)dt z(z)(z) 2 2 1
z Erf (z 1 2 ) / 2 b z 1 2 b z 1 2 t 3(t)(t)dt (2 z 2 )(z) (z) z 2 2 2 2 1 2 1 2 1
z 2 2 2 2 4 2 b z 1 5 z (3 z ) 3 t (t) (t)dt z(3 z )(z)(z) 3 (z;) 2 2 2 1
z 2 2 4 4 2 2 2 2 4 6 2 4 bz 1 3340 15 z (1 ) z (914 5 ) 15 t (t)(t)dt z(155z z )(z)(z) 5 (z;) 2 2 2 1 Putting the above results in eqn. (11), we get the desired result in eqn. (10).
3.2. Moments
th Theorem 3: The k order moment of GABSN(, ,) distribution is given by
2 2 k 1 k k1 k2 k3 k4 k6 E(Z ) E(Z ) E(Z ) E(Z ) E(Z ) E(Z ) E(Z ) C(, ,) 2 2 k th where, E(Z ) is the k moment of Z ~ SN () . (12)
(1 z z 3 ) 2 1 Proof: E ( Z k ) z k ( z ) ( z ) dz C ( , , )
1 k k 1 2 k 2 2 z (z) (z)dz 2 z (z) (z) dz z (z) (z) dz C ( , , ) k 3 k 4 2 k 6 2 z (z) (z) dz 2 z (z) (z) dz z (z) (z) dz
1 k k 1 2 k 2 k 3 k 4 2 k 5 E (Z ) 2 E(Z ) E (Z ) 2 E(Z ) 2 E(Z ) E (Z ) C ( , , )
The moments of Z ~ GABSN(, , ) can be obtained by applying the moments of
Z ~ SN () (Henze, 1986). Thus, we obtained the following results for k 1, 2,3, 4 :
2 2 2 1 b (3 2 ) b (c1 ) 3b (c2 ) E(X ) 3 b 2 2 2 2 3 C(,,) 2(1 ) (1 ) 2(1 )
2 2 2 2 1 3 105 b (3 2 ) b (c1 ) E(X ) 115 2 2 2 C(,,) 2 2 (1 ) (1 )
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2 2 2 3 1 b (3 2 ) b (c1 ) 3b (c2 ) 3b (c3 ) E(X ) 3 15 2 2 2 2 3 2 4 C(, ,) (1 ) 2(1 ) (1 ) 2(1 )
2 2 4 1 15 945 b (c1 ) 3b (c2 ) E(X ) 3 105 2 2 2 3 C(,,) 2 2 (1 ) (1 )
1 3 2 105 2 b (3 22 ) b (c ) Var(X ) C(, ,)115 1 2 2 2 2 [C(, ,)] 2 2 (1 ) (1 ) 2 b 2 (3 22 ) b (c ) 3b 2 (c ) 3 b 1 2 2(1 2 ) (1 2 ) 2 2(1 2 )3
2 4 2 4 6 2 2 4 6 where, c1 15 20 8 ;c2 35 70 56 16 ; c3 315 8 (105126 72 16 .
Remark 1: Using numerical optimization of E(Z) andVar(Z) with respect to, and , the following bounds for mean and variance can be obtained as 2.75863 E(Z) 2.75863 and 0.31173 Var(Z) 8.16228
Remark 2: By taking the limit in the moments of GABSN(, ,) distribution, the moments of GBN(2) distribution can be obtained as
b (3 22 ) 3 (1 2 )3 22 (3 22 )2 E(Z) ; Var(Z) . (1 2 ) (1 2 )3
Remark 3: By taking the limit in the moments of GABSN(, ,) distribution, the moments of GBN(6) distribution can be obtained easily as
b (35 702 564 166 ) 175 (1 2 )7 22 (35 702 564 166 )2 E(Z) ; Var(Z) . 5(1 2 )3 25 (1 2 )7
Remark 4: By taking the limit or in the moments of GABSN(, ,) distribution, we get when,
b(2 2 2 ( 2 16) 3 2 48 2 ) E(Z) ; 2 2b 2 4b 6 15 2
2(2 2 2 2 3 2 16 48 2 )2 2 4b 3 2 16b 30 105 2 Var(Z) . (2 2b 2 4b 6 15 2 )2 2 2b 2 4b 6 15 2
And when then
b(2 2 2 ( 2 16) 3 2 48 2 ) E(Z) , 2 2b 2 4b 6 15 2
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2(2 2 2 2 3 2 16 48 2 )2 2 4b 3 2 16b 30 105 2 Var(Z) . (2 2b 2 4b 6 15 2 )2 2 2b 2 4b 6 15 2
3.3. Skewness and Kurtosis Remark 5: The skewness and kurtosis of GABSN(, ,) distribution is obtained respectively, by using the formulae
3 2 3 2 4 3 2 2 4 E(Z ) 3E(Z )E(Z) 2[E(Z)] and E(Z ) 4E(Z )E(Z) 6E(Z )[E(Z)] 3[E(Z)] 1 3 2 2 E(Z 2 ) [E(Z)]2 E(Z 2 ) [E(Z)]2 which cannot be expressed conveniently as the expression is very vast. The following bounds for skewness and kurtosis can be calculated by numerically optimizing 1 and 2 with respect to , and as 0 1 6.70451 and 1.22732 2 14.1965 .
Remark 6: The skewness and kurtosis of GBN(2) distribution can be obtained easily by taking limit in the results of GABSN(, ,) distribution as
3 2 3 2 2 2 4 2 2 4 (3 2 ) (1 ) (12 25 10 ) 1 3 3 (1 2 )3 22 (3 22 )2
2 2 6 4 2 4 3 2 2 4 6 15 (1 ) 12 (3 2 ) 4 ( ) (9 9 16 4 ) . 2 2 3 (1 2 )3 22 (3 22 )2
Remark 7: The skewness and kurtosis of GBN(6) distribution can be obtained easily by taking limit in the results of GABSN(, ,) distribution as
22 (42 (c )3 25 (1 2 )6 (c ))2 39375 2 (1 2 )14 124 (c ) 4 5002 (1 2 )6 c c 2 4 , 2 2 5 , 1 3 2 21 3 2 2 2 14 2 15625 (1 ) c6 625 (1 ) c6
2 4 6 8 2 4 6 8 where, c4 420 1365 1638 936 208 , c5 21105 126 72 16 , and
22 (35 702 564 166 )2 c 7 . 6 25 (1 2 )7
Note: By taking the limit or in the results of GABSN(, ,) distribution, we obtained the results but we are unable to express the expression conveniently as the expression is very vast. 4. The Stochastic Representation for GABSN(, ,) distribution
Theorem 3: The conditional distribution of W |{ W X } follows GABSN(, ,) distribution, if W ~ ABSN(, ) and X ~ N(0,1) , and are independent.
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Proof :Assume Z W |{W X}. Then, we can have P(W z,W X ) P(Z z) P(W z | W X ) P(W X )
z (1 u u 3 ) 2 1 Using eqn. (5), we get, P(W z,W X ) (u)( u)du 2 2 2 15 6
(1 u u 3 ) 2 1 1 and P(W X ) (u)( u)du C(, ,) 2 2 2 2 2 15 6 2 15 6
z (1 u u 3 ) 2 1 (u)( u)du 2 2 15 2 6 z (1 u u 3 ) 2 1 Therefore, P(Z z) (u)( u)du 1 C(,,) C(,,) 2 2 15 2 6 and the density function of the conditional distribution of Z W |{W X}is
(1 z z 3 ) 2 1 f (z) (z) ( z) ~ GAbSN(, ,) Z C(, ,)
Or, we writeW |{W X} ~ GAbSN(, ,) .
Algorithm We can simulate the data for the proposed distribution with the help of the above stochastic representation by applying the acceptance-rejection technique using the following steps: a) Generate X ~ N(0,1) and W ~ ABSN(, ) (Sharafi et al., 2017) independently.
b) If Z W |{W X}put Z W . Otherwise, go to step a).
5. The Extension of Generalized Alpha Beta Skew Normal Distribution 5.1. Log-Generalized Alpha Beta Skew Normal Distribution In this section, using the idea of (Venegas et al., 2016), we present the definition and some simple properties of log-generalized alpha beta skew normal distribution. Let Z eY , thenY Log(Z) , therefore, the density function of Z is defined as follows:
Definition 3: If the random variable Z has the density function given by [(1 y y 3 ) 2 1] f (z;,,) (y)( y); z 0 , (13) C(,,) z then we say that Z is distributed according to the log-generalized alpha beta skew normal distribution with parameter,, R ,where, y Log(z) and ( y) is the density function of the standard log-normal distribution. We denote it by LGABSN(, ,) . Properties of LGABSN(, ,) :
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[(1 y) 2 1] If 0 , then we get f (z;,) (y)( y) . 2 1 b z 2
This is known as log-generalized ASN() distribution.
[(1 y 3 )2 1] If 0 , then we get f (z;,) (y)( y) . 3 22 15 2 1 b z 2 1 2 This is known as log-generalized beta skew normal LGBSN ( , ) distribution.
[(1 y y3 )2 1] If 0 , then we get f (z;, ) (y) . (2 2 6 15 2 )z
This is known as log-generalized ABSN(, ) distribution.
2(y)( y) If 0 , then we get f (z;) . z This is known as log SN () distribution.
If 0 , then we get the standard log-normal distribution and is given by
(y) f (z) z
If , then we get the log-generalized bimodal normal ( LGBN (2) ) distribution given by
y 2 f (z;) (y)( y) z
If , then we get the log-generalized bimodal normal ( LGBN (6) ) distribution given by
y6 f (z;) (y) ( y) 15z If Z ~ LGABSN(, ,) , then Z ~ LGABSN(, ,) .
5.2. Location Scale Extension The location and scale extension of GABSN(, ,) distribution is as follows. If
Z ~ GABSN(, , ) then Y Z is said to be the location ( ) and scale ( ) extension of
Z and has the density function is given by
3 2 z z 1 1 z z , (14) fZ (z;,,,,) C(,,)
11 where, (z,, ,,) R, and 0 . We denote it byY ~ GABSN (,,,, ) .
6. Parameter Estimation and Applications 6.1. Maximum Likelihood Estimation
Let y1, y2 ,..., yn be a random sample from the distribution of the random variable
Y ~ GABSN (,,,, ) so that the log-likelihood function for (, ,, , ) is given by
3 2 n y y n n (y ) n (y ) l() log 1 i i 1 n logC(,,) nlog( ) log(2 ) i 2 log i i1 2 i1 i1 (15) Taking Partial derivatives of eqn. (15) w.r.t. the parameters, the following normal equations are obtained:
3 (y ) 2 2 D i 3 l( ) n n 2 n (y ) W i 2 i1 1 D i1
(y ) 3(y )3 2D i i 2 4 l() n n (y ) n 2 n (y ) i (y )W i 2 2 2 i i1 i1 1 D i1 l() n( 3 b ) n 2 D y i 2 C(, ,) i1 1 D
b (3 22 ) n3 15 l( ) (1 2 ) n 2 D y 3 i 3 2 C(, ,) i1 1 D
b( 3 2 ) n l() (1 2 )5/ 2 2 n (y ) i (yi )W C(,,) i1
3 (y ) (y ) (.) where, D 1 i i and W (.) . 3 (.) Solving simultaneously of the above equations is not mathematically tractable so one should apply some numerical optimization routine to get the solutions. 6.2. Real Life Applications Here we have considered two datasets which are related to N latitude degrees in 69 samples from world lakes, which appear in Column 5 of the Diversity data set in website: http://users.stat.umn.edu/sandy/courses/8061/datasets/lakes.lsp and the white cells count (WCC) of 202 Australian athletes, given in Cook and Weisberg (1994) for the purpose of
12 data fitting. Using GenSA package in R we have fitted our proposed distribution, i.e.,
GABSN(,,,, )along with the normal N(, 2 ) distribution, the logistic LG(, ) distribution, the Laplace La (, ) distribution, the skew-normal SN(, , ) distribution of
Azzalini (1985), the skew-logistic SLG(,, ) distribution of Wahed and Ali (2001), the skew-Laplace SLa(,,) distribution of Nekoukhou and Alamatsaz (2012), the alpha-skew- normal ASN(, , ) distribution of Elal-Olivero (2010), the alpha-skew-Laplace ASLa(, , ) distribution of Harandi and Alamatsaz (2013), the alpha-skew-logistic ASLG(, , ) distribution of Hazarika and Chakraborty (2014), the alpha-beta-skew-normal ABSN(, , , ) distribution and beta-skew-normal BSN(, , ) distribution of Shafiei et al.
(2016), and the generalized alpha-skew-normal GASN(,,, ) distribution of Sharafi et al. (2017) for comparison purpose. The values of the MLE’s of the parameters for different distributions along with log- likelihood, AIC and BIC are given in Tables 1 and 2. . Table 1: MLE’s, log-likelihood, AIC and BIC for N latitude degrees in 69 samples from world lakes. Parameters log L AIC BIC Distribution N(, 2 ) 45.165 9.549 ------253.599 511.198 515.666 LG(, ) 43.639 ------4.493 -246.65 497.290 501.758 SN(, , ) 35.344 13.7 3.687 -- -- -243.04 492.072 498.774 BSN(, , ) 54.47 5.52 -- -- 0.74 -242.53 491.060 497.760 SLG(,, ) 36.787 -- 2.828 -- 6.417 -239.05 490.808 490.808 La (, ) 43.0 ------5.895 -239.25 482.496 486.964 ASLG(, , ) 49.087 -- -- 0.861 3.449 -237.35 480.702 487.404 SLa(,,) 42.3 -- 0.255 -- 5.943 -236.90 479.799 486.501 ASLa(, , ) 42.3 -- -- -0.22 5.44 -236.08 478.159 484.861 ASN(, , ) 52.147 7.714 -- 2.042 -- -235.37 476.739 483.441 ABSN(, , , ) 53.28 9.772 -- 2.943 -0.292 -234.36 476.719 485.655 GASN(,,, ) 56.319 8.544 -0.672 12.052 -- -230.531 469.062 477.998 GABSN(, ,, , ) 58.333 6.489 -0.616 -6.144 -5.870 -225.369 460.738 471.909
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Figure 3: Plots of observed and expected densities of some distributions for N latitude degrees in 69 samples from world lakes.
Table 2: MLE’s, log-likelihood, AIC and BIC for white cells count (WCC) of 202 Australian athletes. Parameters log L AIC BIC Distribution 2 N(, ) 7.109 1.796 ------404.919 813.838 820.455 La (, ) 6.844 ------1.38 -407.142 818.284 824.901 ASLa(, , ) 6.4 -- -- -0.265 1.263 -400.992 807.984 817.909 LG(, ) 6.996 ------0.991 -401.612 807.224 813.841 SLa(,,) 6.4 -- 0.762 -- 1.287 -400.366 806.732 816.657 BSN(, , ) 6.813 1.687 -- -- -0.061 -399.638 805.276 815.201 ASLG(, , ) 6.413 -- -- -0.21 0.949 -398.943 803.886 813.811 ASN(, , ) 8.195 1.684 -- 0.874 -- -398.393 802.786 812.711 GASN(,,, ) 5.569 2.689 2.242 0.5115 -- -395.545 799.090 812.323 SN(, , ) 5.105 2.691 2.729 -- -- -396.161 798.322 808.247 SLG(,, ) 5.512 -- 1.736 -- 1.331 -395.897 797.794 807.719 ABSN(, , , ) 7.758 1.796 -- 0.813 -0.128 -394.833 797.666 810.899 GABSN(, ,, , ) 9.729 1.508 -0.669 0.411 0.428 -392.788 795.576 812.117
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Figure 4: Plots of observed and expected densities of some distributions for white cells count (WCC) of 202 Australian athletes.
From Tables 1 and 2, it is seen that the proposed generalized alpha beta skew normal GABSN(, ,, , ) distribution provides better fit to the data set under consideration in terms of all criteria, namely the log-likelihood, the AIC as well as the BIC. The plots of observed (in histogram) and expected densities (lines) presented in Figure 3 and 4, also confirms our finding.
6.2.1. Likelihood Ratio test:
Since N(, 2 ), SN(, , ), ABSN(, , ,) , GASN(,, , ) , GBSN(,, , ) and
GABSN(, ,, , ) distributions are nested models, the likelihood ratio (LR) test is used to discriminate between them. The LR test is carried out to test the following hypothesis as shown in Table 3 below. Table 3: The values of LR test statistic for different hypothesis. LR values Hypothesis Dataset 1 Dataset 2
H 0 : 0 vs H1 : 0 10.324 5.514
H 0 : 0vs H1 : 0 17.982 4.09
H0 : 0 vs H1 : 0 35.342 6.746
H 0 : 0 vs H1 : 0 56.460 24.262
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Since all the values of LR test statistic for different hypothesis exceed the 95% critical value that is, 3.841. Thus there is evidence in favors of the alternative hypothesis that the sampled data comes fromGABSN(, ,, , ) distribution. 7. Conclusions In this article the generalized-alpha-beta-skew-normal distribution which includes at most four modes is introduced and some of its basic properties are investigated. Some extensions of the proposed distribution along with some of their properties are discussed. The numerical results of the modelling of two real life data set considered here has shown that the proposedGABSN(, ,, , ) distribution provides better fitting in comparison to the other known distributions.
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