EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6148
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Survival Weighted Pareto Distribution: Order Statistics
and Its Applications

M. I. Khan1,∗, Abdelfattah Mustafa1,2

1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
42351, Kingdom of Saudi Arabia
2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516,
Egypt

Abstract. Nagy et al. [1] proposed an extended Pareto distribution called a survival-weighted
Pareto distribution (SWPD). The proposed model proves to be the best fit for several extensions of
the Pareto distribution. The current study provides quantitative features of SWPD through order
statistics. The SWPD is characterized through the hazard rate function (HRF). The appropriate-
ness of the characterization result is verified empirically by using a real-life data set. Finally, the
measures of inequality are taken into consideration due to the heavy tail properties of the model.

2020 Mathematics Subject Classifications: 60E05, 60E15, 62E10, 62G30

Key Words and Phrases: SWPD, Order statistics, Moments, Characterization, Inequality mea-
sures

1. Introduction

Arnold [2] discussed a profound historical survey on Pareto distribution and explained
its wide use in the context of economic phenomena. The PD provides an extensive appli-
cation in economics, actuarial, finance, social, natural sciences and various fields.
In the statistical arena, dealing with lifetime data is a main concern in several fields,
including engineering, biomedical sciences, finance, and economics. Many continuous dis-
tributions have been introduced for dealing with such data because they can present a
better fit than the based distribution.
The formation of new probability model devoted to serve for a vast range of real-world
problem along with admissible statistical methodologies. In the last thirty years, there
has been growingly interest in the formulation of extended flexible models of distribution
in several areas of significance, where the original distribution fails for complex data.
Several authors developed the weighted model in the literature when a sample is not noted
with equal probability. For reference, [3–11]. The weighted model was first formulated

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6148

Email addresses: izhar.stats@gmail.com (M. I. Khan), amelsayed@mans.edu.eg (A. Mustafa)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 2 of 15

by Fisher [12] and showed its usefulness in ecological sciences, biostatistics, medicine,
pharmacy, and environmental sciences.
However, there are still many situations where classical and weighted models are unsuitable
for real-world data. To address this issue, we formulate the concept of the survival-
weighted model to get the new distribution that enhances the flexibility of the existing
new model technique for better adaption to complex data. The outcomes drawn from
them appear quite sound, genuine and optimal.

2. Preliminaries

The survival-weighted model used in this article possesses the advantages of easy im-
plementation, better fitting performance and solid theoretical foundation.
A random variable X is said to follow a Pareto distribution (PD) if it possesses the fol-
lowing probability density function (PDF)and the cumulative distribution function (CDF)
respectively

f(x, α, β) =
αβα

xα+1
, α, β > 0, x ≥ β, (1)

F (x, α, β) = 1−
(
β

x

)α

, α, β > 0, x ≥ β. (2)

A random variable X has the PDF f(x). Then, the survival weighted distribution (SWD)
of X is given as

fSWD(x) =
S(x)f(x)

E(S(x)
(3)

where S(x) and f(x) are the SF and PDF of the base distribution.
The PDF of the SWPD distribution is generated from Equation (3) as follows

fSWPD(x) =
2αβ2α

x2α+1
, α, β > 0, x ≥ β, (4)

Figure 1: The plots of fSWPD(x) at β = 0.7.

The plots of PDFs are decreasing as shown in Figure 1 at β = 0.7.



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 3 of 15

The CDF of SWPD is stated by

FSWPD(x) = 1−
(
β

x

)2α

, α, β > 0, x ≥ β (5)

The SF and hazard rate function (HRF)

SSWPD(x) =

(
β

x

)2α

, α, β > 0, x ≥ β, (6)

hSWPD(t) =
f(t)

F (t)
=

2α

t
, (7)

where α and β represent the shape and scale parameters, respectively.

Figure 2: The plots of h(t) at β = 0.7.

The h(t) of SWPD distribution decreasing depending on the shape parameter.

The rest of manuscript outlined as follows. Section 3 provides order statistics from SWPD
and its measures, whereas Section 4 contains recurrence relations. Section 5 is devoted to
characterizing the distribution through HRF and supported with an application as well,
while Section 6 introduces measures of inequality. Section 7 discusses some concluding
remarks.

3. Order Statistics

The order statistics (O. S.) plays a key role in several areas of statistics and probabil-
ity. For example, Meteorology, quality control, sports hydrology, several characterizations
of probability distributions, detection of outliers, and so on. The voluminous theory and
application of order statistics are presented by David [13] and Arnold et al. [14].

Let X1, X2, · · · , Xn be a random sample of size n from (4). Let X1:n ≤ X2:n ≤ · · · ≤
Xn:n denote the O.S. Then PDF of r-th O. S. is given by David and Nagaraja [15] for
1 ≤ r ≤ n

fr:n(x) = C(r, n)[F (x)]r−1[1− F (x)]n−rf(x), 0 ≤ x < ∞, (8)



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 4 of 15

where

C(r, n) =
n!

(r − 1)!(n− r)!
.

The PDF of rth r-th O.S. of the SWPD is given by

fr:n(x;α, β) = C(r, n)

[
1−

(
β

x

)2α
]r−1(

β

x

)2(n−r+1)α 2α

x
. (9)

Using binomially expansion of

[
1−

(
β
x

)2α
]r−1

, we have

fr:n(x;α, β) = C(r, n)

r−1∑
i=0

(
r − 1

i

)
(−1)i

(
β

x

)2(n−r+i+1)α 2α

x
. (10)

Figure 3: The PDF of rth O.S., when n = 4, α = 0.7 and β = 2.0.

At r = 1, we get the PDF of rth smallest O.S.

f1:n(x;α, β) =
2nα

x

(
β

x

)2nα

. (11)

Figure 4: The plots of f1:n(x;α, β) at β = 2 and n = 4.



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 5 of 15

The PDF of rth largest O.S. is obtained at r = n, as follows.

fn:n(x;α, β) = n

n−1∑
i=0

(
n− 1

i

)
(−1)i

(
β

x

)2(i+1)α(2α

x

)
. (12)

Figure 5: The plots of fn:n(x;α, β) at β = 2 and n = 4.

3.1. Single Moments

An efficient mechanism for calculating the statistical features such as mean, dispersion,
symmetry, and kurtosis of all order statistics for any distribution, single moments have
assumed a significant interest and are often used as a pivotal tool for estimating future
events. For example, see David [13] and Arnold et al. [14].

Theorem 1. Let X1, X2, · · · , Xn be a random sample from SWPD with CDF and PDF
denoted by F (x) and f(x) respectively and let X(1) ≤ X(2) ≤ · · · ≤ X(n) be the correspond-

ing O.S. Then the expected value of Xr:n which is the kth moments of the rth O.S. for

k = 1, 2, · · · denote µ
(k)
r:n is given by.

µ(k)
r:n = C(r, n)

r−1∑
i=0

(
r − 1

i

)
(−1)i

2αβk

2(n− r + i+ 1)α− k
. (13)

Proof. We have

µk
r:n = E(Xk

r:n) =

∫ ∞

−∞
xkfr:n(x;α, β)dx

= = C(r, n)
r−1∑
i=0

(
r − 1

i

)
(−1)i2αβ2(n−r+i+1)α

∫ ∞

β
xk−1−2(n−r+i+1)αdx.

= C(r, n)

r−1∑
i=0

(
r − 1

i

)
(−1)i2αβ2(n−r+i+1)α

[
xk−2(n−r+i+1)α

k − 2(n− r + i+ 1)α

]∞

β

= C(r, n)
r−1∑
i=0

(
r − 1

i

)
(−1)i

2αβ2(n−r+i+1)α

k − 2(n− r + i+ 1)α

[
1

x2(n−r+i+1)α−k

]∞
β



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 6 of 15

= C(r, n)
r−1∑
i=0

(
r − 1

i

)
(−1)i

2αβk

2(n− r + i+ 1)α− k
,

for 2(n− r + i+ 1)α− k ≥ 0.
This completes the proof.

Remark 1. The kth moments for minimum O.S. is as follows

µ
(k)
1:n =

2nαβk

2nα− k
.

Remark 2. The kth moments for maximum O.S. is as follows.

µ(k)
n:n = n

n−1∑
i=0

(
n− 1

i

)
(−1)i

2αβk

2(k + 1)α− k
.

Equation (13) enables to perform the higher moments and other statistical features for all
O. S. for 1 ≤ n ≤ 4 and varying values of α ∈ (0.6− 4.2) at β = 0.7.

Table 1: Values of µ
(1)
r:n = µr:n.

n = 1 n = 2 n = 3 n = 4
α µ1:1 µ1:2 µ2:2 µ1:3 µ2:3 µ3:3 µ1:4 µ2:4 µ3:4 µ4:4

0.6 4.2000 1.2000 7.2000 0.9692 1.6615 9.9692 0.8842 1.2243 2.0988 12.5927
1.2 1.2000 0.8842 1.5158 0.8129 1.0268 1.7603 0.7814 0.9074 1.1462 1.9650
1.8 0.9692 0.8129 1.1256 0.7714 0.8959 1.2404 0.7522 0.8290 0.9627 1.3330
2.4 0.8842 0.7814 0.9870 0.7522 0.8397 1.0607 0.7385 0.7936 0.8858 1.1190
3.0 0.8400 0.7636 0.9164 0.7412 0.8086 0.9703 0.7304 0.7734 0.8437 1.0125
3.6 0.8129 0.7522 0.8736 0.7340 0.7888 0.9160 0.7252 0.7604 0.8171 0.9489
4.2 0.7946 0.7443 0.8449 0.7289 0.7751 0.8798 0.7215 0.7513 0.7988 0.9068

Table 2: Values of µ
(2)
r:n.

n = 1 n = 2 n = 3 n = 4

α µ
(2)
1:1 µ

(2)
1:2 µ

(2)
2:2 µ

(2)
1:3 µ

(2)
2:3 µ

(2)
3:3 µ

(2)
1:4 µ

(2)
2:4 µ

(2)
3:4 µ

(2)
4:4

0.6 -0.7350 2.94 00 -4.4100 1.1025 6.615 -9.9225 0.8400 1.8900 11.34 -17.01
1.2 2.940 0.840 5.040 0.679 1.163 6.979 0.619 0.857 1.469 8.815
1.8 1.103 0.6785 1.527 0.601 0.833 1.874 0.569 0.698 0.967 2.176
2.4 0.840 0.619 1.061 0.569 0.719 1.232 0.547 0.635 0.802 1.376
3.0 0.735 0.588 0.882 0.551 0.662 0.992 0.535 0.601 0.722 1.083
3.6 0.679 0.569 0.788 0.540 0.627 0.868 0.527 0.580 0.674 0.933
4.2 0.643 0.556 0.730 0.532 0.604 0.793 0.521 0.566 0.642 0.843

The numerical values reported in Tables 1-4 are validated by the following relation

n∑
r=1

µr:n = nE(X), (Arnold et al. [14]).



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 7 of 15

Table 3: Values of µ
(3)
r:n.

n = 1 n = 2 n = 3 n = 4

α µ
(3)
1:1 µ

(3)
1:2 µ

(3)
2:2 µ

(3)
1:3 µ

(3)
2:3 µ

(3)
3:3 µ

(3)
1:4 µ

(3)
2:4 µ

(3)
3:4 µ

(3)
4:4

0.6 -0.229 -1.372 0.915 2.058 -8.232 1 5.488 0.915 5.488 -21.952 14.635
1.2 -1.372 0.915 -3.659 0.588 1.568 -6.272 0.499 0.855 2.281 -9.123
1.8 2.058 0.588 3.528 0.475 0.814 4.885 0.433 0.599 1.028 6.170
2.4 0.915 0.499 1.330 0.433 0.630 1.681 0.407 0.514 0.747 1.992
3.0 0.686 0.457 0.915 0.412 0.549 1.098 0.392 0.470 0.627 1.254
3.6 0.588 0.433 0.743 0.398 0.503 0.863 0.383 0.445 0.562 0.963
4.2 0.534 0.418 0.650 0.389 0.474 0.737 0.377 0.428 0.521 0.809

Table 4: Values of µ
(4)
r:n.

n = 1 n = 2 n = 3 n = 4

α µ
(4)
1:1 µ

(4)
1:2 µ

(4)
2:2 µ

(4)
1:3 µ

(4)
2:3 µ

(4)
3:3 µ

(4)
1:4 µ

(4)
2:4 µ

(4)
3:4 µ

(4)
4:4

0.6 -0.103 -0.360 0.154 -2.161 3.241 -1.389 1.441 -12.965 19.448 -8.335
1.2 -0.36 1.441 -2.161 0.540 3.241 -4.862 0.412 0.926 5.557 -8.335
1.8 -2.161 0.540 -4.862 0.381 0.858 -7.722 0.332 0.528 1.188 -10.692
2.4 1.441 0.412 2.470 0.332 0.570 3.419 0.303 0.42 0.72 4.319
3.0 0.72 0.360 1.080 0.309 0.463 1.389 0.288 0.37 0.556 1.667
3.6 0.54 0.332 0.748 0.295 0.408 0.918 0.279 0.342 0.474 1.066
4.2 0.458 0.315 0.602 0.285 0.375 0.715 0.273 0.324 0.425 0.812

Based on the numerical results in Table 5, it is clear that the relationship between the
variance and the α value is an inverse relationship for all r and n. The variability ordering
between O.S. are verified for the sample size n = 4 by the following relation

V ar(Xi:n) ≤ V ar(Xj:n) for 1 ≤ i ≤ j ≤ n (David and Groeneveld [16]).

Based on Table 6:

• When r = n, the skewness decreases with increasing α (an inverse relationship).

• When r < n, the skewness decreases with increasing α only when α < 4.2 (an inverse
relationship).

• The skewness is positive for all r and n when α > 1.2.

Table 7 reports the leptokurtic and platykurtic kurtoses for all r and n.

Table 5: The variance.

α X1:1 X1:2 X2:2 X1:3 X2:3 X3:3 X1:4 X2:4 X3:4 X4:4

0.6 NA 1.50000 NA 0.16315 3.85442 NA 0.0582 0.3911 6.9350 NA
1.2 1.5000 0.0582 2.7424 0.0177 0.1088 3.8798 0.0083 0.0336 0.1553 4.9537
1.8 0.1632 0.0177 0.2595 0.0063 0.0301 0.3349 0.0032 0.0112 0.0402 0.3988
2.4 0.0582 0.0083 0.0869 0.0032 0.0137 0.1071 0.0016 0.0054 0.0178 0.1233
3.0 0.0294 0.0049 0.0422 0.0019 0.0077 0.0508 0.0010 0.0033 0.0098 0.0573
3.6 0.0177 0.0032 0.0247 0.0012 0.0049 0.0292 0.0007 0.0021 0.0063 0.0327
4.2 0.0117 0.0022 0.0161 0.0009 0.0034 0.0189 0.0004 0.0015 0.0043 0.0209



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 8 of 15

Table 6: The skewness.

α X1:1 X1:2 X2:2 X1:3 X2:3 X3:3 X1:4 X2:4 X3:4 X4:4

0.6 NA NA NA 10.2155 NA NA 4.9204 9.0624 NA NA
1.2 -4.6268 4.9204 NA 3.2636 4.1901 -4.2155 3.0318 2.6979 3.9328 NA
1.8 10.2155 3.2636 9.2695 2.3671 2.7466 8.9269 2.7208 2.0570 2.4869 8.7628
2.4 4.9204 3.0318 4.3499 2.7208 2.2499 4.1723 2.3219 2.1071 1.9748 4.0743
3.0 3.8103 2.3128 3.3521 1.4999 2.2550 3.1559 3.6217 1.3282 1.9670 3.0766
3.6 3.2636 2.7208 2.8797 2.5938 2.1081 2.7119 0.9045 1.6578 1.5117 2.5963
4.2 3.1420 3.0583 2.6535 5.8943 1.9145 2.4035 7.2964 2.9591 1.5911 2.3333

Table 7: The kurtosis .

α X1:1 X1:2 X2:2 X1:3 X2:3 X3:3 X1:4 X2:4 X3:4 X4:4

0.6 -2.984 11.292 -2.989 -246.927 9.737 NA 92.298 -193.42 9.258 -2.996
1.2 11.292 92.297 9.795 26.214 69.617 9.316 22.314 19.067 62.291 9.082
1.8 -246.93 26.214 -207.24 14.265 19.572 -194.05 -45.123 13.216 17.476 -187.79
2.4 92.298 22.314 75.785 -45.123 14.596 70.553 -69.485 10.129 13.239 68.004
3.0 38.319 15.829 31.150 127.779 9.048 29.129 -193.371 -20.352 12.139 28.049
3.6 26.214 -45.123 21.653 249.789 12.950 19.959 436.840 -14.676 19.807 19.108
4.2 16.969 -44.364 18.678 -765.382 46.872 16.046 2373.51 -0.411 -6.663 16.154

3.2. Joint Probability Density Function of Order Statistics

On using binomial expansion, the joint PDF of Xr:n and Xs:n of SWPD is given by

fr,s:n(x, y) = Cr,s:n

r−1∑
i=0

s−r−1∑
j=0

(
r − 1

i

)(
s− r − 1

j

)
(−1)i+jβ2α(n−r+i+1)

(
4α2

xy

)
×

(
β

x

)2α(s+i−r−j)(β

y

)2α(n+j−s+1)

, 1 ≤ r ≤ s ≤ n. (14)

where

Cr,s:n =
n!

(r − 1)!(s− r − 1)!(n− s)!
.

The sketches of the joint PDF for r = 1, 2, 3, s = 2, 3, 4 and n = 4.

4. Recurrence Relations

Calculating the moments of O. S. is a difficult task for some probability models. Re-
cursive computations are considered to meet these difficulties. Applications of recursive
computation based on order statistics have been well demonstrated by notable authors
see, Balakrishnan and Malik [17], Balakrishnan et al. [18], Ali and Khan [19], Kumar et
al. [20, 21], Khan and Mustafa [22], Khan [23] and Akhtar at el. [24, 25] in detail.

Theorem 2. As stated in Theorem 1, the relation for single moments as follows

µ(i)
r:n =

2α(n− r + 1)

i

[
µ(i)
r:n − µ

(i)
r−1:n

]
(15)



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 9 of 15

Figure 6: The joint PDF for r = 1, 2, 3, s = 2, 3, 4 and n = 4.

Proof. We know that

µ(i)
r:n =

∫ ∞

−∞
xifr(x)dx

= Cr:n

∫ ∞

β
xi[F (x)]r−1[1− F (x)]n−rf(x)dx (16)

Using Equation (5) in Equation (16), we have

µ(i)
r:n = Cr:n2α

∫ ∞

β
xi−1[F (x)]r−1[1− F (x)]n−r+1dx. (17)

Integrating Equation (17) by parts and rephrase, the Equation (15) is proved.

5. Characterization

Characterization helps a researcher identify the actual probability distribution under
certain conditions. For a comprehensive study, researchers referred to the contributions of
Ahsanullah et al. [26, 27] and Hamdani [28]. The hazard rate function is used to analyze
extreme values from a probability model of a distribution. An application also presents
the behavior of the graph. We characterize the SWPD through HRF, and the conditions
for characterization are described as follows.



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 10 of 15

Lemma 1. A RV T : Ω → (0,+∞) has a continuous PDF f(t) if and only if the HRF
h(t) satisfies the following equations

f
′
(t)

f(t)
=

h
′
(t)

h(t)
− h(t). (18)

Proof. As stated by the definition of HRF, mentioned in Equation 7, it follows that

h
′
(t)

h(t)
=

f
′
(t)F (t) + f2(t)

F
2
(t)

×
(
F (t)

f(t)

)
=

f
′
(t)

f(t)
+ h(t).

Lemma 1 is proved.

Theorem 3. A RV T : Ω → (β,+∞) has a SWPD (α, β) if and only if the HRF h(t)
satisfies the following equations

h
′
(t)

(h(t))2
= − 1

2α
. (19)

Proof. Necessary part:
The logarithm of fSWP(t) is noted as

ln(fSWP(t)) = 2 ln(α) + 2α ln(β)− (2α+ 1) ln(t) (20)

Differentiate Equation (20) with respect to t, we obtain

f
′
(t)

f(t)
= −(2α+ 1)

t
(21)

Using Equations (7) and (18) the above equation follows that

h
′
(t)

h(t)
=

f
′
(t)

f(t)
+ h(t) = −(2α+ 1)

t

after simplification yields Equation (19).

Sufficiency part:
Suppose that Equation (19) holds. Then the following integration is considered.∫

h
′
(t)

(h(t))2
dt = −

∫
1

2α
dt

1

h(x)
=

t

2α
.

Which is the same as Equation (7).



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 11 of 15

Furthermore, by replacing the HRF in Equation (18) and employing integration, we get∫
f

′
(t)

f(t)
dt = −

∫
(2α+ 1)

t
dt+ C

f(t) = Ct−(2α+1)

Using boundary condition, value of C is 2αβ2α. Thus f(t) is PDF from SWPD(α, β).
This completes the proof.

5.1. Application

This subsection provides the application of characterization result. For this purpose,
we consider the data set that includes given by Fan and Fan [29]. A random number is
used to choose the various random time. For SWPD(α, β), the original dataset is divided
by minimum value (0.33) to convert into the interval (1,α).

Table 8: The time to repair data of a piece of construction equipment.

0.33 0.33 0.50 0.50 0.50 0.95 1.00 1.02 1.17 1.72

1.83 3.20 4.35 5.25 6.52 7.25 8.58 10.25 11.58 13.83

15.93 27.40 27.43 31.93 38.37 40.02 62.77 88.27 92.90

Table 9: Estimated hazard rates of SWPD(α, β) for different parameters .

α
r t 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

4 1.52 1.32 1.97 2.63 3.30 3.95 4.60 5.26 5.92 6.58
8 3.09 0.65 0.97 1.29 1.62 1.942 2.27 2.59 2.91 3.24
13 13.18 0.15 0.23 0.303 0.379 0.455 0.53 0.61 0.68 0.76
21 48.27 0.04 0.062 0.082 0.104 0.124 0.15 0.17 0.19 0.21
27 190.21 0.01 0.015 0.021 0.206 0.032 0.04 0.04 0.05 0.05

Table 9 is endorsed the behavior of hazard rate of SWPD(α, β) as depicted in Figure 2.

6. Measure of Inequality

Since, SWPD(α, β) possess the heavy tail properties. Therefore, it becomes necessary
to study its inequality behavior. Assume that the RV X is a non-negative with continuous
and twice differentiable CDF.
If X ∼ SWPD(α, β), then the Lorenz curve L(p), Bonferroni curve B(p) and Gini index
are defined as

LSWPD(p) =
1

µ

∫ y

0
xf(x)dx, 0 < p < 1



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 12 of 15

= 1− (1− p)1−
1
2α .

Bonferroni curve B(p) is given as

BSWPD(p) =
1

pµ

∫ y

0
xf(x)dx, 0 < p < 1

Gini index:

GSWPD(x) = 1− 1

µ

∫ ∞

0
(1− F (x))2dx =

1

µ

∫ ∞

0
F (x)(1− F (x))dx

=
1

4α− 1
, 4α− 1 > 0.

Table 10: Measures of inequality.

α p = 0.10 p = 0.25 p = 0.50 p = 0.75 p = 0.95 G(x)

1 L(p) 0.0513 0.1340 0.2980 0.5000 0.7764 0.33
B(p) 0.513 0.5359 0.5856 0.6667 0.8172

1.5 L(p) 0.0678 0.1745 0.3700 0.6031 0.8643 0.20
B(p) 0.678 0.6980 0.7400 0.8041 0.9098

2.0 L(p) 0.0759 0.1941 0.4054 0.6464 0.8943 0.14
B(p) 0.759 0.7763 0.8108 0.8618 0.9413

2.5 L(p) 0.0808 0.2056 0.4257 0.6701 0.9090 0.11
B(p) 0.808 0.8223 0.8513 0.8935 0.9568

3.0 L(p) 0.0840 0.2132 0.4388 0.6850 0.9176 0.09
B(p) 0.840 0.8527 0.8775 0.9133 0.9659

3.5 L(p) 0.0864 0.2185 0.4480 0.6952 0.9233 0.076
B(p) 0.864 0.8741 0.8959 0.9269 0.9719

4.0 L(p) 0.0881 0.2225 0.4548 0.7026 0.9273 0.066
B(p) 0.881 0.8901 0.9095 0.9368 0.9761

4.5 L(p) 0.0894 0.2256 0.4510 0.7084 0.9303 0.058
B(p) 0.894 0.9025 0.9199 0.9445 0.9792

5.0 L(p) 0.0990 0.2281 0.4641 0.7128 0.9325 0.053
B(p) 0.990 0.9124 0.9282 0.9504 0.9816

We notice from Table 10 both L(p) and B(p) are increasing as α increases but Gini
coefficient is decreasing for the same value.

7. Conclusion

In this article, we study some features of order statistics for SWPD. We obtained
the recurrence relation for single moments and calculated the expected values, variances,
skewness, and kurtosis for different values of parameters. The model is characterized



M. I. Khan, A. Mustafa / Eur. J. Pure Appl. Math, 18 (2) (2025), 6148 13 of 15

through the hazard rate function and supported with real data. The measure of inequality
is also considered. The results contained in this paper will provide theoretical and practical
insights into the fields of mathematical statistics, finance, actuarial sciences, risk analysis
and medical sciences. The future work related to this research, one may generate new
model using transmutation map and stress- strength of SWPD.

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their
useful suggestions and comments.

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