EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2182-2195
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Order Statistics and Actuarial Measures from Powered
Inverse Rayleigh Distribution

M. I. Khan

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah-
42351, Kingdom of Saudi Arabia

Abstract. Nashaat [25] introduced the powered inverse Rayleigh (PIR) distribution. It provides
a better fit other than (inverse Rayleigh, Rayleigh, and Weibull) distributions. The moments of
order statistics and recurrence relations for the single and double moments have been established.
The computation of the means and variances are enumerated. These computations can be truly
interesting and applied in numerous domains of study. Moreover, cumulative entropy (C.E.) and
actuarial measures (A.M.) are also calculated to address the uncertainty in portfolio optimization.
The usages of C. E. and A.M. are widespread in many real-word applications specifically in physical
sciences and insurance science.

2020 Mathematics Subject Classifications: 60E05, 62E15, 62G30, 62F10

Key Words and Phrases: Order statistics, Entropy, Single and double moments, Recurrence
relations, Actuarial measures

1. Introduction

Trayer [30] discussed inverse Rayleigh (IR) distribution. Voda [31] documented its
wide ranges of applicability in several areas of applied and allied sciences. Since then, IR
distribution is steadily growing and drawing attention by several researchers via different
modifications. The basic idea of the new model is to get the more accurate result of com-
plex data. Some notable works are listed below.
Helbaway and Monem [1] and Sindhua et al. [29] estimated the parameters of IR dis-
tribution for complete and censored samples using the Bayesian approach. Merrovci [23]
presented transmuted IR distribution. Khan [18] introduced modified IR distribution.
Khan and King [21] introduced transmuted modified IR distribution. Haq [15] presented
transmuted exponentiated IR distribution. Rao and Mbwambo [27] established exponen-
tiated IR distribution. Khan [20] obtained moments properties of PIR distribution based
on dual generalized order statistics. Mustafa and Khan [24] introduced the length-biased
PIR distribution. Recently, Khan and Mustafa [19] presented PIR distribution using DUS

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5211

Email address: izhar.stats@gmail.com, khanizhar@iu.edu.sa (M. I. Khan)

https://www.ejpam.com 2182 © 2024 EJPAM All rights reserved.



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2183

transformation.
A random variable (r.v.) X : (Ω→ (0,∞)) is said to have PIR distribution, if its proba-
bility density function (PDF ) is given by

f(x, α, θ) =
2α

θx2α+1
e−

1
θx2α , x > 0, α, θ > 0 (1)

The cumulative distribution function (CDF ) of (1) is

F (x) = e−
1

θx2α , x > 0, α, θ > 0. (2)

Hazard rate function:

HPIRD(x) =
2α

θx2α+1
(3)

Reliability function:

SPIRD(x) = 1− e−
1

θx2α . (4)

The following functional relationship exists from (1)and (2).

f(x) =
2α

θx2α+1
F (x). (5)

The PIR distribution has a tremendous application in finance, stock market and biological
sciences. Note that the PIR distribution involves inverse Rayleigh, Rayleigh, and Weibull
distributions as a sub class.
The moments of order statistics (O.S.) have been enumerated quite significantly for some
probability models: Joshi [17], David [10], Mohie El-Din et al. [13], and Arnold et al.[4].
Goodness of fit tests, Hegazy et al. [16] and Glen et al. [14]. David and Nagaraja [11]
and Arnold et al. [5] have documented and explored the characterization of distribution
using O.S.The application of moments of O.S. can be especially noticed in areas including
reliability theory and quality control processes.
In this manuscript, moments of O.S. are derived from PIR distribution. The tabulation
of descriptive measures based on smallest, largest O.S. and C.E. are tabulated for some
fixed parameters in Section 2. Moreover, recurrence relations based on single and double
moments are extracted in Section 3. Actuarial measures are reported in Section 4. and
Section 5 is reported conclusion.

2. Order Statistics

In quality control processes and reliability theory, the O.S. contributes an important
feature in forecasting the time to fail of a specific item by reviewing few early failures,
Dey et al [12].
A sequence of r.v′s. are arranged in their magnitude of ascending order referred to O.S. Let
X1:n ≤ X2:n ≤ · · · ≤ Xn:n denote the O.S. Then the PDF of kth O.S. Xk:n for 1 ≤ k ≤ n
is reported by David and Nagaraja [11].

fk(x) = Ck:n[F (x)]k−1[1− F (x)]n−kf(x),−∞ < x < ∞ (6)



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2184

where

Ck:n =
n!

(k − 1)!(n− k)!

The PDF of kth O.S. is written as follows.

fk(x) = Ck:n

[
e−

1
θx2α

]k−1[
1− e−

1
θx2α

]n−k 2α

θx2α+1
e−

1
θx2α . (7)

Using binomial expansion of
[
1− e−

1
θx2α

]n−k
, we have

fk(x) = Ck:n

n−k∑
t=0

(
n− k

t

)
(−1)te−

(k+t)

θx2α
2α

θx2α+1
. (8)

For k = 1, we obtain the PDF of kth smallest O.S. as:

f1(x) = n
n−1∑
t=0

(
n− 1
t

)
(−1)te−

(t+1)

θx2α
2α

θx2α+1
. (9)

Similarly, k = n, we obtain the PDF of kth largest O.S. as:

fn(x) = ne−
n

θx2α
2α

θx2α+1
. (10)

2.1. Moments of kth O. S.

In this subsection, we derive the moments of O.S. when parent population is PIR
distribution.

Theorem 1. Let X1, X2, . . . , Xn be a random sample (R.S.) of size n from PIR distri-
bution and let X1:n, X2:n, . . . , Xn:n mark the corresponding the O.S. Then ith moments of

the kth O.S. for i = 1, 2, . . . denoted by µ
(i)
k:n is given by

µ
(i)
k:n = Ck:n

n−k∑
t=0

(
n− r
t

)
(−1)t(k + t)

i
2α

−1

(
1

θ

) i
2α

γ

(
1− i

2α

)
, i = 1, 2, 3, 4. (11)

Proof: We know that

µ
(i)
k:n =

∫ ∞

−∞
xifk(x)dx

= Ck:n

n−k∑
t=0

(
n− k

t

)
(−1)t

∫ ∞

0
xie−

(k+t)

θx2α
2α

θx2α+1
dx. (12)

Letting u = (k+t)
θx2α in (12), yields (11).



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2185

Remark 1. The kth moments for smallest O.S. from (11) is as follows.

µ
(i)
1 = n

n−1∑
t=0

(
n− 1
t

)
(−1)t(t+ 1)

i
2α

−1

(
1

θ

) i
2α

Γ

(
1− i

2α

)
First and second order moments of kth smallest O.S. can be obtained at i = 1, 2.

µ
(1)
1 = n

n−1∑
t=0

(
n− 1
t

)
(−1)t(t+ 1)

1
2α

−1

(
1

θ

) 1
2α

Γ

(
1− 1

2α

)
and

µ
(2)
1 = n

n−1∑
t=0

(
n− 1
t

)
(−1)t(t+ 1)

1
α
−1

(
1

θ

) 1
α

Γ

(
1− 1

α

)
Therefore, the variance for kth smallest O.S. can be obtained as follows.

V ar(X1) = µ
(2)
1 − [µ

(1)
1 ]2

Table 1: Values of µ
(i)
1:n for smallest O.S. when n = 4

µ
(1)
1:n

α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 0.898 0.782 0.721 0.680 0.651
3.0 0.913 0.814 0.76 0.725 0.698
3.5 0.925 0.837 0.79 0.758 0.735
4.0 0.933 0.856 0.814 0.785 0.763

µ
(2)
1:n

2.5 0.818 0.620 0.527 0.470 0.430
3.0 0.842 0.669 0.584 0.531 0.493
3.5 0.861 0.706 0.629 0.580 0.544
4.0 0.876 0.737 0.666 0.619 0.586

µ
(3)
1:n

2.5 0.756 0.499 0.391 0.329 0.288
3.0 0.785 0.555 0.453 0.393 0.351
3.5 0.808 0.600 0.505 0.446 0.405
4.0 0.827 0.638 0.548 0.492 0.452

µ
(4)
1:n

2.5 0.711 0.408 0.295 0.235 0.196
3.0 0.740 0.466 0.356 0.294 0.253
3.5 0.764 0.514 0.408 0.346 0.305
4.0 0.785 0.555 0.453 0.393 0.351



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2186

Table 1 reveals that for fixed value of θ moments are increasing when α is increasing.

Table 2: Values of mean, variance, C.V, skewness, and kurtosis for the smallest O.S. when n = 4

Mean = µ
(1)
1:n

α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 0.898 0.782 0.721 0.680 0.651
3.0 0.913 0.814 0.76 0.725 0.698
3.5 0.925 0.837 0.79 0.758 0.735
4.0 0.933 0.856 0.814 0.785 0.763

Variance
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 0.0117 0.0089 0.0076 0.0067 0.0062
3.0 0.0083 0.0066 0.0058 0.0052 0.0049
3.5 0.0062 0.0051 0.0046 0.0042 0.0039
4.0 0.0048 0.0041 0.0037 0.0034 0.0032

C.V.
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 12.057 12.057 12.057 12.057 12.057
3.0 9.995 9.995 9.995 9.995 9.995
3.5 8.536 8.536 8.536 8.536 8.536
4.0 7.450 7.450 7.450 7.450 7.450

Skewness
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 0.809 0.809 0.809 0.809 0.809
3.0 0.734 0.734 0.734 0.734 0.734
3.5 0.682 0.682 0.682 0.682 0.682
4.0 0.643 0.643 0.643 0.643 0.643

Kurtosis
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 4.426 4.426 4.426 4.426 4.426
3.0 4.178 4.178 4.178 4.178 4.178
3.5 4.021 4.021 4.021 4.021 4.021
4.0 3.914 3.914 3.914 3.914 3.914

Table 2 exhibits that variances, skewness and kurtosis are decreasing when α is in-
creasing except momentsat fixed value of θ.

Remark 2. The kth moments for largest O.S. from (11) is as follows.

µ(i)
n = n

i
2α

(
1

θ

) i
2α

Γ

(
1− i

2α

)
.



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2187

Table 3: Values of µ
(i)
n:n for largest O. S. when n = 4.

µ
(1)
n:n

α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 1.536 1.337 1.233 1.164 1.113
3.0 1.422 1.267 1.184 1.129 1.088
3.5 1.348 1.221 1.152 1.106 1.071
4.0 1.296 1.188 1.13 1.09 1.06

µ2
n:n

2.5 2.593 1.965 1.671 1.489 1.362
3.0 2.15 1.706 1.49 1.354 1.257
3.5 1.896 1.555 1.385 1.276 1.197
4.0 1.733 1.457 1.317 1.225 1.159

µ3
n:n

2.5 5.096 3.362 2.636 2.218 1.94
3.0 3.545 2.507 2.047 1.772 1.585
3.5 2.823 2.098 1.763 1.559 1.416
4.0 2.413 1.86 1.598 1.435 1.319

µ4
n:n

2.5 13.917 7.993 5.779 4.591 3.84
3.0 6.751 4.253 3.245 2.679 2.309
3.5 4.565 3.072 2.437 2.068 1.820
4.0 3.545 2.507 2.047 1.772 1.585

Table 3 shows that for fixed value of θ moments are decreasing when α is increasing.



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2188

Table 4: Values of mean, variance, C.V, skewness, and kurtosis for the largest O. S. when n = 4

Mean = µ
(1)
n:n

α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 1.536 1.337 1.233 1.164 1.113
3.0 1.422 1.267 1.184 1.129 1.088
3.5 1.348 1.221 1.152 1.106 1.071
4.0 1.296 1.188 1.13 1.09 1.06

Variance
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 0.2329 0.1765 0.1501 0.1338 0.1223
3.0 0.1269 0.1007 0.088 0.08 0.0742
3.5 0.0792 0.0649 0.0578 0.0533 0.05
4.0 0.0538 0.0453 0.0409 0.0381 0.036

C.V.
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 31.414 31.414 31.414 31.414 31.414
3.0 25.051 25.051 25.051 25.051 25.051
3.5 20.873 20.873 20.873 20.873 20.873
4.0 17.907 17.907 17.907 17.907 17.907

Skewness
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 3.535 3.535 3.535 3.535 3.535
3.0 2.806 2.806 2.806 2.806 2.806
3.5 2.425 2.425 2.425 2.425 2.425
4.0 2.189 2.189 2.189 2.189 2.189

Kurtosis
α θ = 1 θ = 2 θ = 3 θ = 4 θ = 5

2.5 48.092 48.092 48.092 48.092 48.092
3.0 24.678 24.678 24.678 24.678 24.678
3.5 17.534 17.534 17.534 17.534 17.534
4.0 14.166 14.166 14.166 14.166 14.166

The behavior of Table 4 is that descriptive measures are decreasing when α is increasing
at the fixed value of θ.

2.2. The Joint PDF of kth and lth O.S.

The joint PDF of Xk:n and Xl:n is given by (Arnold et al. [4]) for 1 ≤ k ≤ l ≤ n.

fk,l(x, y) = Ck,l:n[F (x)]k−1[F (y)− F (x)]l−k−1[1− F (y)]n−lf(x)f(y),−∞ < x < y < ∞.
(13)



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2189

Equation (13) can be re-written using binomial expansion.

fk,l(x, y) = Ck,l:n

n−l∑
t=0

l−k−1∑
z=0

(
n− s
t

)(
l − k − 1

z

)
(−1)t+z[F (x)]k+z[F (y)]l−k+t−z

× f(x)

F (x)

f(y)

F (y)
.

Therefore, the joint PDF of Xk,l:n from PIR distribution.

fk,l(x, y) =
4α2

θ2
Ck,l:n

n−l∑
t=0

l−k−1∑
z=0

(
n− l
t

)(
l − k − 1

z

)
(−1)t+ze−

(
k+t

θx2α
+ l−k+t−z

θy2α

)

× 1

x2α+1
.

1

y2α+1
.

2.3. Cumulative Entropy

There are several types of entropies that exist in literature. Each one is employed for
a specific situation. The cumulative entropy (C.E.) is the most prominent version of the
entropy reported by Crescenzo and Longobardi [9] in (14).

CE(X) = −
∫ ∞

0
F (x)lnF (x)dx (14)

The C.E. for (1) is.

CE(X) =
1

2α

(
1

θ

) 1
2α

Γ

(
1− 1

2α

)

Table 5: The nature of C.E. for PIR distribution.

θ
α 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.0 2.22 1.57 1.29 1.11 0.99 0.91 0.85 0.79 0.74 0.71
1.5 0.567 0.45 0.40 0.36 0.33 0.31 0.30 0.28 0.27 0.26
2.0 0.369 0.31 0.28 0.26 0.25 0.24 0.23 0.22 0.21 0.21
2.5 0.265 0.23 0.212 0.20 0.19 0.18 0.18 0.17 0.17 0.17
3.0 0.213 0.19 0.18 0.17 0.16 0.16 0.15 0.15 0.15 0.14

The value of C.E. is decreasing when α is increasing for the fixed value of θ.



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2190

3. Recurrence Relations based on O.S.

The recurrence relations based on O.S. and its applications have been well documented
via Balakrishnan and Malik [7], Balakrishnan et al. [8], Arnold and Balakrishnan [3], Ali
and Khan [2] and Samual and Thomas [28] in detail. To calculate the moments of O.S. is
a tedious task for some distribution. For this fact, recursive computation approaches are
repeatedly desired.

Theorem 2. As stated in Theorem 1, we have the following single moments relation:

µ
(i)
k:n =

(n− k + 1)

θ(i− 2α)

[
µ
(i−2α)
k:n − µ

(i−2α)
k−1:n

]
(15)

Proof: We know that

µ
(i)
k:n =

∫ ∞

−∞
xifk(x)dx

µ
(i)
k:n = Ck:n

∫ ∞

0
xi[F (x)]k−1[1− F (x)]n−kf(x)dx. (16)

Using (5) in (16), we have

µ
(i)
k:n = Ck:n

1

θ

∫ ∞

0
xi−2α[F (x)]k−1[1− F (x)]n−k+1dx. (17)

Integrating (17) by parts and simplifying yields (15).

Theorem 3. For 1 ≤ k ≤ l ≤ n, n ∈ N , we have the following product moment relations.

µ
(i1,i2)
k,l:n =

(n− l + 1)

θ(i2 − 2α)

[
µ
(i1,i2−2α)
k,l:n − µ

(i1,i2−2α)
k,l−1:n

]
(18)

Proof: We start from (13),

µ
(i1,i2)
k,l:n = Ck,l:n

∫ ∞

0

∫ ∞

x
xi1yi2fk,l(x, y)dydx (19)

or,

µ
(i1,i2)
k,l:n = Ck,l:n

∫ ∞

0
xi1 [F (x)]k−1f(x)Wxdx (20)

where

Wx =

∫ ∞

x
yi2 [F (y)− F (x)]l−k−1[1− F (y)]n−lf(y)dy

or,

Wx =
1

θ

∫ ∞

x
yi2−2α[F (y)− F (x)]l−k−1[1− F (y)]n−l+1dy.

Now integrating the above equation by parts, we get,

Wx =
1

θ

{
(n− l + 1)

(i2 − 2α)

∫ ∞

x
yi2−2α[F (y)− F (x)]l−k−1[1− F (y)]n−lf(y)dy

− l − k − 1

i2 − 2α

∫ ∞

x
yi2−2α[F (y)− F (x)]l−k−2[1− F (y)]n−l+1f(y)dy

}
.

Putting the values of Wx in (20), directly yields (18).



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2191

4. Actuarial Measures (A.M.)

The A.M. plays a leading role in insurance science via uncertainty. Due to its usefulness
in portfolio optimization, the interested readers refer to Panjer [26], Artzrner [6] and
Landsman [22]. We derive some important risks as follows.

4.1. Value at Risk:

It is represented with a confidence level q (typically 90%, 95% or 99%). The VaR of
R.V. X is the qth quantile of (2).

V aRq(x) = F−1(q). (21)

Therefore, the V aRq(x) of PIR distribution is given by.

Xq = [−θln (q) ]−
1
2α . (22)

The value of VaR is increasing at the different level of q for fixed values of α and θ.

4.2. Tail value at Risk:

The expected value of the loss, which is greater than the VaR is called Tail value at
Risk (TV aR).
By definition

TV aRq(x) =
1

1− q

∫ ∞

V aRq

xf(x)dx. (23)

Therefore, the TV aRq(x) from (23) is.

TV aRq(x) =
(θ)−

1
2α

1− q
γ

(
1− 1

2α
,

1

θ(V aRq)
2α

)
. (24)

4.3. Tail Variance:

The variability of the risk along the tail of distribution is known as Tail Variance (TV ).
It is determined as.

TVq(x) = E
[
X2|X > xq

]
− [TV aRq]

2. (25)

Therefore, theTV(X) of the PIR distribution is addressed in (26)

TVq(x) =
(θ)−

1
α

1− q
γ

(
1− 1

α
,

1

θ(V aRq)
2α

)
−

[
(θ)−

1
2α

1− q
γ

(
1− 1

2α
,

1

θ(V aRq)
2α

)]2
(26)

where

E
[
X2|X > xq

]
=

1

1− q

∫ ∞

V aRq

x2f (x) dx =
(θ)−

1
α

1− q
γ

(
1− 1

α
,

1

θ(V aRq)
2α

)



M. I. Khan / Eur. J. Pure Appl. Math, 17 (3) (2024), 2182-2195 2192

Table 6: V aRq(x) at θ = 1, 2, and 3 at different level of q.

θ = 1
α\q 75% 80% 85% 90% 95% 99%

0.5 3.45 4.55 6.25 9.09 20.0 100
1.0 1.86 2.13 2.50 3.02 4.47 10.0
1.5 1.51 1.66 1.84 2.09 2.71 4.64
2.0 1.36 1.46 1.58 1.74 2.11 3.16
2.5 1.28 1.35 1.44 1.55 1.82 2.51
3.0 1.23 1.28 1.36 1.44 1.65 2.15

θ = 2
α\q 75% 80% 85% 90% 95% 99%

0.5 1.72 2.22 3.03 4.76 10.0 50.0
1.0 1.31 1.49 1.74 2.18 3.16 7.07
1.5 1.20 1.30 1.45 1.68 2.15 3.68
2.0 1.15 1.22 1.32 1.48 1.78 2.66
2.5 1.12 1.17 1.25 1.37 1.58 2.19
3.0 1.10 1.14 1.20 1.29 1.47 1.92

θ = 3
α\q 75% 80% 85% 90% 95% 99%

0.5 1.16 1.49 2.04 3.13 6.67 33.3
1.0 1.08 1.22 1.43 1.77 2.58 5.78
1.5 1.05 1.14 1.27 1.46 1.88 3.22
2.0 1.04 1.11 1.20 1.32 1.61 2.40
2.5 1.03 1.08 1.15 1.26 1.46 2.02
3.0 1.02 1.07 1.13 1.21 1.37 1.79

4.4. Total Variance Premium:

The combination of TVq and TV aRq is called the Total Variance Premium (TVP). It
is definedas follows.

TV Pq(X) = TV aRq + δTVq (27)

where 0 < δ < 1.
Substituting the expressions (24) and (26) into (27) completes the proof.

5. Conclusion

The current study explores the moments of the O.S. from the PIR distribution. The
numerical computations are reported based on the O.S. Cumulative entropy is evaluated.
The expressions for single and double moments are setup. The actuarial measures are also
tabulated.



REFERENCES 2193

Acknowledgements

The author wishes to extend his sincere gratitude to the Deanship of Scientific Research
at the Islamic University of Madinah for support provided to the Post-Publication Program
(3).

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