Proceedings On Some New Classes of Pearsonian Distributions

Section on Physical and Engineering Sciences – JSM 2011

1 On Some New Classes of Pearsonian Distributions 1

M. Shakil1, B. M. G. Kibria2, J. N. Singh3 Miami Dade College, Hialeah Campus, Hialeah, FL 33012, USA 2 Florida International University, Miami, Florida 33199, USA 3 Barry University, Miami Shores, FL 33161, USA

Abstract This paper develops some new classes of continuous probability distributions based on the generalized Pearson differential equation. Some characteristics of the new distributions are defined. Some distributional relationships are established. It is hoped that the proposed attempt will be helpful in designing a new approach of unifying different families of distributions based on generalized Pearson differential equation. Key Words: Burr distributions, generalized inverse Gaussian distributions, Pearson generalized differential equation. 1. Pearson Distribution Various systems of distributions have been constructed to provide approximations to a wide variety of distributions, see, e.g., Johnson et al. (1994). These systems are designed with the requirements of ease of computations and feasibility of algebraic manipulations. To meet the requirements, there must be as few parameters as possible in defining a member of the system. One of these systems is the Pearson system. A continuous distribution belongs to this system if its pdf (probability density function) f satisfies a differential equation of the form

xa 1 df x   , 2 f x  dx b x  cx  d

(1)

where a , b , c , and d are real parameters such that f is a pdf . The shapes of the pdf depend on the values of these parameters, based on which Pearson (1895, 1901) classified these distributions into a number of types known as Pearson Types I – VI. Later in another paper, Pearson (1916) defined more special cases and subtypes, known as Pearson Types VII - XII. Many well-known distributions are special cases of Pearson Type distributions which include Normal and Student t distributions (Pearson Type VII), Beta distribution (Pearson Type I), Gamma distribution (Pearson Type III), among others. 2. Generalized Pearson Distribution In recent years, some researchers have considered a generalization of (1), known as generalized Pearson differential equation (GPE), given by j m 1 df x   j 0 a j x  n , j f x  d x  j 0 b j x

(2)

where m , n  N /0 and the coefficients a j and b j are real parameters. The system of continuous univariate pdf  s

generated by GPE is called a generalized Pearson

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2 system, which includes a vast majority of continuous pdf  s , by proper choices of these parameters. For example:  Roy (1971) studied GPE, when m  2, n  3, b0  0 , to derive five frequency curves whose parameters depends on the first seven population moments.  Dunning and Hanson (1977) used GPE in his paper on generalized Pearson distributions and nonlinear programming.  Cobb et al. (1983) extended Pearson's class of distributions to generate multimodal distributions by taking the polynomial in the numerator of GPE of degree higher than one, and the denominator, say, v x  , having one of the following forms: (I) (II) (III) (IV)

v x   1,    x   , v x   x , 0  x   , v x   x 2 , 0  x   , v x   x 1  x , 0  x  1 .

 Chaudhry and Ahmad (1993) studied another class of generalized Pearson's distributions when

m  4, n  3, b0  b1  b2  0,

a a4   2 , 0  2  , b 3  0 . 2 b3 2 b3

 Rossani and Scarfone (2009) have studied GPE in the following form

a  a1 x  a2 x 2 1 d f x   0 , f x  d x b0  b1 x  b2 x 2 and used it to generate generalized Pearson distributions in order to study charged particles interacting with an electric and/or a magnetic field.  Recently, Shakil and Kibria (2010), and Shakil, Kibria and Singh (2010a, b) have defined some new classes of generalized Pearson distributions for different choices of the parameters m , n  N /0 and the coefficients a j and b j in GPE.

3. Some New Classes of Generalized Pearson Distribution We give 3 new different classes of distributions as solutions of the GPE (2). The derivations of the proposed distributions need the applications of many special functions and formulas, for which the interested readers are referred to Abramowitz & Stegun (1970), and Gradshteyn & Ryzhik (2000), and Chaudhry and Zubair (2002), among others.

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3 3.1 Generalized Pearson Distribution - I: We consider the following differential equation

df X ( x)  a0  a1 x  a 2 x 2   f X ( x), b1  0 ,   dx b x 1  

(3)

which is a special case of the GPE (2) when m  2, n  1 , and b0  0 . The solution to the differential equation (3) is given by





(4) f X ( x)  C x exp   x 2   x ,   0,   0,   0, x  0 , a a a where    2 ,   0 ,    1 , b1  0 , and C is the normalizing constant 2 b1 b1 b1 given by

C

 2      1 / 2     1  exp   2 /  8    D (  1)   / 

2



,   0,   0,   0 , (5)

where D p (z ) denotes the parabolic cylinder function. The cdf of the Pearsonian model (4) is given by

  1 k    k   2      2 k  1,  x  , FX ( x )     1    (6) k!     k 0     where     2 k  1,  x  denotes the incomplete gamma function. For simplicity, taking   1 /  2 2  in (4), the possible shapes of the p.d.f. are provided for some C



selected values of the parameters in Figure 1. It is clear from Figure 1 that the distributions of the random variable X are positively (that is, right) skewed and unimodal. (a) (b)

Figure 1: PDF Plots of X for (a)   1 ,   0.5 ,   0.2, 0.5, 1, 2 (left), and (b)   1 ,   1,   0.2, 0.5, 1, 2 (right). The kth moment is given by



E Xk



  2    ,  0,   0,   0 .   /  2  

 k    1  D ( k    1)  /

   1 

 2 k

D (  1)

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4 It can easily be verified that the p.d.f. (4) is unimodal with the mode given by

x

   2  8 4

,   0,   0,   0 .

The characteristic function is easily given by



g X (t )  E e i t X



   it   1 2 i   exp   t  t  D (  1)    8  4     2   ,   0,   0,   0 ,  D (  1)  / 2 

 

where i  given by



1 is the imaginary number, and i 2   1 . The Shannon entropy is easily





H [ X ]   ln(C )   E  X    E X 2   E  ln X ,   0,   0,   0 ,

where

 1    1  j     2     2 j  1     2 j  1  ln    , E  ln X      1  C    j 0 j!    and C denotes the normalizing constant given by (5), and   .  denotes digamma j

function. The survival and hazard functions for our newly proposed distribution are respectively given by

S  x   1  FX  x   1  and

f X  x h x    1  FX  x 

1

C

  1

C

  1

  1 k  k! k 0   

C x  exp k     1  k! k 0  

k      2      2 k  1,  x   ,   

 x    2 

2

x

        2 k  1,  x     k

,

where   0,   0,   0, x  0 , and C denotes the normalizing constant given by (5). Distributional Relationships: It is observed that the product densities like  The product of exponential and Rayleigh pdf’s,  The product of gamma and Rayleigh pdf’s,  The product of gamma and Rice pdf’s,  The product of gamma and half-normal pdf’s  Several other well known continuous probability densities like Rayleigh and halfnormal, among others, can easily be derived from (4) by simple transformation of the random variable X or by using special values of the parameters   0,   0 and   0 . 3.2 Generalized Pearson Distribution – II: We consider the GPE (2) in the following form p 2p df X ( x)  a0  a p x  a 2 p x   dx b p 1 x p 1 

  f X ( x), b p  1  0 , x  0 ,  

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5 where m  2 p , n  p  1, a1  a2    a p  1  a p  1    a2 p  1  0 , and

b 0  b1  b2    b p  0 . The solution to the differential equation (7) is given by





f X ( x)  C x  1 exp   x p   x  p , x  0,   0 ,   0 ,       , (8) a2 p a p  bp  1 a0 where    ,  ,  , b p  1  0 , p  0 , and C is the p bp  1 p bp  1 bp  1 normalizing constant given by 

1 p  2 p C    2    K 2  





,

(9)

p



where   0 ,   0 ,       , and p  0 , and K  2  

 denotes the

p

modified Bessel function of third kind. Here  and p denote the shape parameters, and  and  represent the scale parameters. The following are normalizing constants in special cases of (8): 





p   p C       p C



  p

, when   0 ,   0 ,   0 , and p  0 ;

p         p

, when   0 ,   0 ,   0 , and p  0 .

The cdf of the new distribution is easily given by

FX ( x ) 



1 2 



2 p K 

2







k 0

  1 k     k! 

k

   pk , xp p 

 

   ,(10) 

p



 ,  xp;   p

 

2 





2p



  

K 2  



,

(11)

p





where   0 ,   0 ,      , p  0 , and   ,  x p ;    denotes the  p  generalized incomplete gamma function. The possible shapes of the p.d.f. (8) are provided for some selected values of the parameters in Figure 2. It is clear from Figure 2 (a, b), the distributions of the random variable X are positively (that is, right) skewed with longer and heavier right tails.

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6 (a)

(b)

Figure 2: PDF Plots of X for (a)   1,   1,   0, p  1, 2, 3, 4 (left), and (b)   1,   0.5, p  1,   1, 0, 1, 2 (right). It can easily be verified that the pdf (8) is unimodal with the mode given by

x

p



1 



 1   4  p 2 2

2 p

,   0,   0, p  0 .

The kth moment,  k , for some integer k  0 , is given by

k  E  X

k







K k  2   k   2 p p    ,   0 ,   0 , k  0, p  0 .    K 2  





p

Pearson’s measure of skewness,  1 , and kurtosis,  2 : These are respectively given by

1 

3

 2 

3/ 2

, and  2 

4 ,  2  2

where  2 ,  3 and  4 denote the second, third and fourth central moments, respectively. For some selected values of the parameters, numerical values of skewness (  1 ) and kurtosis (  2 ) are provided in the Tables 1 to 3 below. It is evident from these

computations that the skewness,  1 , is positive which implies that distribution of the random variable X is positively skewed. Moreover, it is observed from these tables that, for all selected values of the parameters, the kurtosis  2  3 , implying that the distribution is heavier tailed, except for the parameters   1,   1,   2, p  4 , for which the kurtosis,  2  2.908023518  3 , implying that the distribution is lighter tailed.

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7 Table 1: For the parameters   1,   1,   1, p  1, 2, 3, 4

3

4

1

2

0.4513147514 0.02995803988 0.00600444965 0.00191004265

1.378485339 0.04148812322 0.00683665637 0.00201724356

2.307929056 0.972860493 0.62948272 0.467261395

12.14486579 4.298398749 3.379542649 3.08636673

p 2 1 2 3 4

0.3369028741 0.09824456746 0.04497725004 0.02556556224

Table 2: For the parameters   1,   1,   0, p  1, 2, 3, 4

p 1 2 3 4

2

3

4

1

2

0.7199622288 0.1190888005 0.0490032393 0.0268297530

1.179718748 0.0355783309 0.0061367510 0.0018516337

4.696586296 0.0560258304 0.0077972574 0.0021709036

1.931140681 0.865723448 0.565719725 0.421337293

9.060723594 3.950449013 3.247074666 3.015831478

Table 3: For the parameters   1,   1,   2, p  1, 2, 3, 4

p 1 2 3 4

2

3

4

1

2

2.145032369 0.1597920446 0.0559569540 0.0289209801

4.143916965 0.0421575868 0.0058280869 0.0016246050

26.06644560 0.0880167684 0.0095417143 0.0024323380

1.319046996 0.659998588 0.440296115 0.330314986

5.665185238 3.447109766 3.047321414 2.908023518

The Shannon entropy is easily given by



H [X ]   E X

p

   E  X     1  E  ln X   ln (C) , p

where C is the normalizing constant given by Eq. (9). The survival and hazard rate functions are, respectively, given by

S  x   1  FX  x   1 

 ,  x p;  p 

 

2 

2



 2 p K

  





,

(12)

p

and 

fX  x  h x    1  FX  x 



p    2 p x  1 exp   x p   x  p   2    K 2  





p

1

    ,  x p ;    p  2 



 2 p K p

where x  0,   0 ,   0 ,       , and p  0 .

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8 Distributional Relationships: Special Cases: Note that when p  1 and    , the equation (8) reduces to the pdf of the generalized inverse Gaussian (GIG) distribution. Also, it is easy to see that a number of other distributions such as Inverse Gaussian (with

1 1 p  1,    ), Reciprocal Inverse Gaussian (with p  1,   ), Hyperbolic (with 2 2 p

1 p  1,   1 ), Hyperbola (with p  1,   0 ), Generalized Gamma (with     ,    0,   p k , where   0, k  0 , and probability density functions for other well1 k known distributions such as Chi-Squared (with   ,   0 , p  1,   , where 2 2 1 k  0 is an integer), Chi (with   ,   0 , p  2,   k , where k  0 is an 2 integer), Erlang (with   0 , p  1,   c , where c  0 is an integer), Exponential (with p  1,   1,   0 ), Gamma (with p  1,   0 ), Weibull (with 

1 p   ,   0,     where   0 ), Rayleigh (with p  2,   2,   0 ,   1 1 ), Maxwell-Boltzman (with p  2,   3,   0 ,   ), Half Normal  2 2 2 2 p

1 (with p  2,   1,   0 ,   ), Log-Normal (with     ,   0,   p k , 2 2  where   0, k  0 , and letting k   ), Inverse Gamma (with p  1,   0,    , 1

2 where   0 ), Inverse Half Normal (with p  2,   0,   1,   , where    0 ), among others, can be derived as special cases of the Eq. (8). Thus we refer the proposed distribution having the pdf (8) as a new (or a generalization) of GIG distributions. It is also worth noting that the proposed family of generalized GIG is closed under power transformation. That is, if X ~ GeneralizedGIG   ,  ,  , p  , then

 p  Y  X s ~ GeneralizedGIG   ,  , ,  , where s  0 . One can use this property s s   of generalized GIG in information analysis. (B) Further, we see that the following distributions are special cases of the newly proposed distribution.  pth Root Reciprocal Inverse Gaussian Distribution.  Root Reciprocal Inverse Gaussian Distribution.  Good’s Generalized Inverse Gaussian Distribution.  For   1, p  2 , the equation (8) reduces to the p.d.f. of the GIG Distribution of Chaudhry and Ahmad (1993).  pth Root of the Generalized Inverse Gaussian Distribution.  Taking p  1 and     , where 0     , in equation (8), we obtain the Wesolowski’s and Chou and Huang’s GIG Distributions, with the p.d.f. given by

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9

f X ( x) 

1     2   

 2



x    1 exp   x   x 1



K  2  



,

x  0,   0 ,   0 ,   0,

which reduces to Seshadri and Wesolowski’s GIG when   2 a and   2 b , where a  0 , b  0 .  For p  1 and suitably changing the parameters, the Eq. (8) reduces to the pdf of Gneiting’s Normal Scale Mixtures Distributions.  Relationships to some other distributions: It is easy to see that, by a simple transformation of the variable x or by taking special values of the parameters  ,  ,  , p, the following product probability density functions are also special cases of our newly proposed distribution. (1) The product of the PDFs of Half Normal and Inverse Half Normal Distributions; (2) The product of the PDFs of Rayleigh and Inverse Rayleigh Distributions; (3) The product of the PDFs of Maxwell and Inverse Maxwell Distributions; (4) The product of the PDFs of Chi and Inverse Chi Distributions; (5) The product of the PDFs of Gamma and Inverse Gamma Distributions; (6) The product of the PDFs of Weibull and Inverse Weibull Distributions; and (7) The product of the PDFs of Root Reciprocal Exponential and Maxwell Distributions; among others. 3.3 Generalized Pearson Distribution – III: We consider the GPE (2) in the following form

a0  a p x p df X ( x)   p 1 b x  b dx p 1 x  1

  f X ( x), b 1  0, b p  1  0 , x  0 ,  

(14)

when m  p , n  p  1, a1  a2    a p  1  0 , and b 0  b2    b p  0 . The solution to the differential equation (14) is given by



f X ( x)  C x   1    x p



where   b 1 ,   b p  1 ,  



, x  0,   0 ,   0 ,   0,   0, and p  0 ,(15)

a0  b 1 b1

, 

a0 b p  1  a p b 1 p b 1 bp  1

, b 1  0, b p  1  0 ,

and C is the normalizing constant given by 



p   p   p C ,      ,    p p  

(16)

where   0 ,   0 ,   0,   0 , p  0 , and  . denotes beta function. By the definition of beta function, the parameters in (16) should be chosen as such that   Using the substitution t p  z , and the negative binomial series representation

889

 p

.

Section on Physical and Engineering Sciences – JSM 2011

10 k  1k c k w k  1  c  k  w k 1  w    ,  w  1,  c  k! k 0 k! k 0 

c

where c k  c (c  1)  (c  k  1) 

 c  k  denotes the Pochhammer symbol, the  c 

cdf of the random variable X can easily be expressed as 

FX ( x ) 

 p p   





    ,    p p

k 0

k  k      p k     1    k x     k !   p k   

   ,  

(17)



p  p    x         ,    p p

2

    p  F1  , ;  1; x  ,   p p 

where x  0,   0 ,   0 ,   0,   0, p  0 ,  

 p

(18)

, and 2 F1 . denotes the Gauss

hypergeometric function. The possible shapes of the p.d.f. (15) are provided for some selected values of the parameters in Figure 3 (a, b, c) below. From these graphs, it is evident that the distribution of the RV X is right skewed. (a)

(b)

(c)

Figure 3: PDF Plots of X for (a)   1,   1,   2,   2, p  2, 4, 5, 8 (left); (b)   1,   1,   2, p  3,   2, 2.5, 4, 5 (center); and (c)   1,   1,   2, p  2,   1.25, 1.50, 2, 3 (right). The mode of the proposed distribution is given by

    1  x     p    1

1 p

  ( say) ,   0 ,   0 ,   0,   0, p  0 ,

890

(19)

Section on Physical and Engineering Sciences – JSM 2011

11 which exists provided   1 , and  p   . Clearly, it is unimodal. The kth moment,

 k , of the random variable X , for some integer k  0 , is easily expressed in terms of the beta function as 

   k  E X k    x k f X ( x) dx      0

 k  k    ,  p   p ,     ,    p p

k p

(20)

where k  0,   0 ,   0 ,   0,   0, p  0 , and  p   . The Shannon entropy is easily given by the following equation:

H [ X ]  E [ ln ( f X  X ]        1                  p p           ln    

              p   

      p  B  ,       p     p , p   

(21)

where   0 ,   0 ,   0,   0, p  0 ,  p   , and and   .  denotes the digamma function. The survival and hazard rate functions for our newly proposed distribution are respectively given by 

p  p    x    S  x   1  FX  x   1      ,    p p

2

    p  F1  , ;  1; x  ,   p p 

(22)

and    p



p   h x  

f X x   1  FX  x 

  p x   1

    ,       x p p p









p  p    x    1     ,    p p

891

2

    p  F1  , ;  1; x  p p   

,

(23)

Section on Physical and Engineering Sciences – JSM 2011

12 where x  0,   0 ,   0 ,   0,   0, p  0 , and  p   . Distributional Relationships: It is easy to see that a number distributions such as the generalized beta of the second kind (GB2), Burr’s types III and XII, Lomax, Fisk, Pareto, half-student’s t, log-logistic, Singh-Maddala, Dagum, q-exponential, two parameter Kappa distributions, three parameter Kappa distributions, and probability density functions for other well-known distributions such as beta of the second kind, chi-squared, exponential, F, gamma, generalized gamma, half-normal, log-normal, inverted beta, Maxwell, Rayleigh, and Weibull, are special cases or can be derived as limiting cases of our newly proposed distribution, including others as given below: (i) Feller-Pareto distribution: For a random variable X , the distribution with the p.d.f. as given below is known as the Feller-Pareto distribution:

x  0 1 f X x  = ( )   B ( 1 ,  2 )  1

Let 

1

1



=  , ( x  0 )



1

1 [1  (



2 1   ( x  0 )   B ( 1, 2 )



2

x  0



1



  2

) ]1

1 [

1

1





 ( x  0 )

  2

]1

=  y p . Then it is easy to see that the 

1 1   ( yp ) 2   p  y p 1 , p.d.f. of Y is f Y  y  = p  1  2  B ( 2 , 1 ) (   y ) where B (  2 ,  1 )  B (  1 ,  2 ) . Let  2 =

f Y y 

p B(



 p

 p

 p

and  1 =  

 p

. Then





, 

p

 p

y 1 )

1 (    y p )

,

which is the distribution given in Eq. (15) on with Y = X . (ii) Generalized F distribution with the p.d.f. given by

f X ( x  ,  ,  , ) 

   x     1



  ,   1   x 

892





 

, x  0,  ,  ,  ,   0 .

Section on Physical and Engineering Sciences – JSM 2011

13 (iii) Pareto IV or Gamma mixture of Weibull distributions with the pdf given by

f X ( x  , ,  ) 

1

1   x  

, x  0,  ,  ,   0 .



(iv) Inverted beta distribution: For two independent random variables

X 1 ~ gamma  , 1 and X 2 ~ gamma  , 1 , the distribution of the ratio W 

X1 X2

is known as the inverted beta distribution with the p.d.f. given by

f

W

( IB : w ,  ,  ) 

     w  1

     

  1  w   

, w  0,   0 ,   0 .

This is the distribution given in equation (15) with W  X , and by changing the parameters suitably and taking the special values of the parameters. (v) Beta-prime distribution: For a random variable X ~ beta m , n , the distribution of the ratio Z 

f

Z

X is known as the beta-prime distribution with the p.d.f. given by 1 X

(z) 

  m  n  z m 1

  m   n   1  z 

mn

, z  0, m  0 , n  0 .

This is the distribution given in equation (15) with Z  X , and by changing the parameters suitably and taking the special values of the parameters. (vi) Pearson’s type VI distribution with the p.d.f. given by

f X ( x) 

x a 1

 a , b   1  x 

ab

, x  0, a  0 , b  0 .

(vii) rth reciprocal of each of the generalized beta of the second kind (GB2), Burr’s Types III and XII, Singh-Maddala, Dagum and Pareto. (viii) rth root reciprocal of each of the generalized beta of the second kind (GB2), Burr’s Types III and XII, Singh-Maddala, Dagum and Pareto. (ix) For  

1 b , p  b ,   ,   d ,   c b , the equation (15) reduces to the d d

Majumder and Chakravarty’s Distribution. (x) Compounding of the gamma distributions. (xi) Compounding of the Weibull and gamma. (xii) Compounding of the generalized gamma and inverted Weibull distributions when they have the same shape parameters.

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14 (xiii) Gamma mixture of generalized gamma distributions which also coincides with the generalized F. (xiv) Ratios of generalized gamma distributions with the same power parameters which also coincide with the generalized F distribution. Thus the newly proposed distribution with the p.d.f. given by (15) defines a more flexible family and therefore is a natural generalization of Burr type and many other life distributions. In view of this excellent property, we may refer to this new distribution as the generalized Burr (GB) distribution.

4. Concluding Remarks

This paper derives some new classes of continuous probability distribution based on the generalized Pearson differential equation. Some characteristics of the new distributions are defined. It is hoped that the proposed attempt will be helpful in designing a new approach of unifying different families of distributions based on generalized Pearson differential equation. Acknowledgements The first author is thankful to the Miami Dade College (Hialeah Campus) for providing STEM grant to attend the 2011 JSM. We also acknowledge Dr. M. Ahsanullah and Dr. G. G. Hamedani for their comments and suggestions on the subject which helped a lot in the preparation of the paper. References Abramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York. Chaudhry, M. A., and Ahmad, M. (1993). On a probability function useful in size modeling. Canadian Journal of Forest Research, 23(8), 1679–1683. Chaudhry, M. A., and Zubair, S. M. (2002). On A Class of Incomplete Gamma Functions, with Applications. Chapman & Hall/CRC, Boca Raton. Cobb, L., Koppstein, P., and Chen, N. H. (1983). Estimation and moment recursion relations for multimodal distributions of the exponential family. Journal of the American Statistical Association, 78(381), 124-130. Dunning, K., and Hanson, J. N. (1977). Generalized Pearson distributions and nonlinear programming. Journal of Statistical Computation and Simulation, Volume 6, Issue 2, 115 – 128. Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products (6th edition). Academic Press, San Diego. Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, (volume 1, second edition). John Wiley & Sons, New York.

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15 Pearson, K. (1895). Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London, A186, 343-414. Pearson, K. (1901). Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew of variation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 197, 343-414. Pearson, K. (1916). Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew of variation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 216, 429-457. Rossani, A., and Scarfone, A. M. (2009). Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field. Physica, A, 388, 2354-2366. Roy, L. K. (1971). An extension of the Pearson system of frequency curves. Trabajos de estadistica y de investigacion operativa, 22 (1-2), 113-123. Shakil, M., and Kibria, B. M. G. (2010). On a family of life distributions based on generalized Pearson differential equation with applications in health statistics. Journal of Statistical Theory and Applications, 9 (2), 255-282. Shakil, M., Singh, J. N., and Kibria, B. M. G. (2010a). On a family of product distributions based on the Whittaker functions and generalized Pearson differential equation. Pakistan Journal of Statistics, 26(1), 111-125. Shakil, M., Kibria, B. M. G., and Singh, J. N. (2010b). A new family of distributions based on the generalized Pearson differential equation with some applications. Austrian Journal of Statistics, 39 (3), 259–278.

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