Extended skew generalized normal distribution

METRON - International Journal of Statistics2011, vol. LXIX, n. 3, pp. 265-278

KANAK CHOUDHURY – MOHAMMAD ABDUL MATIN

Extended skew generalized normal distribution

Summary - The skew normal distributions’ family represents an extension of thenormal family to which a parameter (λ) has been added to regulate the skewness ofthe distribution. In recent years, not only the skewness but the kurtosis is also ofmore concern in representing the features of the distribution. In this study a moreflexible distribution, extended skew generalized normal distribution, is developed torepresent the skewness as well as the kurtosis. This distribution is potentially usefulfor the data that has more skewness and kurtosis. Some statistical properties of thedistribution have also been studied.

Key Words - Skewness; Kurtosis; Skew normal; Skew curved normal; Skew general-ized normal.

1. Introduction

In statistical literature a general tendency is observed towards more flexiblemethods in representing the features of data as adequately as possible and inreducing the unrealistic assumptions. For continuous observations within aparametric approach one very important aspect is the normality assumptionthat underlines the most of the methods of analysis. However, in real life, weare very frequently dealing with phenomena whose outcomes behave in a verynon-normal fashion.

To ameliorate such a situation Azzalini (1985) introduced the followinglemma that is central to the development of skewed distributions.

Lemma 1. If f0 is a one-dimensional probability density function symmetric about0, and G is a one-dimensional distribution function such that G ′ exists and is adensity symmetric about 0, then

f (z) = 2 f0 (z) G {w (z)} ; −∞ < z < ∞, (1)

is a density function for any odd function w (·).Received May 2009 and revised August 2011.

266 KANAK CHOUDHURY – MOHAMMAD ABDUL MATIN

If f0 and G be the density and distribution function of standard normalrandom variable z respectively then (1) becomes

f (z) = 2φ (z)� {λz} ; −∞ < z < ∞, (2)

where w (z) is considered as λz, λ is a constant. This is called skew normaldistribution that contains the normal distribution as a special case. Thus theskew normal distributions’ family represents an extension of the normal familyto which a parameter has been added to regulate the skewness of the distribution.

The introduction of the skew normal distribution led Azzalini (1986), Henze(1986), Azzalini and Dalla-Valle (1996), Branco and Dey (2001), Arnold andBeaver (2002), Arellano-Valle et al. (2004) and others to study the distribution.In an oblique reference to these studies, Matin and Bagui (2008) have studiedthe small-sample behavior of different estimators of the logistic regression modelparameters with skew-normally distributed explanatory variables.

One remarkable property of the skew normal distribution is that as theskewing parameter tends to infinity the distribution behaves like a half normaldistribution. In practice, for moderate values of the skewing parameter, nearlyall probabilities gather either on the positive or negative numbers that determinedby the sign of the skewing parameter (Arellano-Valle et al. 2004). To mitigatesuch a limitation Arellano-Valle et al. (2004) introduced a class of distributions,skew generalized normal (SGN) distribution, which includes the skew normaldistribution and the normal distribution as a special case.

In recent years, not only the skewness but the kurtosis is also of moreconcern in representing the features of a distribution. In this study a moreflexible distribution, extended skew generalized normal (ESGN) distribution, isdeveloped to represent the skewness as well as the kurtosis. This is an extensionof the skew generalized normal distribution that includes the SGN, SN andnormal distributions. In addition to a skewing parameter it also includes twomore extra parameters which can be explained as the parameters that incorporatethe kernel of the second and fourth moment of a random variable.

Gomez et al. (2007) developed a general family of skew symmetric dis-tributions which were generated by the cumulative distribution function of thenormal distribution. In this family of distribution, it is seen that a significantimprovement for asymmetry and kurtosis parameters has been found for theother skew symmetric distributions (skew t5-normal, skew Laplace-normal etc.).However, no improvement has been found for the skew normal-normal distribu-tion over the skew normal distribution for asymmetry and kurtosis parameters.In reality, skew normal-normal distribution and skew-normal distribution bothare the same.

In Section 2, we have defined the probability density function of the ex-tended skew generalized normal distribution along with its very simple prop-erties. Section 3 contains some important properties that are related to the

Extended skew generalized normal distribution 267

half normal and chi-squared distribution. Moments, skewness, kurtosis andlocation-scale extension of the distribution are also introduced in this section.In addition, the Fisher’s Information matrix has been discussed. Some condi-tional and prior distributions are discussed in Section 4. Finally, in order tohave a better understanding of the extended skew generalized normal density,numerical data illustrations are given in Section 5.

2. Extended skew generalized normal distribution

Definition 1. A random variable X has the extended skew generalized normaldistribution if its density is given by

f (x |λ1, λ2, λ3) = 2φ (x)�(

λ1x√1 + λ2x2 + λ3x4

), x ∈ R, (3)

where φ (·) and �(·) are the probability density function and cumulative prob-ability density function of normal distribution respectively, λ1 ∈ R, λ2 � 0, andλ3 � 0. This density is denoted by ESGN (x; λ1, λ2, λ3).

For any x > 0, we have

P (|X | � x)=∫ x

−x2φ (u)�

(λ1u√

1 + λ2u2 + λ3u4

)du

=∫ x

02φ (u)�

(λ1u√

1 + λ2u2 + λ3u4

)du

−∫ 0

x2φ (−u)�

(− λ1u√

1 + λ2u2 + λ3u4

)du

=∫ x

02φ (u)

[�

(λ1u√

1+λ2u2+λ3u4

)+ �

(− λ1u√

1 + λ2u2 + λ3u4

)]du

=∫ x

02φ (u) du = P (|Z | � x) .

Moreover, the above identity for x = ∞ shows that (3) is, indeed, abona fide probability density function. Figure 1 illustrates some of the possibleshapes of (3) under various choices of (λ1, λ2, λ3).


Den

sity

Den

sity

0.5

0.4

0.3

0.2

0.1

0.0

0.5

0.4

0.3

0.2

0.1

0.0

−4 −2 0 2 4 −4 −2 0 2 4x x

(a) (b)

Figure 1. Examples of the ESGN Distribution: Density Curve for (a) (λ1, λ2, λ3) = (1, 1, 1) (SolidLine) and (λ1, λ2, λ3) = (−1, 1, 1) (Dashed Line), (b) (λ1, λ2, λ3) = (1, 3, 3) (Solid Line)and (λ1, λ2, λ3) = (−1, 3, 3) (Dashed Line).

Some of the basic properties of the ESGN density arise directly from (3).

Properties 1.

1. f (x |λ1 = 0, λ2, λ3) = φ (x) for all x ∈ R, λ2 � 0, and λ3 � 0.2. For λ3 = 0 for all x ∈ R the ESGN distribution becomes the SGN distri-

bution.3. f (x |λ1, λ2 = 0, λ3 = 0) = 2φ (x)� (λ1x) for all x ∈ R i.e. the ESGN

distribution becomes the SN distribution.4. If X ∼ ESGN (λ1, λ2, λ3) then −X ∼ ESGN (−λ1, λ2, λ3).5. f (x | − λ1, λ2, λ3) + f (x |λ1, λ2, λ3) = 2φ (x) for all x ∈ R.6. limλ1→+∞ f (x |λ1, λ2, λ3) = 2φ (x) I {x�0} and limλ1→−∞ f (x |λ1, λ2, λ3)=

2φ (x) I {x�0} for all λ2, λ3.

The cumulative distribution function of the ESGN (λ1, λ2, λ3) can be ex-pressed as

F (x |λ1, λ2, λ3) =∫ x

−∞2φ (t)�

(λ1t√

1 + λ2t2 + λ3t4

)dt, (4)

that immediately leads to the following properties:

Properties 2.

1. F (x |λ1 = 0, λ2, λ3) = �(x) for all x ∈ R, λ2 � 0 and λ3 � 0.2. F (−x |λ1, λ2, λ3) = 1 − F (x | − λ1, λ2, λ3).

Note that, in studying any distribution, one of the important matters isrelated with the generation of random samples from that distribution. TheESGN distribution is a log-concave distribution (Caramanis and Mannor 2007).So, random samples can be generated from this distribution by using adaptiverejection sampling of Gilks and Wild (1992).


3. Some important properties

3.1. Transformed distribution

Like as the Azzalini’s (1985) SN the absolute value of the ESGN randomvariable follows a half-normal distribution. In particular, this implies that all themoments of the ESGN distribution are finite while the even moments coincidewith those of the standard normal distribution.

Proposition 1. Let X ∼ ESGN (λ1, λ2, λ3) and Y ∼ N (0, 1). Then |X | and |Y |are identically distributed that is |X | d= |Y | ∼ H N (0, 1) where H N (0, 1) denotesthe standard half-normal distribution.

Proof. We know that Z = |Y | has density 2φ (z) I {z > 0} where I {z > 0} isan indicator variable. By (3) the density of W = |X | is given by

fW (w)= fX (w) + fX (−w)

= 2φ (w)

[�

(λ1w√

1 + λ2w2 + λ3w4

)+ �

(− λ1w√

1 + λ2w2 + λ3w4

)]

= 2φ (w)

[�

(λ1w√

1 + λ2w2 + λ3w4

)+ 1 − �

(λ1w√

1 + λ2w2 + λ3w4

)]= 2φ (w) for w > 0,

and since fW (w) = 0 for w � 0, we have fW (w) = fz (w) for all w ∈ R.

The following Proposition 2 is a direct consequence of Proposition 1. Thesquare of a standard normal variate follows a chi-square distribution with 1degree of freedom. Similar property holds for the skew normal density (seeAzzalini 1985); which is also true for the ESGN distribution.

Proposition 2. Let X ∼ N (0, 1), Y ∼ ESGN (λ1, λ2, λ3) and let h be an evenfunction. Then h (X) and h (Y ) are identically distributed. In particular, Y 2 ∼χ2 (1).

A small-scale simulation experiment has been conducted where randomsample has been generated from ESGN distribution for fixed values of λ1, λ2

and λ3 using adaptive rejection sampling procedure. To find the degrees offreedom of χ2-distribution, the log-likelihood function of χ2-distribution hasbeen fitted for the squared sample. Since the parameter of the chi-square distri-bution, that is, the degrees of freedom is a discrete value, we have consideredthe maximum value of the likelihood function for those degrees of freedomwhich have integer value. It is repeated 500 times for fixed values of λ1, λ2

and λ3 then taking average value of the degrees of freedom of χ2-distribution,it is found that the degree of freedom is 1 (one). The similar result has beenfound for different values of λ1, λ2 and λ3 and different sizes of samples.


3.2. Moments, skewness and kurtosis

Taking h (x) = xk for k even in Proposition 2 indicates that the evenmoments of ESGN and standard normal distributions are identical. It impliesthe existence of the odd moments however no explicit expressions are availablefor them. The odd moments are obtained by using the following formula:

E(X2k+1

)=∫ ∞

−∞2x2k+1φ (x)�

(λ1x√

1 + λ2x2 + λ3x4

)dx . (5)

The main advantage of ESGN distribution is that it not only takes intoaccount the skewness but also the kurtosis. For X ∼ SN (λ) the skewness andkurtosis are ranges in between

−.9953 �√

β1 � .9953 and 3.000 � β2 � 3.8692.

For skew curved normal distribution the skewness and kurtosis are ranges inbetween (Arellano-Valle et al. 2004)

−.3760 �√

β1 � .3760 and 3.000 � β2 � 4.332.

We have found, for ESGN distribution, the skewness and kurtosis are rangesin between

−.9953 �√

β1 � .9953 and 3.000 � β2 � 7.2377.

It is thus observed that the ESGN distribution has the identical skewnessas of SN however, has wider range of kurtosis compared to the SN as well asof the SGN. The upper limit of the kurtosis of the ESGN distribution is muchhigher than the kurtosis of the SGN distribution for λ1 which becomes equalfor larger λ1 (Figure 2). The lower limit of the kurtosis of the SGN and theESGN distributions are almost equal for λ1. For any λ2, as λ3 increases thekurtosis of the ESGN distribution increases. Again, for any λ3, as λ2 increasesthe kurtosis of the ESGN distribution decreases. However, for λ3 equals tozero, the kurtosis of the ESGN distribution increases as λ2 increases.

The Fisher’s information matrix. The Fisher’s information matrix for thedistribution of the random variable X ∼ ESGN (λ1, λ2, λ3) can be written as

I (θ) =( Iλ1λ1 Iλ1λ2 Iλ1λ3

Iλ2λ2 Iλ2λ3Iλ3λ3

),


0

1

2

3

4

0 25 50 75 100−100 −75 −50 −2

Kur

tosi

s

Lamda 1

Maximum Kurtosis of SGN for lamda 1

Minimum Kurtosis of SGN for lamda 1

Maximum Kurtosis of ESGN for lamda 1

Minimum Kurtosis of ESGN for lamda 1

Figure 2. Kurtosis of ESGN and SGN Distributions for λ1.

where θ = (λ1, λ2, λ3). Algebraic operations led to the following expressionsfor the elements of I (θ)

Iλ1λ1 = 2∫ ∞

−∞x2(

1 + λ2x2 + λ3x4)�(x)�2 (K1)/φ (K1)dx

Iλ2λ2 = λ21

2

∫ ∞

−∞x6(

1 + λ2x2 + λ3x4)3 �(x)�2 (K1)/φ (K1)dx

Iλ3λ3 = λ21

2

∫ ∞

−∞x10(

1 + λ2x2 + λ3x4)3 �(x)�2 (K1)/φ (K1)dx

Iλ1λ2 = −∫ ∞

−∞x3�(x)(

1 + λ2x2 + λ3x4)2

�(K1)

φ (K1)

·[φ (K1)

√1 + λ2x2 + λ3x4 + λ1x (φ (K1) − �(K1))

]dx

Iλ1λ3 = −∫ ∞

−∞x5�(x)(

1 + λ2x2 + λ3x4)2

�(K1)

φ (K1)

·[φ (K1)

√1 + λ2x2 + λ3x4 + λ1x (φ (K1) − �(K1))

]dx

Iλ2λ3 = 1

2

∫ ∞

−∞λ1x7�(x)(

1 + λ2x2 + λ3x4)3

�(K1)

φ (K1)

·[3φ (K1)

√1 + λ2x2 + λ3x4 + λ1x (φ (K1) − �(K1))

]dx

where K1 = λ1x/√

1 + λ2x2 + λ3x4. No explicit expressions are possible forthe elements of I (θ). However, it is possible to find the value of integralsnumerically. We have found that the above Fisher’s information matrix is not


singular for non-zero λ1. If λ1 = 0, the Fisher’s information matrix will besingular as the skew normal distribution (Azzalini, 1985).

3.3. Location-scale extension

The basic principle used in Definition 1 can also be extended for location-scale family.

Definition 2. The location-scale extended skew generalized normal variable isdefined as Z = μ + σ X , where X ∼ ESGN (λ1, λ2, λ3), μ ∈ R and σ > 0. Itsdensity function is given by

f (z|θ) = 2

σφ

(z − μ

σ

)�

⎛⎜⎜⎜⎜⎝

λ1(z − μ)/σ√1 + λ2

(z − μ

σ

)2

+ λ3

(z − μ

σ

)4

⎞⎟⎟⎟⎟⎠ , (6)

where φ (·) and �(·) are the probability density function and cumulative dis-tribution function of the normal density respectively and θ = (μ, σ, λ1, λ2, λ3).This density is denoted by Z ∼ ESGN (μ, σ ; λ1, λ2, λ3). For λ1 = 0, the familyof densities (6) becomes N

(μ, σ 2

). For λ3 = 0, (6) coincides with the skew

generalized normal density. Again, for λ2 = 0 and λ3 = 0, (6) coincides withSN

(μ, σ 2, λ1

). Note that since the model depends on three parameters and

for the location-scale extension it depends on five parameters, we need largesample to estimate the parameters with reasonable precision.

4. Some conditional and prior distributions

In Bayesian inference prior probabilities of the parameters that helps tomake inference about the parameters is considered. In the following, we willshow that the inability of knowledge of parameters λ2, λ3 will not influencethe distribution of X where X ∼ ESGN (λ1, λ2, λ3). We will also present theprior distribution of λ1 that is obtained by the Jeffreys’ prior concept.

Proposition 3. Let

Z |(Y1 = y1,Y2 = y2,Y3 = y3)∼ ESGN (y1,y2,y3)=2φ(z)�

(y1z√

1 + y2z2+y3z4

),

where the joint distribution of (Y1,Y2,Y3) is defined as Y1|Y2,Y3 ∼ N(

0, y2y3

),

Y2|Y3 ∼ fY2|Y3 (y2) and Y2 ∼ fY3 . Then Z and Y2,Y3 are independent, whereZ ∼ N (0, 1).


Proof. We have,

f (Z ,Y1|Y2,Y3) = 2φ (z)�

(y1z√

1 + y2z2 + y3z4

)√y3

y2φ

(y1√y2/y3

)

⇒ f (Z |Y2,Y3) = φ (z)∫ ∞

−∞2�

(y1z√

1 + y2z2 + y3z4

)√y3

y2φ

(y1√y2/y3

)dy1

= φ (z)∫ ∞

−∞2φ (t)�

(t z

√y2/y3√

1 + y2z2 + y3z4

)dt

= φ (z)

since the conditional distribution of Z given Y2, Y3 is only a function of Z , soby the definition of conditional distribution Z is independent of Y2, Y3.

From the Proposition 3 following results can be found

f (Y1|Z ,Y2,Y3) =2φ (z)�

(y1z√

1 + y2z2 + y3z4

)√ y3

y2φ

⎛⎝ y1√

y2/y3

⎞⎠

φ (z)

= 2

√y3

y2φ

⎛⎝ y1√

y2/y3

⎞⎠�

(y1z√

1 + y2z2 + y3z4

)

= 2

√y3

y2φ

⎛⎝ y1√

y2/y3

⎞⎠�

⎛⎝ z

√y2/y3√

1 + y2z2 + y3z4

y1√y2/y3

⎞⎠ .

Therefore, f (Y1|Z ,Y2,Y3) ∼ SN(μ = 0, σ =

√y2y3

, λ = z√

y2/y3√1+y2z2+y3z4

).

So, the conditional distribution of Y1 given Z ,Y2,Y3 follows the skew-

normal distribution with location 0, scale√

y2/y3, and shape

(z√

y2/y3

)/√

1 + y2z2 + y3z4. Now,

f (Y1,Y2|Z ,Y3)=2√

y3/y2

φ

⎛⎝ y1√

y2/y3

⎞⎠�

⎛⎝ z

√y2/y3√

1 + y2z2 + y3z4

y1√y2/y3

⎞⎠ fY2|Y3(y2)

⇒ f (Y1|Z ,Y3)

=∫ ∞

02√

y3/y2

φ

⎛⎝ y1√

y2/y3

⎞⎠�

⎛⎝ z

√y2/y3√

1 + y2z2 + y3z4

y1√y2/y3

⎞⎠ fY2|Y3 (y2) dy2.


This proof makes it clear that the inability of learning about Y2 and Y3 fromZ is due precisely to their independence.

Note that it is assumed that Y1 has a zero mean normal distribution withconditional priors fY1|Y2,Y3 (y1) which depends on y2 and y3 through the scaleparameter. In fact, in this case Y2 is also conditional on the prior Y3.

The Jeffreys’ prior of λ1. A probability distribution proportional to the squareroot of the Fisher’s information is known as Jeffreys’ prior (Everitt 1999).Jeffreys (1948) states that if nothing is known about the parameter (say θ) withfinite range the prior distribution should be proportional to dθ . In particular, (i)if θ ranges from –∞ to +∞ the prior probability is still taken as proportional todθ ; (ii) if θ ranges from 0 to ∞ the prior distribution is taken as proportionalto dθ

/θ (Kendall and Stuart 1973).

On this concept, the Jeffreys’ prior for λ1 associated with the ESGN dis-tribution is given by

q j (λ1) ∞I 1/2 (λ1) (7)

where

I (λ1) = Eλ1

(∂

∂λ1ln f (z; λ1, λ2, λ3)

)2

= Eλ1

(∂

∂λ1ln �

(λ1x√

1 + λ2x2 + λ3x4

))2

= Eλ1

⎛⎜⎜⎜⎝ x2(

1 + λ2x2 + λ3x4) φ2

(λ1x√

1 + λ2x2 + λ3x4

)

�2

(λ1x√

1 + λ2x2 + λ3x4

)⎞⎟⎟⎟⎠ .

Therefore,

I (λ1) = 2∫ ∞

−∞x2(

1 + λ2x2 + λ3x4)φ (x)

φ2(

λ1x√1 + λ2x2 + λ3x4

)

�

(λ1x√

1 + λ2x2 + λ3x4

) dx

represents the Fisher’s expected information which (I (λ1)) cannot be writtenin closed form however, the following property of I (λ1) may hold. If λ2 =λ3 = 0, Jeffeys’ prior for λ1 coincide with Jeffreys’ prior for the skew normaldistribution which is derived by Liseo and Loperfido (2006).

Proposition 4. q j (λ1) is symmetric about λ1 = 0.


Proof. We have,

I (λ1) = 2∫ ∞

−∞x2(

1 + λ2x2 + λ3x4)φ (x)

φ2(

λ1x√1 + λ2x2 + λ3x4

)

�

(λ1x√

1 + λ2x2 + λ3x4

) dx

= 2∫ ∞

0

x2(1 + λ2x2 + λ3x4

)φ (−x)φ2( −λ1x√

1 + λ2x2 + λ3x4

)

�

( −λ1x√1 + λ2x2 + λ3x4

) dx

+ 2∫ ∞

0

x2(1 + λ2x2 + λ3x4

)φ (x)φ2(

λ1x√1 + λ2x2 + λ3x4

)

�

(λ1x√

1 + λ2x2 + λ3x4

) dx

= 2∫ ∞

0

x2(1 + λ2x2 + λ3x4

)

·φ (x) φ2

( −λ1x√1 + λ2x2 + λ3x4

)

�

(λ1x√

1 + λ2x2 + λ3x4

){1 − �

(λ1x√

1 + λ2x2 + λ3x4

)}dx

= I (−λ1) .

Proposition 5. q j (λ1) is decreasing in |λ1|.Proof. The monotonicity of q j (·) for λ1 > 0 can be proved by obtaining thefirst derivative;

∂

∂λ1I (λ1)=−2

∫ ∞

0

x3φ(x)φ2(

λ1x√1+λ2x2+λ3x4

)(1 + λ2x2 + λ3x4

)3/2

[2λ1x�

(λ1x√

1 + λ2x2 + λ3x4

)

·{

1 − �

(λ1x√

1 + λ2x2 + λ3x4

)}]

+ φ

(λ1x√

1 + λ2x2 + λ3x4

)[1 − 2�

(λ1x√

1 + λ2x2 + λ3x4

)]dx .

Now, in order to prove that ∂∂λ1

I (λ1) is negative, it suffices to show that forall s > 0, 2s�(s)� (−s) + φ (s) (1 − 2�(s)) > 0. Since, for all s > 0,s�(−s) < φ (s), the derivative of I (λ1) with respect to λ1 will be negativefor all x > 0. Similarly, for all s < 0, 2s�(s)� (−s) + φ (s) (1 − 2�(s)) < 0since, for all s < 0, s�(−s) > φ (s), so the derivative of I (λ1) with respectto λ1 will be negative for all x < 0 which proves the Proposition 6.


5. Numerical illustration

Data on several biomedical variables from 202 athletes have been collectedat the Australian Institute of Sport (see Cook and Weisberg (1994) for moredetails about the data), which are available for download at http://azzalini.stat.unipd.it/SN/index.html. These data contain 11 variables for each 102 male and100 female athletes. The data concerning the percentage of hemoglobin cell inblood of 102 male athletes is considered here. It indicates that the kurtosis ofthe data is very high (Table 1).

Table 1: Summary Statistics for Percentage of Hemoglobin Cell in Blood of Male AustralianAthletes.

n Z S2 √b1 b2

102 15.553 0.873 0.960 5.208

As we have mentioned in Section 3 that the extended skew generalizednormal distribution has wider kurtosis range the ESGN distribution should fitthis data more appropriately. The parameters are estimated by numericallymaximizing the likelihood function,

ln L (θ) = −n

2ln

πσ 2

2− 1

2σ 2

n∑i=1

(zi − μ)2

+n∑

i=1

ln �

⎡⎢⎢⎣ λ1(zi − μ)

/σ√

1 + λ2 (zi − μ)2/σ 2 + λ3 (zi − μ)4

/σ 4

⎤⎥⎥⎦,

with respect to the components of θ = (μ, σ, λ1, λ2, λ3).The log-likelihood value of the ESGN distribution is the highest compared

to other distributions (Table 2) while the AIC value is the smallest. Thisindicates that the ESGN distribution fits the data more appropriately.

Table 2: Maximum Likelihood Parameter Estimates for Percentage of Hemoglobin in Blood ofAustralian Male Athletes Data under Normal, Skew Normal, Skew Generalized Normal and ESGNDistribution.

Parameter N SN SGN ESGN

μ 15.553 14.600 16.164 16.134σ 2 1.000 1.772 1.237 1.202λ1 2.219 −3.696 −2.507λ2 9.644 0.212λ3 2.568

log-likelihood −137.809 −132.894 −130.728 −129.431

AIC 279.618 271.787 269.456 268.861


Prob

abili

ty

0.5

0.4

0.3

0.2

0.1

0.0

14 15 16 17 18 19

ESGNSGNSNNormal

x

Figure 3. Histogram of Percentage of Hemoglobin Cell in Blood of 102 Australian Male Athletes. LinesRepresent Distributions Fitted Using Maximum Likelihood Estimation: ESGN (Black SolidLine); Normal (Blue Dotted Dashed Line); Skew Normal (Red Dashed Line); and Skew Gen-eralized Normal (Turquoise Dashed Line).

It is obvious that fitting any models other than ESGN to these data would beinadequate due to the sample kurtosis value. These features are also illustratedin Figure 3, where a histogram of the data is plotted jointly with the extendedskew generalized normal (ESGN) distribution, skew generalized normal (SGN),skew normal (SN) and normal models.

The asymptotic likelihood ratio tests have been conducted to compare theESGN model against the sub-models mentioned above which concludes thatthe ESGN model is the most appropriate for the particular example analyzedhere.

6. Concluding remarks

The flexibility, increased skewness and kurtosis ranges were able to capturethe features of the data that other distributions missed. The ESGN distributionis found to be the most appropriate distribution for the specific data analyzed.

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KANAK CHOUDHURYDepartment of StatisticsUniversity of ChittagongChittagong-4331 (Bangladesh)[email protected].

MOHAMMAD ABDUL MATINDepartment of StatisticsJahangirnagar UniversitySavar, Dhaka-1342 (Bangladesh)[email protected].

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Extended skew generalized normal distribution