Moments of Powered Rayleigh Distribution by Generalized Order
Statistics
M. I. KHAN
Department of Mathematics, Faculty of Science,
Islamic University of Madinah, Madinah 42351,
SAUDI ARABIA
Abstract: - The power Rayleigh distribution is considered via the power transformation technique. The
suggested model behaves better than the original distribution in dealing with complex data. The key focus of
this paper is to establish some simple recurrence relations for single and product moments of generalized order
statistics from the power Rayleigh distribution. Some numeric computations are carried out at the different
parameters of the power Rayleigh distribution. The moments of order statistics and record values are deduced
from the single and product moments. Further, characterization results are also derived for power Rayleigh
distribution via recurrence relations and conditional expectations.
Key-Words: - Generalized order statistics, order statistics, record value, powered Rayleigh distribution,
recurrence relations, conditional expectation, characterization.
1 Introduction
A mechanism of studying random variables (s)
arranged in ascending order is called generalized
order statistics . [1], was pioneer of this
theory. It contains like, order statistics, record,
progressive censoring, etc., as a particular cases.
The has vast applications in reliability, extreme
value, physical sciences, medical sciences, remote
censoring, and its applicability over several field of
studies are steadily growing.
Let
. Such that
for all based on . Further, let
and be the continuous distribution function
and probability density function of a
random variable . Then ,l) are
named and its joint takes the following
form.
(1)
Here we consider type I of :
Let
1 2 1i
u u u u
. The of
is.
(2)
The joint of
th
r
and
th
s
is given by,
ℎ
,
(3)
The conditional of
given, ,
is.
,
(4)
where,
,
and
ℎ.
If , 1, then reduces to
order statistic and when then
reduces to upper record values.
In recent years, recurrence relations based on
have received considerable attention. Several
authors have established the recurrence relations for
different distributions. For example, [2], [3], [4],
[5], [6], [7], [8] and there are significantly more.
Received: October 9, 2023. Revised: May 6, 2024. Accepted: June 29, 2024. Published: July 29, 2024.
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DOI: 10.37394/23206.2024.23.51
M. I. Khan
E-ISSN: 2224-2880
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The organization of the paper is as follows. Section
2 discusses powered Rayleigh distribution and
statistical properties. Recurrence relations are
addressed in Section 3. Characterization results are
proved in Section 4. The concluding remarks ends
in Section 5.
1.1 Powered Rayleigh Distribution
[9], introduced the Rayleigh distribution . It
has vast applications in reliability, clinical studies,
wireless signal, oceanography, wind energy and
lifetime equipment. Many generalizations of
have been noted in the existing literature.
[10], proposed the powered Rayleigh
distribution via power transformation of .
The is given by,
, (5)
where is the scale parameter and is the shape
parameter.
The is given by.
. (6)
For extensive information on see, [10].
From equation (5) and (6), we note that.
(7)
which will be utilized for deriving the recurrence
relations.
1.2 Statistical Measures of .
The
th
r
moments of is
The statistical properties are given in Table 1
and Table 2 in Appendix.
2 Recurrence Relations
This section addresses the recurrence relations for
single and product moments of from the .
2.1 Single Moments
Theorem 1. Let be a non-negative continuous
follows the . Suppose that and
then.
(8)
Proof: We have from (2),
(9)
Integrating by parts , we get.
which implies that,
.
On using (7), we have,
,
after simplification, Theorem 1 is proved.
Remark 2.1
(i) Setting in (8) result reduced for Rayleigh
distribution as follows,
agreed by [11].
(ii) Setting in (8) result reduced for
order statistic for distribution as follows.
.
(iii) Setting in (8) result reduced for
record values for distribution as follows,
.
(iv) Setting in Remark (iii) result reduced
for record values for as follows,
.
agreed by [12].
2.2 Product Moments
Theorem 2 Let be a non-negative continuous
follows the . Suppose that and
, then,
.
(10)
Proof: From (3), we have
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M. I. Khan
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(11)
where
.
Solving the integral by parts and substituting
the resulting expression in (11), we get:
,
after simplification (10) yields.
3 Characterizations
Characterization of via are presented
below.
Theorem 3 The necessary and sufficient condition
for a to be distributed with given in (5)
is that,
(12)
if and only if
,
(13)
Proof: From (8) necessary part proved. If the
relation given (12) is satisfied, then on organizing
the terms in (12) as given
(14)
Let,
. (15)
Differentiating of (15) both sides, we get:
.
Thus
(16)
On integration left hand side in (16) by parts
and putting the value of ,
which reduces to,
(17)
On using Müntz-Szász theorem [13] to (17), we get
which proves that has the form as in (5).
Theorem 4 Let be a non-negative having
with and for
all , then,
,
(18)
if and only if
, (19)
where
Proof: Necessary part:
From (4) for , we have
(20)
where
Now setting
in (20), we have
(21)
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Again, by setting in (21), we find that.
,
where,
.
The necessary part is verified.
Sufficiency part:
From (4) and (18), we have:
,
(22)
where,
.
Performing differentiation of (22) both sides with
respect to , we acquire
,
where
.
Therefore, by [14]
which confirms that has the form as in (5).
4 Conclusion
The results provided in this article will be useful for
researchers who are working in the domain of
mathematical statistics. It helps to obtain
exploratory analysis based on ordered random
variables.
References:
[1] Kamps, U. A Concept of Generalized Order
Statistics. Stuttgart: Teubner. 1995.
[2] Pawlas, P., and Szynal, D., Recurrence
relations for single and product moments of
generalized order statistics from Pareto,
generalized Pareto, and Burr distributions,
Communications in Statistics- Theory and
Methods, Vol. 30, No. 4, 2001, pp. 739–746.
doi.org/10.1081/STA-100002148.
[3] Ahmad, A. A. and Fawzy, M., Recurrence
relations for single and product moments of
generalized order statistics from doubly
truncated distributions, Journal of Statistical
and Planning Inference, Vol. 117, No. 2,
2002, pp. 241–249. DOI: 10.1016/S0378-
3758(02)00385-3.
[4] Athar, H and Islam, H. M. Recurrence
relations between single and product moments
of generalized order statistics from a general
class of distributions, Metron Vol. LXII,
2004, pp. 327-337.
[5] AL-Hussaini, E.K., Ahmad, A. A. M. and AL-
Kashif, M. A., Recurrence relations for
moment and conditional moment generating
functions of generalized order statistics,
Metrika, Vol. 61, 2005, pp. 199-220. DOI:
10.1007/s001840400332.
[6] Ahmad, A. A., Single and product moments
of generalized order statistics from linear
exponential distribution, Communications in
Statistics- Theory and Methods, Vol. 37,
2008, pp. 1162-1172.
doi.org/10.1080/03610920701713344.
[7] Khan, M. I., Generalized order statistics from
Power-Lomax distribution and
characterization, Applied Mathematics E-
Notes, Vol.18, 2018, pp. 148-155.
[8] Alharbi, YF, Fawzy, MA and Athar, H.,
Expectation properties of generalized order
statistics based on the Gompertz-G family of
distributions, Operations Research and
Decisions, Vol. 33, No. 4, 2023, DOI:
10.37190/ord230401
[9] Rayleigh, J. W. S., On the resultant of a large
number of vibrations of the same pitch and of
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doi.org/10.1080/14786448008626893.
[10] Kilany, N. M. Mahmoud, M. A. W. and El-
Refai, L. H., Power Rayleigh distribution for
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Volume 23, 2024
fitting total deaths of COVID-19 in Egypt,
Journal of Statistics Applications &
Probability, Vol.12, No. 3, 2023, pp.1073-
1085. doi:10.18576/jsap/120316.
[11] Moshref, M. E., EL- Arishy, S. M. and Khalil,
R. E., Moments of Rayleigh distribution based
on generalized order statistics, International
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Statistics, Vol. 28, No. 4, 2012, pp. 75-82.
[12] Mohsin, M., Shahbaz, M. Q. and Kibria, G.,
Recurrence relations for single and product
moments of generalized order statistics for
Rayleigh distribution, Applied Mathematics
and Information Sciences, Vol. 4, No.3, 2010,
pp. 273-279.
[13] Hwang, J. S. and Lin, G. D., Extensions of
Muntz-Szasz theorems and application,
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doi.org/10.1524/anly.1984.4.12.143.
[14] Khan, A. H., Khan, R. U. and Yaqub, M.,
Characterization of continuous distributions
through conditional expectation of functions
of generalized order statistics, Journal of
Applied Probability and. Statistics, Vol.1,
2006, pp. 115-131.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed to the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The author wishes to extend his sincere gratitude to
the Deanship of Scientific Research at the Islamic
University of Madinah for the support provided to
the Post-Publishing Program (3).
Conflict of Interest
The author has no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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Table 1. and .
alpha
1st
2nd
3rd
4th
mean
variance
skewness
Kurtosis
0.5
0.5
0.5
0.75
1.5
0.50
0.25
2.00
9.00
1.0
2
8
48
384
2.00
4.00
2.00
9.00
1.5
4.5
40.5
546.75
9841.5
4.50
20.25
2.00
9.00
2.0
8
128
3072
98304
8.00
64.00
2.00
9.00
2.5
12.5
312.5
11718.8
585937.5
12.50
156.25
2.00
9.00
3.0
18
648
34992
2519424
18.00
324.00
2.00
9.00
3.5
24.5
1200.5
88236.8
8647201.5
24.50
600.25
2.00
9.00
4.0
32
2048
196608
25165824
32.00
1024.00
2.00
9.00
Table 2. and
alpha
1st
2nd
3rd
4th
mean
variance
skewness
Kurtosis
0.5
0.627
0.5
0.47
0.5
0.63
0.11
0.64
3.24
1.0
1.253
2
3.76
8
1.25
0.43
0.63
3.25
1.5
1.88
4.5
12.69
40.5
1.88
0.97
0.63
3.24
2.0
2.507
8
30.08
128
2.51
1.71
0.63
3.24
2.5
3.133
12.5
58.749
312.5
3.13
2.68
0.63
3.25
3.0
3.76
18
101.518
648
3.76
3.86
0.63
3.25
3.5
4.387
24.5
161.208
1200.5
4.39
5.25
0.63
3.24
4.0
5.013
32
240.636
2048
5.01
6.87
0.63
3.25
From Table 1 and Table 2, we conclude that
when alpha increases at fix value of beta, other
characteristics are also increases except skewness
and kurtosis.
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APPENDIX
Statistical measures of PRD.
