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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
MODELING AND ANALYSIS OF SINE POWER RAYLEIGH
DISTRIBUTION : PROPERTIES AND APPLICATIONS
AADIL AHMAD MIR
•
Department of Statistics, University of Kashmir, Srinagar, India
stataadil29@gmail.com
S.P.AHMAD*
•
Department of Statistics, University of Kashmir, Srinagar, India
sprvz@yahoo.com
Abstract
In this manuscript, a new probability model named as Sine Power Rayleigh distribution (SPRD) is
proposed using a Sine-G function as generator. Various statistical properties of this new distribution were
investigated, including the survival function, hazard function, reverse hazard rate, cumulative hazard
function, mills ratio , quantile function, moments, moment generating function, conditional moments
, entropy, and order statistics. The parameters of the proposed distribution were estimated using the
method of maximum likelihood estimation. To assess the model's versatility and applicability, we conduct
analyses on two real life data sets. The outcomes affirm the superior performance of the newly proposed
model SPRD as compared to existing models .
Keywords: Sine G family, Rayleigh distribution, Sine Rayleigh distribution, Reliability Analysis,
Entropy, Order Statistics, Maximum Likelihood Estimation.
1. Introduction
The concept of probability distribution has shown to be quite helpful in managing both small
and large data sets. Probability distribution models are essential and widely utilised in many
domains, including as physics, medicine, business management, engineering, and food. The field
of probability distributions has advanced steadily due to the wide range of domains in which
they are applied.Over the past few decades, researchers have used a variety of ways to introduce
numerous novel probability distributions. New distributions are needed to address the problem
more precisely and effectively, even though there are numerous existing ways for handling
real-world data. From an applied and practical perspective, the new family of distributions
modifies some of the current distributions to make them more flexible, which serves key purposes
in the generalisation of distributions. There are several ways to create new models, including
exponentiation, compounding, and changing and adding constants to well-known distributions.
The Rayleigh distribution (RD), named after Lord Rayleigh [15] is prominent lifetime prob-
ability model concerned with describing skewed data. The probability density function (PDF)
associated with random variable x > 0 having RD with scale parameter 9 is given by
x f x2 \
f(x;9) = 92exp y-2992); x > 0 9 >0
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
and the corresponding cumulative distribution function (CDF) is given as
F(x; в) = 1 - exp Г-2^): x > 0, в > 0
In the statistical literature, numerous extensions of Rayleigh distribution (RD) have been pro-
posed. Surless and Padgett[17] introduced the two parameter Burr type X distribution and named
it as exponentiated Rayleigh distribution (ERD) or generalized Rayleigh distribution. Kundu
and Raqab [11] studied and estimated the parameters of the generalized Rayleigh distribution
using different estimation techniques. Ahmed et al. [2] used the square error loss function and
Al-Bayyati's loss function to perform a Bayesian analysis of RD. Ajami and Jhansi [3] discussed
the parameter estimation of weighted Rayleigh distribution. Ahmad et al. [1] proposed the
Weibull-Rayleigh distribution and studied its characterization and parameter estimation using the
transformed transformer technique. Bhat and Ahmad [6] proposed a new extension of exponenti-
ated Rayleigh distribution and studied its various properties and demostrated its applicability by
considering different datasets. Bhat and Ahmad [5] studied mathematical properties of mixture
of Gamma and Rayleigh distributions. Kilai et al. [8] proposed a new versatile modification
of the Rayleigh distribution for modeling COVID-19 mortality rates. Various researchers have
introduced generalised distributions and their applications, see Mahmood et al. [12] , Muse et al.
[13] and Ahmed et al. [15]. Bhat et al. [7] proposed a new extension of odd lindley power rayleigh
distribution, studied its properties and evaluated parameter estimation techniques using both
classical and Bayesian methods. Bhat and Ahmad [4] recently introduced a new generalization of
the Rayleigh distribution using power transformation technique with PDF and CDF respectively
given by
g(x-вв) = в2x2e xexp^-2^; x > ° ^в > 0 (1)
and the corresponding cumulative distribution function (CDF) is given as
G(x; в, в ) = 1 - exp (- x > 0, в, в > 0 (2)
In the present manuscript, we proposed a new extension of Power Rayleigh distribution (PRD)
using the Sine G family of generated distributions. The proposed distribution is named as Sine
Power Rayleigh distribution (SPRD). It is more flexible and exhibits more complex shapes of
density and hazard rate functions. Also, the proposed model outclass some well established
models in terms of two real life data sets. The rest of the article is unfolded as : In section 2, the
Ratio Transformation (RT) method is discussed. In Section 3, the PDF and CDF of the proposed
model i.e., SPRD are defined. Section 4 deals with the reliability measures of the SPRD. The
expansion of PDF and CDF is discussed in Section 5. Some of important statistical properties are
explored in Section 6. The parameter estimation is discussed in Section 7. The simulation study
and applicability of the model is debated in section 8 and 9 respectively. Finally, some conclusion
are provided in Section 10.
2. SINE G FAMILY OF DISTRIBUTIONS
The CDF and PDF of the Sine G family of distributions proposed by [10] are defined by the
following equations respectively:
' n,
F(x; Z) = sin G(x; Z)
f (x; Z) = П2g(x; Z) cos [ПG(x; Z)
x e R
xe R
(3)
(4)
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
Where G(x; Z) and g(x; Z) in equation (3) and (4) are the CDF and PDF of the base line
distribution with parameter vector Z, respectively.
3. SINE POWER RAYLEIGH DISTRIBUTION (SPRD)
The PDF of the newly proposed probability distribution Sine Power Rayleigh Distribution (SPRD)is
obtained as
f (x; в, в)
пА ив-1
2 в2
х2А
e 2в2 cos
п
2
1 - e
х2А
2в2
x е R+, в, в > 0
(5)
The CDF of the newly proposed probability distribution Sine Power Rayleigh Distribution
(SPRD) is obtained as
F(x; в, в) = sin
п
2
1e
x2?
2в2
x е R+, в, в > 0
(6)
The plots of density function of SPRD for different parameter combinations are presented in
Figure 1 . It is clear from the density function plots that the proposed distribution is unimodal,
decreasing, symmetric and positively skewed.
X
Figure 1: Density plots of SPRD for different combinations of в and в.
4. RELIABILITY ANALYSIS OF THE SINE POWER RAYLEIGH
DISTRIBUTION (SPRD)
This section focuses on obtaining the reliability (survival function), hazard rate (failure rate),
reverse hazard function, cumulative hazard function and mills ratio expressions respectively for
SPRD.
4.1. Survival function
The survival function or reliability function is defined as the probability that a system will survive
beyond a specified time and is obtained for the SPRD as
R(x; в, в) = 1 - F(x; в, в) = 1 - sin
п
x2?
- 1 - e 2в2
(7)
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
4.2. Hazard Rate
The Hazard rate evaluates a lifetime component's likelihood of failure or expiration based on
the completed portion of its life, and consequently, it finds diverse applications in the analysis of
lifetime distributions. Using equation (5) and (7), the expression for the hazard rate of SPRD is
obtained as
h(x; в,9)
f (x; в, 9)
R(x; в, 9)
2p
п в х2в 1 e 292 cos
2 92
_ х2в
П ( 1 - e 292
1 — sin
_x2l
n (1 - e 292 )
(8)
Figure 2 depicts graphs of the hazard rate of the SPRD for different parameter values. Figure
2 suggests that the proposed distribution is quite flexible in nature and can exhibit variety of
shapes such as constant, decreasing, increasing and j-shaped shaped over the parameter space.
x
Figure 2: Hazard rate plots of SPRD for different combinations of в and 9.
4.3. Reverse Hazard function
The concept of reversed hazard rate of a random life is defined as the ratio between the life
probability density to its distribution function . It is expressed as
hr (x; в, 9)
f (x; в, 9)
F(x; в, 9)
„ Лв
п в x^-1e-W
cos
_ x^
n [1 - e 292
sin
, _x^
2(1 - e 292
4.4. Cumulative Hazard function
The cumulative hazard function can be thought of as providing the total accumulated risk of
experiencing the event of interest that has been gained by progressing to time t. The cumulative
hazard function for the SPRD is defined as
ЛspRD(x; в,9) = - logR(x; в,9)
log
1 - sin
n
~2
1-e
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
4.5. Mills Ratio
The mills ratio for the SPRD is defined as
M.R
F(x; в, в)
R(x; в, в)
sin
2(1 - e 2в2
1 — sin
_ х2в
2(1 - e 2в2
(9)
4.6. Quantile function
The quantile function for the SPRD is given by
-ie2log ( 1---sin 1 u
n
(10)
The first quartile (Qi), median (Q2), and third quartile (Q3) can be derived by setting u = ^,^,
and 3 in equation (10) respectively.
5. Expansion of PDF and CDF
Various statistical properties can be easily deduced by using mixture representation of PDF and
CDF of the proposed model.
expansion of cos
_ х2в
2(1 - e 2в2
can be expressed as
cos
n / - 4
2 I 1 - e 2в2
£ £ (1 - e-222
l=0
2l! 221
_x2l\2
Also (1 - e 2в2 can be expressed as
_х2в\21 “ /2А тх2в
1 - e 2в2 J = £ (-1)m( )e 2в2
m=0
m
expansion of sin
_ х2в
П2 ( 1 - e 2в2
sin
n
х2в
2 I 1 - e 2в2
can be expressed as
“ (- 1)p n2P+1 ( _x2t\2p+1
= £0 (2Р + 1)! 22p+T 1 - e 2в2
х2в \ 2p+1
Also (1 - e 2в2 1 can be expressed as
х2в \ 2p+1 TO (2p + 1 \ ^х2в
1 - e-= £ (-1)4 ( 2p + M e-V
/ 4=0 V 4 /
Thus, the PDF and CDF of the proposed model can be written in the mixture representation
respectively as
П TO TO
f (х; в, в) = в х2в-1 £ £
(_ 1 )l+m /2А n2l+1 (т+1)х2в
-e 2в2
l=0 m=0
2l! \m 22l+1
т(х; в, в) = ££ i-Z+L (2 р +1\ п2Р+1 e-
Р=0 4=0 (2Р + 1)!V 4 / 22p+1
(11)
(12)
х
2l
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
6. STATISTICAL PROPERTIES OF SPRD
Some of the mathematical properties such as the rth moment, moment generating function,
conditional moments and associated measures, the entropy and order statistics are derived.
6.1. Moments
The rth moment of the SPRD can be evaluated directly by extending the PDF given in equation (11)
CO
E(Xr) = j xrf (x; в, d)dx ,r = 1,2,..
where f(x ) is the PDF of the SPRD given in equation (11), thus
E(Xr) = в EE (—^_ ( 21 ) I xr+2e-
(_1)Z+m /21 \ n21+!
1=0 m=0
21! \m 221+!
(m+1)x2e
(13)
Using integration via substitution method in equation (13), we perform the following operations.
let (m+^=z =^x=( m+) 2в ,suchthat dx=(mi+T) (z) 1
Thus, simplifyingequation (13) yields
E(Xr ) = (2,2) * EE цр (m) ( mU ) *+1 r( £ +1)
1=0 m=0
(14)
where,
O
CO O
CO , r
Г( + 1) = / z( ^ +) e_zdz
20
setting r = 1 in equation (14) the mean of the model is computed as
1 O O (_ 1 )/ + и
E(X) = (202) 2? EE ( )
1=0 m=0
21!
2 A n21+1 ( 1
m
22/+1 U + 1
2?
+1
Г( 2в + 1)
(15)
Similarly for r = 2, 3 and 4 in equation (14) ,the second, third and fourth moment about origin are
respectively calculated as
E(X2)
1 O O
(2d2)1E E
1=0 m=0
E(X3) = (202)2в E E
1=0 m=0
O
CO
E(X4)
(202)2 E E
1=0 m=0
CO O
(_1)1+m
2Л
(_1)1+m
2л
(_1)1+m
2Л
21
m
21
m
21
m
, п21+х ( 1
1 221+1 Vm +1
п21+х / ' 1 '
221+1 1 vm +1
, п21+х ( 1
1 221+1 Vm +1
в+i
2?
+1
в + 1
1
Г( в +1)
3
Г( 2в +1)
2
Г( в + 1)
(16)
(17)
(18)
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
6.2. Moment Generating function of SPRD
we can calculate moment generating function based on the rth moment of SPRD as given by
TO tr
MX(t) = E r!E(Xr)
r=0 Г
(19)
mx (t) = (ie2)2? eee
r=01=0 m=0
tr (_1)7+m /21
r! 27! Vm
n
-27+1
1
227+1 I m + 1
2? + 1
r( 2? +1)
(20)
6.3. Conditional moments and associated measures
In this section, the expression for conditional moments is acquired. But first we will introduce an
important lemma which will be applied in the next section.
Lemma 1. Let us suppose a random variable X follows SPRD (?, в) with PDF given in equation
(11) and let фг(z) = /0 xrf (x; ?, в) dx denotes the rth incomplete moment, then we have
r TO TO
фг(z) =(2в2) 2 E E
1=0 m=0
(_1)1+m
27!
21
m
n27+l
227+1
1 \ 2?+1
m +1/ Y
r Л (m + 1)z2?
2? + )' 202
(21)
b
where 7(я, b) = f za-1 e_zdz denotes the lower incomplete gamma function.
0
Proof: Using the PDF of SPRD given in equation (11), we have
z
фг (z) = J xrf (x; ?, 0)dx
0
? E E (_1)7+m 721
02 7=0 «Ьс^7^ U
n27+l
227+1
z
/ xr+2? 1 e
0
(m+1)x2?
2в2
(22)
On Simplification, we obtain
Фг (z) = (202) 2? E E
7=0 m=0
(_1)7+m /27\ П
27!
t-27+1
m 227+1
TO TO
1 \ 2?+1
m + 1 / Y
r Л (m + 1)z2?
2? + y' 202
(23)
Setting r=1 in equation (23) will yield first incomplete moment as given by
1 TO TO
Ф1 (z) = (2в2) 2? E E
7=0 m=0
(_1)7+m /27\ П
27!
r27+1
m 227+1
1 \ 2? +1
m +1) Y
1 + Л (m + 1 )z2?
2? + У' 2в2
(24)
6.3.1 Lorenz and Bonferroni inequality Curves
The Lorenz and Bonferroni inequality curves are an important application of the first incomplete
moment. For a given probability distribution, they are defined by
= E(X) lxf(x; ?,e) dx = вд
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
v—'TO v—к
Ul=0 и
(-1)l+m
(2l) +1 // ± + Л (m+1)t2e
(m) 221+1 \m+1J l\\2fi + 1)' 2Q2
>1=0 Um=0 2l!
(-1)l+m (2l) n2l+1 ( 1 \2в
Ul=0 Um=0 27! (m) 22l+1 l m+1 J
+1
Г( 2? + 1)
L
P
Similarly,
BP
1
PE(X)
•t
xf (x; в, Q) dx
Ф1(t)
pE(X)
bp
(—1)l+m (2l) n2l+1
Ul=0 Um=0 27! (m) 2й+1
1
m+1
2?+1
7
p u=o uto=o
(-1)l+m (2l) п27+1
2l! (m) 227+1
1
m+1
K2? +1
(m+1)t2? ^
2Q2 j
2?+1
Г( 2? + 1)
6.3.2 rth Conditional Moment and rth Reversed Conditional Moment of SPRD
The rth conditional moment of the SPRD is calculated by
E [Xr |x > t]
1
R(t) Jt
xr f (x; в, Q) dx
1
R(t)
[E(Xr) - фг(t)]
where R(t) is the reliability of SPRD at time t.
Inserting the value of equation (7), (14) and (23), we obtain
(2q2)2в П=0UU (m)ПТ+- (m+T)2в+1 [r (2? +1) - 7 ((2e + l) ,)
E [Xr |x > t] =
(-1)p+q
■t2?
(2p+1) Й+ e-^
-| V-'TO
1 - up=0 Uq=0 (2p+1)! ( ’q ) 22P+1 1
Similarly, the rth reversed conditional moment of the SPRD is defined by
Фг (t)
E [Xr|x < t]
F(t) Уо
xrf (x; в, Q) dx
F(t)
CO
E [Xr |x < t]
_ r
(2Q2)2в U“=0
(-1)l+m (2l) n2l+1
2Л (m) 22Т+Г
1
m+1
2j3 +1
7
2? + 1
(-1)p+q (2p+1) n2p+1 e
Up=0 Uq=0 (2p+1)! ( q ) 22P+1 e
qt2?
2Q2
(m+1)t2?
2Q2 J
6.3.3 Mean Residual Life (MRL) and Mean Waiting Time (MWT)
The MRL is defined as
H(t)
1
ЗД
E(t) — J xf (x; ?, Q) dx
t
1
щ[Е0) - ф1«] - t
After inserting the value of equation (7), (15) and (24), we obtain the required expression for
mean residual life as
Ф(0
(2Q2)2? U=o U
0 Um=0 2l!
(-1)f+m (2U n2‘
(2l)п^Ц (^_) 2?+1 Гг (1 + 13 7 ((x + 1) (m+1)t2<i
(m) 227+1 ^m+1 j |/ ^2? + ) \2p + ^ , 2Q2
qt2?
(-i)p+q (2p+i) ПрЦe-
1 Up=0 uq=0 (2p+1)! ( q ) 22P+1 e
t
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
The MWT is defined as
n(t) = t - jQ xf (x;e9) dx =t —
ц(t) = t
(292)2в ET=0 ЕЖ=о
(—1)l+m (2l) п2+
21! (m) "22Г+Т"
_X_ ^ 25+1Y
m+1 J '
2)3 + 1
у-^ж v-^ж ( — 1)p+q (2p+1) n2p+1 e
Ep=0 Eq=0 (2p+1)! ( q ) 22P+1 e
qj2^
2d2
(m+1)t2e \
2d2 )
6.4. Renyi entropy
The entropy of a random variable is defined as the average amount of information lost during
a random experiment. The Renyi entropy, which Alfred Renyi introduced [16] and generalises
Shannon's measure of information, is defined as
1 f “
Rn = -----log/ fn (x; в, 9) dx, n > 0, n = 1
1 — n J—ж
Using the PDF given in equation (11), we have
Rn
1
1—n
log
ву
9
ж ж
ЕЕ
l=0 m=0
( —1)l+m
2Г!
21
m
n2l+t
221+1
n ж
/ xn(2e x) e
n(m+1)x2e
292
0
R
n
A -•(9 Y i (s lo ЧE C) Й’
П(2в—1)+1
292 N 2в / п(2в —1) +1
П (m +1) J V 2в
6.5. Order Statistics of SPRD
The order statistics connected to the SPRD is devoted in this section. Let x(r;n) be the rth order
statistics with the random sample x^), x(2), x(3), ...x(n) derived from the SPRD having the PDF
f (X;,в,9) and CDF F(X;в,9). Therefore, the PDF and CDF of x(r;n) say f(r;n)(x) and F(r;n)(x)
are respectively defined as
f(r;n) (x)
1
[F(x; в, 9)]r—1 [1 — F(x; в, 9 )]n—rf (x; в, 9)
B(n, n — r + 1)
(25)
j=r \J
F(r;n)(x) = Е J [F(x;в,9)У [1 — F(x;в,9)]
n—j
(26)
Using equation (5) and equation (6) in equation (25) and equation (26), the PDF and CDF of rth
ordered statistics for the SPRD are derived and are expressed as
n
f(r;n) ( x)
в x2в ПП (2x^e 292 cos Г / x2P\l n (1 — e 292 j
B(n,n — r +1)
sin
1e
x2в
292
r—1
1 sin
n
2
1e
x2l
292
n—r
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MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
F(r;n) (x) = E ( .■
J=r
sin
П t - x2?
2 I 1 - e 292
1 — sin
п t - x23
- ( 1 - e 292
n —J
where B(a, b) = ^a+b^ is the beta function.
7. Estimation of Parameters
The goal of this study is to estimate the unknown parameters в and 9 of the SPRD using Maximum
Likelihood Estimation (MLE). we assume that Xi, x2,..., xn be a random sample of n observations
drawn from the SPRD (?, 9) with unknown parametric vector 0 = (?, 9)T.
7.1. Maximum Likelihood Estimation (MLE)
Here, Maximum Likelihood Estimation (MLE) approach is used to obtain the estimators of the
unknown parameters of SPRD ( в, 9) . The likelihood function is given by
L(0)
пв
W
x2 в n
П xf-1
cos
k=1
x23
п (1 - e-292)
For the parametric vector (0) = (в, 9)T , the logarithm likelihood function is expressed as
-in n
l = n log (2) + n log(3) - 2n log(9) - ^ E X2/ + (2в - 1) E logXk
2 29 k=1 k=1
+ E log cos
k=1
П , £
2 I 1 - e 292
(27)
J
n
e
The elements of the score vector U(0) =(U?, U9) are obtained by partially differentiating
Equation (27) with respect to the model parameters and are given by
M
¥
nn
-+2 E ln(xk)
в k=1
1
292
n
E x? ln(xk)
k=1
п
402
n
E tan
k=1
п
2
1e
x23
xk
292
x23
e 292 x^3 ln(xk)
dl
d9
2n 1
+ Й E x? + 4 E tan
п
9
93
k=1
293
k=1
x23
П I k
2 I 1 - e 292
n
n
x23
e- m2 x^3
The likelihood estimates of the model parameters can be obtained by setting the score vector U(0)
= 0 . Since, the above equations are non-linear and hence the model parameters are estimated
using Newton-Raphson algorithm.
8. SIMULATION ILLUSTRATION
In this section, we carry out simulation study using R software to examine the behaviour of MLE's
for various sample sizes.We generate the random samples of size 25,75,150,300 and 500 from
712
Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
SPRD and repeat the process for 1000 times in R software.Various combinations of parameters
are chosen as (1.5,1.35) and (0.5,2.2) with relation to the standard order (в,в). The average
MLE values, bias, and related empirical mean squared errors (MSEs) were determined for each
scenario. Tables 1 exhibits the ML estimates, bias and MSE. We observe from table 1 that the
agreement between theory and practice improves as the sample size n increases. MSE and bias of
the estimators suggest that the estimators are consistent and the maximum likelihood estimator
of the parameters perform quite well and the results are precise and accurate. The MSE decreases
with increasing sample size under all conditions.
Table 1: MLE,Bias and MSE for the parameters в and в
sample size Parameters MLE Bias MSE
n в в в в в в в в
25 1.5 1.35 1.58963 1.38116 0.21193 0.15622 0.07685 0.04192
75 1.52863 1.36292 0.11586 0.08563 0.02170 0.01211
150 1.51474 1.35744 0.07911 0.05752 0.00999 0.00543
300 1.50528 1.35236 0.05462 0.03945 0.00459 0.00248
500 1.50487 1.35130 0.04267 0.03108 0.00278 0.00153
25 0.5 2.2 0.53233 2.39244 0.07177 0.40725 0.00960 0.36239
75 0.50767 2.24504 0.03767 0.20412 0.00222 0.06987
150 0.50439 2.22579 0.02799 0.14628 0.00126 0.03659
300 0.50299 2.21458 0.01852 0.10195 0.00054 0.01658
500 0.50085 2.20587 0.01432 0.07776 0.00034 0.00967
9. APPLICATION
This section is devoted to illustrate the flexibility, adaptability, and suitability of the SPRD, by
means of two real data sets . We compare the proposed distribution with the following models :
• Power Rayleigh distribution (PRD) With PDF given as
f (x;вв) = J>*2e-1exp(-2^2); в,в >0
• Weighted Rayleigh Distribution (WRD) with PDF given as
хв+1 exp (-2в2)
f (x; в, в)
вв+22вГ (§ + 1)'
в, в > 0
Rayleigh distribution (RD) with PDF given as
f(x;в) = e2exp (-2^); в >0
Here, several goodness of fit criterion such as -2ll, Akaike Information Criterion (AIC), Bayesian In-
formation Criterion (BIC), Akaike Information Criterion Corrected (AICC), Kolmogorov-Smirnov
(KS) and P value statistics are used. The statistic with the lowest value of -2ll, AIC, BIC, AICC,K-S
and maximum value of P value is considered the best fit.
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MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
9.1. Data Set 1
Data set 1:The first data is on the breaking stress of carbon fibres of 50 mm length (GPa). The
data has been previously used by [4] and [14] . The data is as follows:
0.39, 0.85, 1.08,1.25,1.47,1.57,1.61,1.61, 1.69, 1.80,1.84,1.87,1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41,
2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95,
2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60,
3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90
Table 2: Estimates (standard errors), -211, AIC, BIC, AICC, K-S statistic and P-valuefor Data-set 1.
Model в 6 —2ll AIC BIC AICC K-S P-value
SPRD 1.6366 (0.1595) 5.8515 (1.2057) 171.6825 175.6825 180.0618 175.8730 0.0791 0.8029
PRD 1.7205 (0.1654) 4.8502 (1.0369) 172.1352 176.1352 180.5145 176.3256 0.0823 0.7625
WRD 2.5727 (0.7452) 1.3551 (0.1234) 175.7107 179.7107 184.0900 179.9012 0.1104 0.3963
RD 2.0491 (0.1261) 196.4168 198.4168 200.6065 198.4793 0.2265 0.0022
Histogram of data
Figure 3: Fitted density plots for dataset 1
9.2. Data set 2
Data set 2: Consider the following data set in Johnson and Kotz [9] and represent the survival
times (in years) after diagnosis of 43 patients with a certain kind of leukemia.
0.019, 0.129, 0.159, 0.203, 0.485, 0.636, 0.748, 0.781, 0.869,1.175, 1.206, 1.219, 1.219, 1.282, 1.356,
1.362, 1.458, 1.564, 1.586, 1.592, 1.781, 1.923, 1.959, 2.134, 2.413, 2.466, 2.548, 2.652, 2.951, 3.038, 3.6,
3.655, 3.745, 4.203, 4.690, 4.888,5.143, 5.167, 5.603, 5.633, 6.192, 6.655, 6.874
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MODELING AND ANALYSIS OF SINE POWER RT&A, No 1 (77)
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS Volume 19, March 2024
Table 3: Estimates (standard errors), -2ll, AIC, BIC, AICC,K-S statistic and P-valuefor Data-set 2.
Model в 0 —2ll AIC BIC AICC K-S P-value
SPRD 0.5887 (0.0736) 1.6864 (0.2041) 162.9906 166.9906 170.5130 167.2906 0.0869 0.901
PRD 0.6198 (0.0766) 1.3094 (0.1647) 163.2203 167.2203 170.7427 167.5203 0.0903 0.8744
WRD 0.0010 (0.3799) 2.2409 (0.2728) 181.9592 185.9592 189.4816 186.2592 0.2423 0.0128
RD 2.2415 (0.1709) 181.9277 183.9277 185.6889 184.0252 0.2421 0.0128
Histogram of data
<D
□
0 1
1 -----------------1-----------------1------------------1-----------------г
2 3 4 5 6
A
7
data
Figure 4: Fitted density plots for dataset 2
The results obtained in Table 2 and Table 3 reveal that SPRD has the least value of all the
comparison criterions, hence SPRD can be considered a strong competitor to other distributions
compared here for fitting data. The plots of the fitted models are displayed in figure 3 and 4.
Also, from these plots , it is evident that SPRD provides a close fit to the two data sets.
10. CONCLUSION
In this paper, a new life time distribution namely Sine Power Rayleigh distribution (SPRD) is
proposed and studied. The SPRD model is an expansion that incorporates the Sine-G family
of distributions introduced by [10] resulting in a novel trigonometric distribution. The new
distribution is more flexible and its hazard rate function exhibits complex shapes. The study
derives various properties of the proposed distribution, including the survival function, hazard
rate function, reverse hazard function, cumulative hazard function, moments, moment generating
function, quantile function, Lorenz and Bonferroni inequality curves, Renyi entropy and order
statistics.The parameters of the proposed distribution are estimated using the maximum likeli-
hood method and a simulation study is conducted to assess the performance of the maximum
likelihood estimators (MLEs) for these parameters. Furthermore, the effectiveness of the proposed
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Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER
RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
RT&A, No 1 (77)
Volume 19, March 2024
distribution is evaluated by applying it to two distinct real life datasets and comparing it with well
known standard distributions such as the Rayleigh distribution, Power Rayleigh distribution and
Weighted Rayleigh distribution. The results demonstrate that the Sine Power Rayleigh distribution
(SPRD) surpasses its competitors in terms of fitting the two datasets.
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