Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

3020 
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Recurrence Relations for Moments of Generalized Order 

Statistics from Area-biased Rayleigh Distribution and its 

Characterization 

A. A. A-Rahman1, I. B. Abdul-Moniem 2, M. B. Moemen 1* 

1. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical 

Research, Cairo University, Egypt. 

2. Department of Statistics, Higher Institute of Management Sciences in Sohag, Sohag, 

Egypt., 

* Corresponding Author: mo.baha2010@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

            In this paper, the derivation of recurrence relations for single and product 

moments from generalized order statistics using the Area-biased Rayleigh distribution 

is presented. This includes specific cases for order statistics and records. Additionally, 

the Area-biased Rayleigh distribution is characterized through a recurrence relation for 

single moments.  

Keywords: Generalized order statistics -Single and product moments - Recurrence 

relations - Area biased Rayleigh distribution - Characterization. 

 

1. Introduction 

            Recurrence relations are mathematical formulas that link each term in a series to the 

preceding terms, thereby simplifying calculations and revealing data patterns. These relations 

find applications in moments of order statistics and generalized order statistics, such as 

parameter estimation, hypothesis testing, and deriving conclusions about the underlying 

distribution. Many researchers have employed the generalized order statistics (GOS) in their 

research, such as Kamps and Gather (1997), Keseling (1999), Cramer and Kamps (2000), 

Ahsanullah (2000), Pawlas and Szynal (2001), Ahmed (2007), Ahmed and Fawzy (2003), 

Khan et al. (2007), AL-Hussaini et al. (2005), Kumar (2011), Mahmoud and Ghazal (2012). 

Recently, Abdul-Moniem [(2014, 2019, 2022)] has contributed significantly to this field, 

presenting various studies on recurrence relations for moments of generalized order statistics 

from different distributions, extending well-known life distribution families, and exploring new 

characterizations of these extended distributions. These studies have resulted in deriving new 

recurrence relations for moments, particularly in the context of weighted distributions such as 

the length-biased Maxwell distribution and extended distributions like those based on the 

Marshall–Olkin family. Additionally, these works have enhanced the understanding of life 

distribution properties and provided novel characterizations that improve the applicability of 

these statistical models in various practical scenarios. Mohsin et al. (2010) further contributed 

by deriving a recurrence relation for single moments of generalized order statistics from the 

Rayleigh distribution, enhancing understanding of distribution properties and offering new 

characterizations for practical applications. 

mailto:mo.baha2010@gmail.com


Communications on Applied Nonlinear Analysis 

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A random variable (RV) Z is considered to have Area-biased Rayleigh distribution (ABRD) if 

its probability density function (PDF) has the following form: 

 

𝑓(𝑧) =
2𝑧3

𝛽4
𝐞𝐱𝐩 (− 

𝑧2

𝛽2
) ;  𝛽 > 0 , 𝑧 ≥ 0   

 

(1) 

 

The survival function (SF) corresponding to equation (1) is expressed as: 

 

�̅�(𝑧) = (1 +
𝑧2

𝛽2
 ) 𝐞𝐱𝐩 (−

𝑧2

𝛽2
 ) ;  𝛽 > 0 , 𝑧 ≥ 0   

 

(2) 

 

Substituting from (2) in (1), we get: 

 

�̅�(𝑧) = (
𝛽2

2𝑧
+

𝛽4

2𝑧3
 ) 𝑓(𝑧) 

 

(3) 

 

Bashir and Rasul, (2018) provided further details on this distribution and its applications. 

Kamps (1995) developed the concept of generalized order statistics (GOS), which encompasses 

various order models of random variables. for simplicity, Let, F throughout represent a 

continuous distribution function with a density function f. The random variables 

𝑍(1, 𝑛, �̃�, 𝑘), . . . , 𝑍(𝑛, 𝑛, �̃�, 𝑘) are referred to as generalized order statistics based on F, if their 

joint probability density function takes the following form 

𝑘 (∏ 𝛾𝑗

𝑛−1

𝑗=1

) (∏[�̅�(𝑧𝑖)]𝑚𝑖

𝑛−1

𝑖=1

𝑓(𝑧𝑖)) [�̅�(𝑧𝑛)]𝑘−1𝑓(𝑧𝑛) , 

for �̅�−1(1) > 𝑧1 ≥ 𝑧2 ≥. . . ≥ 𝑧𝑛 > �̅�−1(0), with parameters 𝑛 ∈  𝑁, 𝑛 ≥ 2, 𝑘 > 0 , �̃� =

(𝑚1, 𝑚2, . . . , 𝑚𝑛−1) ∈ 𝑅𝑛−1, 𝑀𝑟 = ∑ 𝑚𝑖
𝑛=1
𝑖=𝑟  , such that 𝛾𝑟 = 𝑘 + 𝑛 − 𝑟 + 𝑀𝑟 > 0,  for all 𝑟 ∈

{1,2, . . . , 𝑛 − 1}.For 𝛾𝑖 ≠  𝛾𝑗  , 𝑖 ≠ 𝑗 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑗 ∈ (1,2, . . . , 𝑛 − 1)  the probability density 

function (PDF) of  𝑍(𝑟, 𝑛, �̃�, 𝑘) is expressed (by Cramer and Kamps.(2000))  as follows 

 

𝑓𝑍(𝑟,𝑛,�̃�,𝑘)(𝑧) = 𝐶𝑟−1𝑓(𝑧) ∑ 𝑎𝑖(𝑟)𝑟
𝑖=1 [�̅�(𝑧)]𝛾𝑖−1                                   (4) 

 

The joint PDF for  𝑍(𝑟, 𝑛, �̃�, 𝑘) and  𝑍(𝑠, 𝑛, �̃�, 𝑘), 1 ≤ 𝑟 < 𝑠 ≤ 𝑛  is given as 

 

𝑓𝑍(𝑟,𝑛,�̃�,𝑘),𝑍(𝑠,𝑛,�̃�,𝑘)(𝑧, 𝑡) = 𝐶𝑠−1 (∑ 𝑎𝑖
(𝑟)(𝑠)𝑠

𝑖=𝑟+1 [
𝐹(𝑡)

𝐹(𝑧)
]

𝛾𝑖

) (∑ 𝑎𝑖(𝑟)𝑟
𝑖=1  [�̅�(𝑧)]𝛾𝑖)

𝑓(𝑧)𝑓(𝑡)

𝐹(𝑧)𝐹(𝑡)
  ,             

(5) 

 

where 𝑧 < 𝑡                                                                                                                                             

and                                      



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𝑎𝑖(𝑟) = ∏
1

𝛾𝑗 − 𝛾𝑖

𝑟

𝑗=1
𝑗≠𝑖

,   1 ≤ 𝑖 ≤ 𝑟 ≤ 𝑛 , 

𝑎𝑖
(𝑟)

(𝑠) = ∏
1

𝛾𝑗 − 𝛾𝑖
, 𝑟 + 1 ≤ 𝑖 ≤ 𝑠 ≤ 𝑛 .

𝑠

𝑗=𝑟+1
𝑗≠𝑖

 

It should be noted that when  𝑚1 = 𝑚2 =. . . = 𝑚𝑛−1 = 𝑚 ≠ −1, 

 

 𝑎𝑖(𝑟) =
(−1)𝑟−𝑖

(𝑚 + 1)𝑟−1(𝑟 − 1)!
(

𝑟 − 1

𝑟 − 𝑖
),                                          (6) 

 

and 

𝑎𝑖
(𝑟)(𝑠) =

(−1)𝑠−𝑖

(𝑚 + 1)𝑠−𝑟−1(𝑠 − 𝑟 − 1)!
(

𝑠 − 𝑟 − 1

𝑠 − 𝑖
)                        (7) 

 

Therefore, the PDF of  𝑍(𝑟, 𝑛, �̃�, 𝑘)  given in (4) reduces to 

 

𝑓𝑍(𝑟,𝑛,𝑚,𝑘)(𝑧) =
𝐶𝑟−1

(𝑟−1)!
[�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)]                             (8) 

 

and joint PDF of  𝑍(𝑟, 𝑛, �̃�, 𝑘) and  𝑍(𝑠, 𝑛, �̃�, 𝑘) given in (5) reduces to 

 

𝑓𝑍(𝑟,𝑛,𝑚,𝑘),𝑍(𝑠,𝑛,𝑚,𝑘)(𝑧, 𝑡) =  
𝐶𝑠−1 

(𝑟 − 1)! (𝑠 − 𝑟 − 1)!
[�̅�(𝑧)]𝑚𝑓(𝑧)𝑔𝑚

𝑟−1[𝐹(𝑧)] 

× {ℎ𝑚[𝐹(𝑡)] − ℎ𝑚[𝐹(𝑧)]}𝑠−𝑟−1[�̅�(𝑡)]𝛾𝑠−1𝑓(𝑡),                     𝑧 < 𝑡                    (9) 

Whereas 

𝐶𝑟−1 = ∏ 𝛾𝑖
𝑟
𝑖=1  , 𝛾𝑖 = 𝑘 + (𝑛 − 𝑖)(𝑚 + 1), 

ℎ𝑚(𝑧) = {

−1

𝑚 + 1
(1 − 𝑧)𝑚+1,   𝑚 ≠ −1

− ln(1 − 𝑧)           ,   𝑚 = −1
 

                                       and                                        𝑔𝑚(𝑧) = ℎ𝑚(𝑧) − ℎ𝑚(0) ,   𝑧 ∈ [0,1). 

We also set 𝑍(0, 𝑛, 𝑚, 𝑘) = 0. If  𝑚 = 0, 𝑘 = 1  then 𝑍(𝑟, 𝑛, 𝑚, 𝑘) simplifies to the 

(𝑛 − 𝑟 + 1)𝑡ℎ order statistics, 𝑍𝑛−𝑟+1:𝑛 from the sample  𝑍1, 𝑍2, … , 𝑍𝑛. When 𝑚 = −1 then 

𝑍(𝑟, 𝑛, 𝑚, 𝑘) simplifies to the kth record values as noted (by Pawlas. and Szynal. (2001)). 

 

In this research, we provide explicit expressions and some recurrence relations for single and 

product moments of GOS from ABRD. Its many deductions and special cases are also studied. 

A recurrence relation for single moments was utilized to characterize ABRD. 

 

2. Recurrence relation for single moments of GOS 

The single moments of GOS for ABRD are as the follows 



Communications on Applied Nonlinear Analysis 

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𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)] =
𝐶𝑟−1 

(𝑟 − 1)!
∫ 𝑍𝑗

∞

0

[�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚
𝑟−1 [𝐹(𝑧)] 𝑑𝑧                             

=  
𝐶𝑟−1 

 (𝑚 + 1)𝑟−1(𝑟 − 1)!
∫ 𝑍𝑗

∞

0

[�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧) [1 − (�̅�(𝑧))
𝑚+1

]
𝑟−1

 𝑑𝑧

=
𝐶𝑟−1 ∑ (𝑟−1

𝑤
)(−1)𝑤𝑟−1

𝑤=0

 (𝑚 + 1)𝑟−1(𝑟 − 1)!
∫ 𝑍𝑗

∞

0

[�̅�(𝑧)]𝛾𝑟+𝑤(𝑚+1)−1𝑓(𝑧)  𝑑𝑧

=
𝑗 𝐶𝑟−1 

(𝑟 − 1)! (𝑚 + 1)𝑟−1
∑

(𝑟−1
𝑤

)(−1)𝑤

[𝛾𝑟 + 𝑤(𝑚 + 1)]

𝑟−1

𝑤=0

∫ 𝑍𝑗−1

∞

0

[�̅�(𝑧)]𝛾𝑟+𝑤(𝑚+1)  𝑑𝑧 

using equation (2), we get 

 

𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)]

=
𝑗 𝐶𝑟−1 

(𝑟 − 1)! (𝑚 + 1)𝑟−1
∑

(𝑟−1
𝑤

)(−1)𝑤

[𝛾𝑟 + 𝑤(𝑚 + 1)]

𝑟−1

𝑤=0

 ∫ 𝑍𝑗−1

∞

0

[(1 +
𝑧2

𝛽2
)]

𝛾𝑟+𝑤(𝑚+1)

 

𝒆𝒙𝒑 (−(𝛾𝑟 + 𝑤(𝑚   + 1))
𝑧2

𝛽2
)    𝑑𝑧 

By applying the binomial expansion, we get 

𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)]

=
𝑗 𝐶𝑟−1 

(𝑟 − 1)! (𝑚 + 1)𝑟−1
∑ ∑

(𝑟−1
𝑤

)(𝛾𝑟+𝑤(𝑚+1)
𝑣

)(−1)𝑤

𝛽2𝑣[𝛾𝑟 + 𝑤(𝑚 + 1)]

𝛾𝑟+𝑤(𝑚+1)

𝑣=0

𝑟−1

𝑤=0

 ∫(𝑧2)𝑣+
𝑗−1

2

∞

0

𝒆𝒙𝒑 (−(𝛾𝑟

+ 𝑤(𝑚 + 1))
𝑧2

𝛽2
 )    𝑑𝑧 

 

By substitution, 𝑦 =
𝛾𝑟+𝑤(𝑚+1)

𝛽2 𝑧2 in the integration process, we obtain 

𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)]

=
𝑗𝛽𝑗  𝐶𝑟−1 

2(𝑟 − 1)! (𝑚 + 1)𝑟−1
∑ ∑

(𝑟−1
𝑤

)(𝛾𝑟+𝑤(𝑚+1)
𝑣

)(−1)𝑤

[𝛾𝑟 + 𝑤(𝑚 + 1)]𝑣+
𝑗
2

+1

𝛾𝑟+𝑤(𝑚+1)

𝑣=0

𝑟−1

𝑤=0

 ∫(𝑦)𝑣+
𝑗
2

−1

∞

0

𝒆𝒙𝒑(−𝑦 ) 𝑑𝑦 

Therefore 

𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)]

=
𝑗𝛽𝑗  𝐶𝑟−1 

2(𝑟 − 1)! (𝑚 + 1)𝑟−1
∑ ∑

(𝑟−1
𝑤

)(𝛾𝑟+𝑤(𝑚+1)
𝑣

)(−1)𝑤

[𝛾𝑟 + 𝑤(𝑚 + 1)]𝑣+
𝑗
2

+1

𝛾𝑟+𝑤(𝑚+1)

𝑣=0

𝑟−1

𝑤=0

𝛤 (𝑣

+
𝑗

2
)                  (10) 

Remark 2.1 Setting 𝑚 = 0, 𝑘 = 1 in equation (10), we derive the single moments of order 

statistics for ABRD as following form 



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𝐸[𝑍𝑟:𝑛
𝑗

] =
𝑗𝑛! 𝛽𝑗

 2(𝑟 − 1)! (𝑛 − 𝑟)!
∑ ∑

(𝑟−1
𝑤

)(𝑛−𝑟+𝑤+1
𝑣

)(−1)𝑤

[𝑛 − 𝑟 + 𝑤 + 1]𝑣+
𝑗
2

+1

𝑛−𝑟+𝑤+1

𝑣=0

𝑟−1

𝑤=0

𝛤 (𝑣 +
𝑗

2
) 

Then 

𝐸[𝑍𝑟:𝑛
𝑗

] =
𝑗𝑛! 𝛽𝑗

 2(𝑛−𝑟)!
∑ ∑

 (𝑛−𝑟+𝑤+1)−𝑣−
𝑗
2.(𝑛−𝑟+𝑤)!(−1)𝑤

𝑤!(𝑟−1−𝑤)!(𝑛−𝑟+𝑤+1−𝑣)!𝑣!

𝑛−𝑟+𝑤+1
𝑣=0

𝑟−1
𝑤=0 𝛤 (𝑣 +

𝑗

2
)       (11) 

 

Using equation (11), some numerical results for the mean and variance of order statistics 

from ABRD, as presented in the following tables. 

Table 1: Mean of order statistics for ABRD 

 

parameter 
n 

β=3.5 β=3 β=2.5 β=2 β=1.5 β=1.3 β=1 β=0.8 β=0.6 β=0.4 R 

4.653 3.988 3.323 2.659 1.994 1.728 1.329 1.063 0.798 0.532 1 1 

3.701 3.172 2.644 2.115 1.586 1.375 1.057 0.846 0.634 0.423 1 
2 

5.604 4.804 4.003 3.202 2.402 2.082 1.601 1.281 0.961 0.64 2 

3.258 2.793 2.327 1.862 1.396 1.21 0.931 0.745 0.559 0.372 1 

3 4.587 3.932 3.276 2.621 1.966 1.704 1.311 1.048 0.786 0.524 2 

6.113 5.24 4.366 3.493 2.62 2.27 1.747 1.397 1.048 0.699 3 

2.984 2.558 2.131 1.705 1.279 1.108 0.853 0.682 0.512 0.341 1 

4 
4.081 3.498 2.915 2.332 1.749 1.516 1.166 0.933 0.7 0.466 2 

5.093 4.365 3.638 2.91 2.183 1.892 1.455 1.164 0.873 0.582 3 

6.453 5.531 4.609 3.687 2.765 2.397 1.844 1.475 1.106 0.737 4 

2.791 2.392 1.994 1.595 1.196 1.037 0.797 0.638 0.478 0.319 1 

5 

3.757 3.22 2.683 2.147 1.61 1.395 1.073 0.859 0.644 0.429 2 

4.568 3.915 3.263 2.61 1.958 1.697 1.305 1.044 0.783 0.522 3 

5.443 4.665 3.888 3.11 2.333 2.022 1.555 1.244 0.933 0.622 4 

6.705 5.747 4.789 3.832 2.874 2.491 1.916 1.533 1.149 0.766 5 

2.644 2.267 1.889 1.511 1.133 0.982 0.756 0.604 0.453 0.302 1 

6 

3.523 3.02 2.517 2.013 1.51 1.309 1.007 0.805 0.604 0.403 2 

4.223 3.62 3.017 2.413 1.81 1.569 1.207 0.965 0.724 0.483 3 

4.912 4.211 3.509 2.807 2.105 1.825 1.404 1.123 0.842 0.561 4 

5.708 4.893 4.077 3.262 2.446 2.12 1.631 1.305 0.979 0.652 5 

6.905 5.918 4.932 3.946 2.959 2.565 1.973 1.578 1.184 0.789 6 

- 

Note that: the results presented in Table1 are consistent with the properties of order statistics  

described by David and Nagaraja (2004), specifically ∑ 𝜇𝑖:𝑛 = 𝑛𝜇1:1
𝑛
𝑖=1  



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Table 2: Variance of order statistics for ABRD 

parameter 
n 

β=3.5 β=3 β=2.5 β=2 β=1.5 β=1.3 β=1 β=0.8 β=0.6 β=0.4 R 

2.852 2.096 1.455 0.931 0.524 0.394 0.233 0.149 0.084 0.037 1 1 

1.614 1.186 0.823 0.527 0.296 0.223 0.132 0.084 0.047 0.021 1 
2 

2.281 1.676 1.164 0.745 0.419 0.315 0.186 0.119 0.067 0.03 2 

1.18 0.867 0.602 0.385 0.217 0.163 0.096 0.062 0.035 0.015 1 

3 1.305 0.958 0.666 0.426 0.24 0.18 0.106 0.068 0.038 0.017 2 

1.993 1.464 1.017 0.651 0.366 0.275 0.163 0.104 0.059 0.026 3 

0.953 0.7 0.486 0.311 0.175 0.131 0.078 0.05 0.023 0.012 1 

4 
0.958 0.704 0.489 0.313 0.176 0.132 0.078 0.05 0.028 0.013 2 

1.14 0.837 0.581 0.372 0.209 0.157 0.093 0.06 0.033 0.015 3 

1.815 1.333 0.926 0.592 0.333 0.25 0.148 0.095 0.053 0.024 4 

0.811 0.596 0.414 0.265 0.149 0.112 0.066 0.042 0.024 0.011 1 

5 

0.773 0.568 0.395 0.253 0.142 0.107 0.063 0.04 0.023 0.01 2 

0.839 0.616 0.428 0.274 0.154 0.116 0.068 0.044 0.025 0.011 3 

1.034 0.759 0.527 0.338 0.19 0.143 0.084 0.054 0.03 0.014 4 

1.691 1.242 0.863 0.552 0.311 0.233 0.138 0.088 0.05 0.022 5 

0.713 0.524 0.364 0.233 0.131 0.098 0.058 0.037 0.021 0.0093 1 

6 
0.657 0.483 0.335 0.215 0.121 0.091 0.054 0.034 0.019 0.0086 2 

0.679 0.499 0.346 0.222 0.125 0.094 0.055 0.035 0.02 0.0089 3 

0.762 0.56 0.389 0.249 0.14 0.105 0.062 0.04 0.022 0.0099 4 



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0.958 0.704 0.489 0.313 0.176 0.132 0.078 0.05 0.028 0.013 5 

1.599 1.175 0.816 0.522 0.294 0.221 0.131 0.084 0.047 0.021 6 

 

 

 2.1  The rth TL-moments and rth L-moments 

The rth TL-moments are given in the following formula (by Elamir. and Seheult. (2003)). 

 

 𝐿𝑟
(𝑠,𝑡)

=
1

𝑟
∑(−1)𝑘

𝑟−1

𝑘=0

(
𝑟 − 1

𝑘
)  𝐸(𝑍𝑟+𝑠−𝑘:𝑟+𝑠+𝑡) ,                                                                                (12) 

 

Whereas r, s and t take the values 1,2,3, ...We have noted that the rth L-moments can be obtained 

by taking s= t = 0. 

Using (12) and (11) with j=1, n= r+s+t and r= r+s-k the rth TL-moments can be obtained as 

follows 



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 𝐿𝑟
(𝑠,𝑡)

=
1

𝑟
∑(−1)𝑘

𝑟−1

𝑘=0

(
𝑟 − 1

𝑘
) ∑ ∑

𝛽. (𝑟 + 𝑠 + 𝑡)!  (𝑡 + 𝑘 + 𝑤 + 1)−𝑣−
1
2. (𝑡 + 𝑘 + 𝑤)! (−1)𝑤

2𝑤! (𝑟 + 𝑠 − 𝑘 − 1 − 𝑤)! (𝑡 + 𝑘 + 𝑤 + 1 − 𝑣)! 𝑣! (𝑡 + 𝑘)!

𝑡+𝑘+𝑤+1

𝑣=0

𝑟+𝑠−𝑘−1

𝑤=0

× 𝛤 (𝑣 +
1

2
) 

Then 

 

     𝐿𝑟
(𝑠,𝑡)

=
1

𝑟
∑ ∑ ∑

𝛽. (𝑟 + 𝑠 + 𝑡)! (𝑟 − 1)! (𝑡 + 𝑘 + 𝑤 + 1)−𝑣−
1
2. (𝑡 + 𝑘 + 𝑤)! (−1)𝑤+𝑘

2𝑤! 𝑘! (𝑟 − 1 − 𝑘)! (𝑟 + 𝑠 − 𝑘 − 1 − 𝑤)! (𝑡 + 𝑘 + 𝑤 + 1 − 𝑣)! 𝑣! (𝑡 + 𝑘)!
       (13)  

𝑡+𝑘+𝑤+1

𝑣=0

𝑟+𝑠−𝑘−1

𝑤=0

𝑟−1

𝑘=0

× 𝛤 (𝑣 +
1

2
)                                                                                                                             

The initial four TL-moments can be obtained from (13) by setting r = 1,2,3 and 4 

respectively. 

 

The generalized TL-moments ratios, such as the coefficient of variation, coefficient of 

skewness and coefficient of kurtosis, are calculated from the initial four generalized TL-

moments. These ratios are defined as 𝜏1
(𝑠,𝑡)

=
𝐿1

(𝑠,𝑡)

𝐿2
(𝑠,𝑡)  , 𝜏3

(𝑠,𝑡)
=

𝐿3
(𝑠,𝑡)

𝐿2
(𝑠,𝑡)  𝑎𝑛𝑑  𝜏4

(𝑠,𝑡)
=

𝐿4
(𝑠,𝑡)

𝐿2
(𝑠,𝑡)    

respectively. The rth L-moments can be obtained by taking s= t= 0   in equation (13) as 

 

𝐿𝑟 =
1

𝑟
∑ ∑ ∑

𝛽. (𝑟)! (𝑟 − 1)! (𝑘 + 𝑤 + 1)−𝑣−
1
2. (𝑘 + 𝑤)! (−1)𝑤+𝑘

2𝑤! 𝑘! (𝑟 − 1 − 𝑘)! (𝑟 − 𝑘 − 1 − 𝑤)! (𝑘 + 𝑤 + 1 − 𝑣)! 𝑣! (𝑘)!
𝛤 (𝑣

𝑘+𝑤+1

𝑣=0

𝑟−𝑘−1

𝑤=0

𝑟−1

𝑘=0

+
1

2
)        (14)   

 

The initial four L-moments can be obtained from equation (14) by setting r =1,2,3 and 4 

respectively. Using equation (13), some numerical results for 

 𝐿1
(𝑠,𝑡)

, 𝐿2
(𝑠,𝑡)

, 𝐿3
(𝑠,𝑡)

 𝐿4
(𝑠,𝑡)

, 𝐿1, 𝐿2, 𝐿3, 𝐿4, 𝜏1
(𝑠,𝑡)

, 𝜏1
(𝑠,𝑡)

, 𝜏3
(𝑠,𝑡)

, 𝜏4
(𝑠,𝑡)

, 𝜏1,𝜏2𝑎𝑛𝑑 𝜏3   are obtained in 

Table.3. 

Table 3 

(0,0) (2,0) (1,0) (0,2) (0,1) (2,2) (1,1) (𝑠, 𝑡) parameter’s 

0.532 0.699 0.64 0.372 0.423 0.522 0.524 𝐿1
(𝑠,𝑡)

 

𝛽 = 0.4 

0.109 0.078 0.087 0.063 0.076 0.039 0.058 𝐿2
(𝑠,𝑡)

 

0.008 0.015 0.013 -0.006 -0.003 0.001 0.002 𝐿3
(𝑠,𝑡)

 

0.012 0.008 0.009 0.005 0.006 0.001 0.003 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 



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0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 

0.798 1.048 0.961 0.559 0.634 0.783 0.786 𝐿1
(𝑠,𝑡)

 

𝛽 = 0.6 

 

0.163 0.117 0.131 0.094 0.114 0.059 0.087 𝐿2
(𝑠,𝑡)

 

0.011 0.022 0.02 -0.009 -0.005 0.002 0.004 𝐿3
(𝑠,𝑡)

 

0.019 0.013 0.014 0.007 0.009 0.002 0.005 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 

1.063 1.397 1.281 0.745 0.846 1.044 1.048 𝐿1
(𝑠,𝑡)

 

𝛽 = 0.8 

0.217 0.155 0.174 0.125 0.152 0.079 0.116 𝐿2
(𝑠,𝑡)

 

0.015 0.03 0.027 -0.012 -0.006 0.002 0.005 𝐿3
(𝑠,𝑡)

 

0.025 0.017 0.018 0.0095 0.012 0.003 0.007 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.05 𝜏4 

1.329 1.747 1.601 0.931 1.057 1.305 1.311 𝐿1
(𝑠,𝑡)

 

𝛽 = 1 

0.272 0.194 0.218 0.157 0.19 0.098 0.145 𝐿2
(𝑠,𝑡)

 

0.019 0.037 0.033 -0.015 -0.008 0.003 0.006 𝐿3
(𝑠,𝑡)

 

0.031 0.021 0.023 0.012 0.016 0.004 0.008 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 

1.728 2.27 2.082 1.21 1.375 1.697 1.704 𝐿1
(𝑠,𝑡)

 

𝛽 = 1.3 

0.353 0.253 0.283 0.204 0.247 0.128 0.188 𝐿2
(𝑠,𝑡)

 

0.024 0.048 0.043 -0.019 -0.011 0.004 0.008 𝐿3
(𝑠,𝑡)

 

0.04 0.027 0.03 0.015 0.02 0.005 0.011 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 

1.994 2.62 2.402 1.396 1.586 1.958 1.966 𝐿1
(𝑠,𝑡)

 

𝛽 = 1.5 

0.408 0.291 0.327 0.235 0.285 0.148 0.217 𝐿2
(𝑠,𝑡)

 

0.028 0.055 0.05 -0.022 -0.012 0.004 0.009 𝐿3
(𝑠,𝑡)

 

0.046 0.031 0.035 0.018 0.023 0.005 0.013 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 



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2.659 3.493 3.202 1.862 2.115 2.61 2.621 𝐿1
(𝑠,𝑡)

 

𝛽 = 2 

0.544 0.389 0.436 0.313 0.38 0.197 0.289 𝐿2
(𝑠,𝑡)

 

0.038 0.074 0.066 -0.029 -0.016 0.006 0.012 𝐿3
(𝑠,𝑡)

 

0.062 0.042 0.046 0.024 0.031 0.007 0.017 𝐿4
(𝑠,𝑡)

 

4.89 8.99 7.346 5.94 5.571 13.263 9.067 𝜏1 

0.069 0.19 0.152 -0.094 -0.043 0.03 0.042 𝜏3 

0.114 0.108 0.106 0.076 0.082 0.036 0.059 𝜏4 

 

Theorem 2.1. let Z be a random variable has a probability density function given by equation 

(1). For an integer j such that j > 0, the moments of Z satisfy the following recurrence relation: 

𝐸[𝑍𝑗(𝑟, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝛽2

2𝛾𝑟
 {𝐸[𝑍𝑗−2(𝑟, 𝑛, �̃�, 𝑘)] + 𝛽2𝐸[𝑍𝑗−4(𝑟, 𝑛, �̃�, 𝑘)]} 

(15) 

Proof. We have from Lemma 2.3 (by Athar. and Islam. (2004)) that 

𝐸[𝜉{𝑍(𝑟, 𝑛, �̃�, 𝑘)}] − 𝐸[𝜉{𝑍(𝑟 − 1, 𝑛, �̃�, 𝑘)}] = 𝐶𝑟−2 ∫ 𝜉 ‘(𝑧)

𝛽

𝜃

 ∑ 𝑎𝑖(𝑟)

𝑟

𝑖=1

[�̅�(𝑧)]𝛾𝑖   𝑑𝑧 

If we let 𝜉(𝑧) = 𝑧𝑗   , then 

𝐸[𝑍𝑗(𝑟, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, �̃�, 𝑘)]

=  𝑗𝐶𝑟−2 ∫ 𝑧𝑗−1

𝛽

𝜃

 ∑ 𝑎𝑖(𝑟)

𝑟

𝑖=1

[�̅�(𝑧)]𝛾𝑖   𝑑𝑧                  (16) 

On using (3) in (16), we get 

 

𝐸[𝑍𝑗(𝑟, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝐶𝑟−1

𝛾𝑟
 ∫ 𝑧𝑗−1 (

𝛽2

2𝑧
+

𝛽4

2𝑧3
)

∞

0

 ∑ 𝑎𝑖(𝑟)

𝑟

𝑖=1

[�̅�(𝑧)]𝛾𝑖−1 𝑓(𝑧) 𝑑𝑧 

 

𝐸[𝑍𝑗(𝑟, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝛽2𝐶𝑟−1

2𝛾𝑟
 ∫ (

1

𝑧
+

𝛽2

𝑧3
)𝑧𝑗−1 𝐶𝑟−1 

∞

0

  𝑓(𝑧) ∑ 𝑎𝑖(𝑟)

𝑟

𝑖=1

[�̅�(𝑧)]𝛾𝑖−1 𝑑𝑧 

After simplification, the recurrence relation in equation (15) is derived. 

 

Corollary 2.2 When 𝑚1 = 𝑚2 = ⋯ = 𝑚𝑛−1 = 𝑚 ≠ −1  the recurrence relations for single 

moment of GOS for ABRD are provided as follows 



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𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, 𝑚, 𝑘)]

=
𝑗𝛽2

2𝛾𝑟
 {𝐸[𝑍𝑗−2(𝑟, 𝑛, 𝑚, 𝑘)] + 𝛽2𝐸[𝑍𝑗−4(𝑟, 𝑛, 𝑚, 𝑘)]} 

 

(17) 

Proof.  This can easily be deduced from (15) given the relation (6). 

 

Remark 2.3 By setting 𝑚 = 0, 𝑘 = 1  in Theorem 2.1., we derive the recurrence relations for 

a single moment of order statistics from ABRD 

𝐸(𝑍𝑟:𝑛
𝑗

) − 𝐸(𝑍𝑟−1:𝑛
𝑗

)

=
𝑗𝛽2

2(𝑛 − 𝑟 + 1)
 [𝐸(𝑍𝑟:𝑛

𝑗−2
)

+ 𝛽2𝐸(𝑍𝑟:𝑛
𝑗−4

)]                                                                         (18) 

 

Remark 2.4 Setting 𝑚 = −1, 𝑘 = 1 in Theorem 2.1., we derive the recurrence relations for 

upper record values as 

𝐸[𝑍𝑗(𝑟, 𝑛, −1,1)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, −1,1)]

=
𝑗𝛽2

2
{𝐸[𝑍𝑗−2(𝑟, 𝑛, −1,1)] + 𝛽2𝐸[𝑍𝑗−4(𝑟, 𝑛, −1,1)]} 

(19) 

3. Recurrence relation for product moments of GOS 

Theorem 3.1 let Z be a random variable has a probability density function given by equation 

(1). For integer i, j such that i, j >0, the product moments of Z satisfy the following recurrence 

relation: 

 

𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝛽2

2𝛾𝑠
 {𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗−2(𝑠, 𝑛, �̃�, 𝑘)]

+ 𝛽2𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗−4(𝑠, 𝑛, �̃�, 𝑘)]} 

(20) 

 

Proof. We have from Lemma 3.2 (by Athar. and Islam. (2004)) that 

 

𝐸[𝜉{𝑍 (𝑟, 𝑛, �̃�, 𝑘). 𝑍(𝑠, 𝑛, �̃�, 𝑘)}] − 𝐸[𝜉{𝑍(𝑟, 𝑛, �̃�, 𝑘). 𝑍(𝑠 − 1, 𝑛, �̃�, 𝑘)}]

= 𝐶𝑠−2 ∫ ∫
𝜕

𝜕𝑡
𝜉(𝑧, 𝑡) ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

𝛽

𝑧

𝛽

𝜃

𝑓(𝑧)

�̅�(𝑧)
 𝑑𝑡𝑑𝑧 

If we let 𝜉(𝑧, 𝑡) = 𝑧𝑖𝑡𝑗   , then 

 



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𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝐶𝑠−1

𝛾𝑠
 ∫ ∫  𝑧𝑖𝑡𝑗−1 ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

∞

𝑧

∞

0

𝑓(𝑧)

�̅�(𝑧)
 𝑑𝑡𝑑𝑧 

In view of equation (3), note that 

�̅�(𝑡)

𝑓(𝑡)
= (

𝛽2

2𝑡
+

𝛽4

2𝑡3
 ) =

𝛽2

2
𝑡−1 (1 +

𝛽2

𝑡2
 ) 

Therefore, 

 

𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝐶𝑠−1

𝛾𝑠
 ∫ ∫  𝑧𝑖𝑡𝑗−1 ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

∞

𝑧

∞

0

𝑓(𝑧)

�̅�(𝑧)
 (

�̅�(𝑡)

𝑓(𝑡)
.
𝑓(𝑡)

�̅�(𝑡)
) 𝑑𝑡𝑑𝑧 

 

𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠, 𝑛, �̃�, 𝑘)] − 𝐸[𝑍𝑖(𝑟, 𝑛, �̃�, 𝑘). 𝑍𝑗(𝑠 − 1, 𝑛, �̃�, 𝑘)]

=
𝑗𝐶𝑠−1

𝛾𝑠
 ∫ ∫  𝑧𝑖𝑡𝑗−1

𝛽2

2
𝑡−1 (1

∞

𝑧

∞

0

+
𝛽2

𝑡2
 ) ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

𝑓(𝑧)

�̅�(𝑧)
 .

𝑓(𝑡)

�̅�(𝑡)
𝑑𝑡𝑑𝑧

=
𝑗𝛽2𝐶𝑠−1

2𝛾𝑠
 ∫ ∫  𝑧𝑖𝑡𝑗−2 ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

∞

𝑧

∞

0

𝑓(𝑧)

�̅�(𝑧)
 .

𝑓(𝑡)

�̅�(𝑡)
𝑑𝑡𝑑𝑧

+
𝑗𝛽2𝐶𝑠−1

2𝛾𝑠
 ∫ ∫  𝑧𝑖𝑡𝑗−4𝛽2 ∑ 𝑎𝑙

(𝑟)(𝑠) [
�̅�(𝑡)

�̅�(𝑧)
]

𝛾𝑙

∑ 𝑎𝑙(𝑟)[�̅�(𝑧)]𝛾𝑙

𝑟

𝑙=1

𝑠

𝑙=𝑟+1

∞

𝑧

∞

0

𝑓(𝑧)

�̅�(𝑧)
 .

𝑓(𝑡)

�̅�(𝑡)
𝑑𝑡𝑑𝑧 

After simplification, the recurrence relation in equation (20) is derived. 

 

Corollary 3.2 For  𝑚1 = 𝑚2 = ⋯ = 𝑚𝑛−1 = 𝑚 ≠ −1 the recurrence relations for product 

moments of the Generalized Order Statistics (GOS) for ABRD are derived from equation (20) 

in relation to equation (7) is given as 

 

𝐸[𝑍𝑖(𝑟, 𝑛, 𝑚, 𝑘). 𝑍𝑗(𝑠, 𝑛, 𝑚, 𝑘)] − 𝐸[𝑍𝑖(𝑟, 𝑛, 𝑚, 𝑘). 𝑍𝑗(𝑠 − 1, 𝑛, 𝑚, 𝑘)]

=
𝑗𝛽2

2𝛾𝑠
 {𝐸[𝑍𝑖(𝑟, 𝑛, 𝑚, 𝑘). 𝑍𝑗−2(𝑠, 𝑛, 𝑚, 𝑘)]

+ 𝛽2𝐸[𝑍𝑖(𝑟, 𝑛, 𝑚, 𝑘). 𝑍𝑗−4(𝑠, 𝑛, 𝑚, 𝑘)]} 

(21) 

Proof. This follows directly from equation (20), considering the relation given in equation 

(7). 

Remark 3.3 By setting  𝑚 = 0, 𝑘 = 1 in equation (21), we derive the recurrence relations for 

product moments of order statistics.as 



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𝐸(𝑍𝑟,𝑠:𝑛
𝑖,𝑗

) − 𝐸(𝑍𝑟,𝑠−1:𝑛
𝑖,𝑗

)

=
𝑗𝛽2

2(𝑛 − 𝑠 + 1)
 [𝐸(𝑍𝑟,𝑠∶𝑛

𝑖,𝑗−2
)

+ 𝛽2𝐸(𝑍𝑟,𝑠∶𝑛
𝑖,𝑗−4

)]                                                        (22) 

Remark 3.4 setting  𝑚 = −1 in equation (21), gives us the recurrence relations for product 

moments of kth record values as 

𝐸 [(𝑍𝑟
(𝑘)

)
𝑖

(𝑍𝑠
(𝑘)

)
𝑗

] − 𝐸 [(𝑍𝑟
(𝑘)

)
𝑖

(𝑍𝑠−1
(𝑘)

)
𝑗

]

=
𝑗𝛽2

2𝑘
{𝐸 [(𝑍𝑟

(𝑘)
)

𝑖

(𝑍𝑠
(𝑘)

)
𝑗−2

] + 𝛽2𝐸 [(𝑍𝑟
(𝑘)

)
𝑖

(𝑍𝑠
(𝑘)

)
𝑗−4

]}             (23)    

4. Characterization 

Theorem 4.1 Let Z be a non-negative random variable with a continuous distribution 

function F (z) such that 

F (0) = 0 and 0 < F (z) < 1 for all z > 0, then 

𝐸[𝑍𝑗(𝑟, 𝑛, 𝑚, 𝑘)] − 𝐸[𝑍𝑗(𝑟 − 1, 𝑛, 𝑚, 𝑘)]

=
𝑗𝛽2

2𝛾𝑟
 {𝐸[𝑍𝑗−2(𝑟, 𝑛, 𝑚, 𝑘)] + 𝛽2𝐸[𝑍𝑗−4(𝑟, 𝑛, 𝑚, 𝑘)]} 

 

if and only if   �̅�(𝑧)

= (1

+
𝑧2

𝛽2
 ) 𝒆𝒙𝒑 (−

𝑧2

𝛽2
 )                                                                                           (24) 

Proof: The necessity part follows directly from equation (17). Conversely, if the recurrence 

relation given in equation (24) is satisfied, then by using equation (8), we establish that 

𝐶𝑟−1

(𝑟 − 1)!
 ∫ 𝑍𝑗  [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

−
𝐶𝑟−2

(𝑟 − 2)!
 ∫ 𝑍𝑗  [�̅�(𝑧)]𝛾𝑟−1−1𝑓(𝑧)𝑔𝑚

𝑟−2 [𝐹(𝑧)] 𝑑𝑧

∞

0

 

=
𝑗 𝛽2𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−2 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

+
𝑗 𝛽4𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
∫ 𝑍𝑗−4 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

 

Integrating the first term in left-hand side by parts, we get 

 



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𝑗𝐶𝑟−1

𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−1 [�̅�(𝑧)]𝛾𝑟  𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

−
𝑗 𝛽2𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
∫ 𝑍𝑗−2 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

 

−
𝑗 𝛽4𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−4 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧 = 0

∞

0

 

then 

𝑗𝐶𝑟−1

𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−1 [�̅�(𝑧)]𝛾𝑟−1  �̅�(𝑧) 𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

−
 𝑗 𝛽2𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
∫ 𝑍𝑗−2 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

 

 

−
 𝑗 𝛽2𝐶𝑟−1

2𝛾𝑟(𝑟 − 1)!
 ∫ 𝛽2 𝑍−2𝑍𝑗−2 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧 = 0

∞

0

 

this implies that 

 

𝑗𝐶𝑟−1

𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−1 [�̅�(𝑧)]𝛾𝑟−1  �̅�(𝑧) 𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧

∞

0

−
𝑗 𝐶𝑟−1

𝛾𝑟(𝑟 − 1)!
∫

𝛽2

2
[𝑧−1 + 𝛽2 𝑍−3]𝑍𝑗−1 [�̅�(𝑧)]𝛾𝑟−1𝑓(𝑧)𝑔𝑚

𝑟−1 [𝐹(𝑧)] 𝑑𝑧 = 0 

∞

0

 

therefore, 

𝑗𝐶𝑟−1

𝛾𝑟(𝑟 − 1)!
 ∫ 𝑍𝑗−1 [�̅�(𝑧)]𝛾𝑟−1   𝑔𝑚

𝑟−1 [𝐹(𝑧)] {�̅�(𝑧) −
𝛽2

2
[𝑧−1 + 𝛽2 𝑍−3]𝑓(𝑧)} 𝑑𝑧

∞

0

= 0                                         (25) 

Now applying a generalization of the Muntz-Szasz theorem (by Hwang. and Lin. (1984)) to 

equation (25), we get 

�̅�(𝑧) −
𝛽2

2
[𝑧−1 + 𝛽2 𝑍−3]𝑓(𝑧) = 0 

 

�̅�(𝑧)

𝑓(𝑧)
=

𝛽2

2
[𝛽2 𝑍−3 + 𝑧−1] 

Hence, 

𝑓(𝑧)

�̅�(𝑧)
= [

2𝑧3

𝛽2(𝛽2 + 𝑧2)
] 

Integrating both sides from 0 to t, we get 



Communications on Applied Nonlinear Analysis 

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3034 
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                                                                    �̅�(𝑡) =   (1 +
𝑡2

𝛽2 ) 𝒆𝒙𝒑 (−
𝑡2

𝛽2 )                 (26) 

5. Conclusion 

In this research, we employed the Area -biased Rayleigh distribution to derive recurrence 

relations for both single and product moments of generalized order statistics (GOS). 

Additionally, we explored specific cases related to order statistics and record values. The 

recurrence relation for single moments contributes to characterizing the properties of the Area 

-biased Rayleigh distribution. 

 

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