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MODELING AND ANALYSIS OF SINE POWER RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS
AADIL AHMAD MIR •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
S.P.AHMAD* •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
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 probability 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 ( x2 \ f (x; 9 ) = 92 exp w) ; x > 0, 9 > 0
and the corresponding cumulative distribution function (CDF) is given as
x2
F(x; 0) = 1 - exp 20^; x > 0, 0 > 0
In the statistical literature, numerous extensions of Rayleigh distribution (RD) have been proposed. 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 exponentiated 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;fi,0) = fix2fi-1exp(-202); x > 0, fi,0 > 0 (1)
and the corresponding cumulative distribution function (CDF) is given as
( X2fi \
G(x; fi, 0 ) = 1 - exp (- ^ J; x > 0, fi, 0 > 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:
F(x; Z) = sin G(x; Z)
f (x; Z) = ng(x; Z) cos n2 G(x; Z)
x G R x <E R
(3)
(4)
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; ß, e)
ПА T2ß-1 - x
2ß
2 e2
x2ß 1e 2e2 cos
П ! - 4
— ( 1 - e 2e2
t g , ß, e > 0
(5)
The CDF of the newly proposed probability distribution Sine Power Rayleigh Distribution
(SPRD) is obtained as
F(t; ß, e ) = sin
f (1 - e
т2А 2e2
t g r+, ß, e > 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.
b= 1.05,0 = 0.85 b= 1.05,0 = 1.9 b = 0.6,0= 1.3 b= 1.5,0= 1.3 b = 0.25,0 = 0.65
Figure 1: Density plots of SPRD for different combinations of ft and 9.
0
2
3
4
5
x
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(t; ß, e) = 1 - F(t; ß, e) = 1 - sin
П (1 - e
T2ß
2e2
(7)
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; p,Q)
f (x; p, Q ) x2p n Q2 x2p-1 e-~2b2 cos ï(l - e x2p\l 2Q2
R(x; p, Q 1 - sin n (1 x2p - - e 2q2 )
(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 ft 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; p, Q )
f (x; p, Q) F(x; p, Q)
a x2p f px2p-1e 2^cos r ( x2p\] n 1 1 - e 2q2 J
sin r / x2p\l fil - e J
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
Asprd(x; p,Q) = - logR(x; p,Q) = - log M - sin
f (1 - e
2Q2
4.5. Mills Ratio
The mills ratio for the SPRD is defined as
M.R
F(x; ß, 9) R(x; ß, 9)
sin r / x2ß\1 f ( 1 - e 292
1 - sin r ( x2ß \] f 1 1 - e 292 J
4.6. Quantile function
The quantile function for the SPRD is given by
-2d2log M - n sin-1 u
(9)
(10)
The first quartile (Q1), median (Q2), and third quartile (Q3) can be derived by setting u = 1, 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
2(1 - e 292
can be expressed as
cos
n / - 4 — ( 1 - e 292
E
/=0
I „21
(-1)1 2l ! 22/
1 — e 292
\21
Also (1 — e 2e2 can be expressed as
x2ß\
2/
1 - e 292 = E (-1)
m=0
, 2A mx2ß
m' )e 2^ m
expansion of sin
f ( 1 - e 292
sin
n
x2ß
- ( 1 - e 292
can be expressed as
~ (_ 1)p n2p+i ( _x2l\2p+1
E -~7VT _„„ , -, I 1 - e 292
p=0 (2p + 1)! 22P+1
_x2ß \ 2P+1
Also (1 — e 2e2 ) can be expressed as
2ß \ 2p+1 c
1 - e-292 ] = E (-1)q
q=0
2 p + 1\ - qx
q
__2ß
e 292
Thus, the PDF and CDF of the proposed model can be written in the mixture representation
respectively as
f (x; ß, 9) = 92 x2ß- EE 2/!
(_ 1 )l+m (2l\ n2/ + 1 (m+1)x2ß \ ' I \ e 292
/=0 m=0
m 22l+1
F(x; ß, 9)= EE
(—1)p+q /2p + 1\ n2P+x
p=0 q=0
(2p + 1)!^ q J 22P+1
qx'
2ß
-e 292
(11)
(12)
co c
Aadil Ahmad Mir , S.P.Ahmad
MODELING AND ANALYSIS OF SINE POWER RT&A, No 1 (77) RAYLEIGH DISTRIBUTION : PROPERTIES AND APPLICATIONS_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; ft, 9)dx ,r = 1,2,.. 0
where f(x ) is the PDF of the SPRD given in equation (11), thus
ft (-')'+" P) Cxr+2ft-1 -(m^
E(xr) = 92EE ^Q292 (13)
Using integration via substitution method in equation (13), we perform the following operations.
let = z x =(2+1)2ft, such that dx = 2ft ()* (z)215-1
Thus, simplifyingequation (13) yields
E^^iE-^®^ (m+r)*H+1) (14)
where,
CO , r
r( 2ft + 1) = I z( ^+ e-zdz 20
setting r = 1 in equation (14) the mean of the model is computed as
1 c c (_ 1)'+m (2') n2'+1 ( 1 ) 2ft +1 1
E(x) =(292) 2ft E mi ^—lyi— © (mrr) r( 2ft +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) = (292)ft ii^Q^ (^f^ + 1) -
E(x3) = (292) • E E ^Q ^ (m^y'+1 r(2ft +1) (17)
^ 2 3 C C (-1)'+m fn\ n2'+v 1 \5ft + „, 3
1 221+1 \m +1
n2'+1 . ( 1 N
221+1 1 K.m +1,
. n2'+1 ( 1
E(X4) = (292)ft tt^Oi+r (m+T f^ft + 1) (18)
6.2. Moment Generating function of SPRD
we can calculate moment generating function based on the rth moment of SPRD as given by
œ ,r
MX(t) = E rrE(Xr) (19)
r=0 T
œ œ œ tr (_ i)l+m /2/\ „21+1/ 1 \ 2s+1 r
Mx(t) = (282)'EEErr^GO^ (mil)' r(W +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 (', 0) with PDF given in equation (11) and let q>T(z) = /0 xr f (x; ', 0) dx denotes the rth incomplete moment, then we have
^ 4 (-1)l+m (2l\ n2li1 ( 1 \ 2'+1 (( r \ (m + 1)z2' \ „„
fr(z) = (20 )2' E E Hr-(„)wn (mm) 2'+V,-ÏT-) (21)
b
where 7 (a, 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 Ttt a a\A ' œ œ (-1)l+m (2l\ n2li1 Z r+28-1 - (m+12x2' ^r(z) = j xJ(x; ', e)dx = £ E E [m) x ' e 202
Vt (Z) = , xrf (x; ', 0)dx = £ E [D^J+T I xr+2'-1 e-^^ (22)
0 l=0 m=0 2l ! \m) 22li1 o
On Simplification, we obtain
^ 4 ^^ (-1)l+m fn\ K21+1 ( 1 N 2?+1 (( r \ (m + 1)z2? \ „„
,fr(z) = (^ZZ^-y^(mrr) H(2? + V,(23)
Setting r=1 in equation (23) will yield first incomplete moment as given by
, * .-i (-1)l+m (2l\ n2l+1 ( 1 \ 2?+1 (( 1 \ (m + 1)z2? \
f1(z) = (2fl2)2'£E0 ^ U)2« (mrr) n(2?+'V-w-) (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
r - 1 ft xf (x;? , 0) dx- ^
E(X) *~E(X)
Lœ Lœ (-1)7+m ^Un2^ fO^1 Yff± + A (m+1)t2ß\ L7=0 Lm=0 27! (m) 227+1 \m+V YV V2ß + 202 /
LP
JP — 1 1
(-1)'+m /2K n2^+1 f 1 ^ 2ß + 1 r(^ + 1) L7=0 Lm=0 27! (m) 22l+1 U+U 1( 2ß + ^
Similarly,
3P = [txf(x; ß, 0) ^ (t)
B
P
pE(X) Jo ' pE(X)
Lœ Lœ (-1)7+m (27) n27+1 (O 2TÎ + 1 Y ((^ + (m+1)t2ß ) L7=0 Lm=0 27! (m) "257+1 ^m+1^ ^2ß + 202 y
p Lœ Lœ (-1)7+m (2l) n27H r^Vß + 1 r( i + 1)
p L7=0 Lm=0 27! (m) 22l+1 \m+1j i( 2ß + 1)
6.3.2 rth Conditional Moment and rth Reversed Conditional Moment of SPRD
The rth conditional moment of the SPRD is calculated by
1 r œ 1
E [Xr |x > t] = Jt xrf (x; ß, 0) dx = [E(Xr) - ^(t)]
where R(t) is the reliability of SPRD at time t. Inserting the value of equation (7), (14) and (23), we obtain
(202) 2-ß vœ Lœ (-1)l+m (2l) n2l+1 ( 1 ) 2ß + 1 \r ( r + A Y (( r + A (m+1)t2ß ' E ,Xr I I20 J P L7=0 Lm=0 27! (m) ^m+lj 2ß + ^ - [2ß + LJ , 202 /
E [X |x > t] —
(-1)p+q (2p+1) n2P+1 e-^
1 Lp=0 Lq=0 (2p+1)! ( q ) 2P+1e 2Q
Similarly, the rth reversed conditional moment of the SPRD is defined by
E [Xr|x < t] = /Vf (x;p,Q) dx =
(2Q2) (-1)'+m (2/) n21+1 f 1 \ 4+1 ^ ff r + -A (m+1)t2p \ (2Q ) p Ll=Q Lm=0^T!-(m) 221+1 ^m+T ) ^\2p + ^-2Q2-J
E [Xr |x < t]
t2ß
Lœ Lœ (-1)p+q (2p+1) n?f+_p-^ Lp=0 Lq=0 (2p+1)! ( q ) 22p+1 e 20
6.3.3 Mean Residual Life (MRL) and Mean Waiting Time (MWT)
The MRL is defined as
v(t)
R(t)
E(t) -Jt xf (x; ß, 0) dx
t
R(t)
[E(t) - ^(t)] - t
After inserting the value of equation (7), (15) and (24), we obtain the required expression for mean residual life as
v(t)
(202) * 0=0 nœ=0 ÖS+1 (mrr)+1 \r (+ 1) - Y ((+ 1) , ^)
t2ß
t
(-1)p+q (2p+1) nfPiie-
1 Lp=0 ^q=0 (2p+1)! ( q ) 22P+1 e 20
1
1
The MWT is defined as
F(t) = t - Ft) j\f (x;', 0) dx = t - ^ (202)2' Eœ Eœ (-1)l+m (2l) n2li1 ( 1 \ 4 + 1 ^ (( 1 + A (m+1)t2?
(20 ) P El=0 Em=0 2l! (m) 22+1 ^m+lj ^ + -
V(t) = t--"l?-
Eœ Eœ (-1)p+q (2p+1) e-q20r EP=0 Eq=0 (2p+1)! ( q ) 22p+1 e 20
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 œ
R« = 1-log/ fn (x;?, 0) dx, « > 0, « = 1
1 — n J-œ
Using the PDF given in equation (11), we have
1 ( E E (-1)l+m (2l\ n2l+! y J^(2?-D„-
0
Rn=1-n**{oyIE,,E=0i=^P 5+r) Ix«^'-^
(\ « «(2?-1)+1 ee (-1)l+m /2A n^W 202 N 2? /n(2? - 1) + 1
E0 ¿0 2l! \m) 22l+0 U (m +1) J U 2?
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;,?,0) and CDF F(X;?,0). Therefore, the PDF and CDF of x(r;n) say f(r;n)(x) and F(r;n)(x) are respectively defined as
1
f(r;n)(x) = B(n,n - r + 1) [F(x;? ,0)]r-1 [1 - F(x;?,0)]n-rf (x;?,0) (25)
F(r,n)(x) = t(n) [F(x;? ,0)]j [1 - F(x;?,0)]n-j (26)
j=T ^''
1n-j
^ j j [F (x;F,v )J [^ F (x;y,v)
j=r
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
f(r;n)(x)
? x2? n <2x2?-1e-202 cos
, _x2l
n ( 1 - e 202
B(n,n - r + 1)
sin
n l x2?
2 11 - e-^
r1
1 sin
n l x2? '
2 11 - e-^
nr
F(r;n)(x) = £ ( i
]=r
sin
- ( 1 - e 202
1 — sin
f (1 - e-
x2ß 202
n-j
where B(a, b) = is the beta function.
7. Estimation of Parameters
The goal of this study is to estimate the unknown parameters and 9 of the SPRD using Maximum Likelihood Estimation (MLE). we assume that x1, x2,..., xn be a random sample of n observations drawn from the SPRD (ft,9) with unknown parametric vector © = (ft,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 ( , 9) . The likelihood function is given by
L(0)
nß 202
x2ß n =1 202
n
k=1
2ß-1
cos
x2ß
n (1 - e-& )
For the parametric vector (0) = (ft, 0)T , the logarithm likelihood function is expressed as
1 n n
l = n log (2) + n log(ft) - 2n log(0) - 2-2 £ xf + (2ft - 1) £ logxfc 2 20 k=1 k=1
+ £ log cos k=1
2 Í 1 - e-^
(27)
The elements of the score vector U(©) =(Up, U9) are obtained by partially differentiating Equation (27) with respect to the model parameters and are given by
dl n
1
dß = ß + 2 £ ln(xk) - 212 £ xf ln( xk) - jOl £ tan
n
k=1
x2
— Í 1 — e 202
x2
e 202 x ff ln( xk )
dl -2n 1 n 2ß n n
+ ñS £ x* + ñp £ tan 0 k=1 20 k=1
30
0
n
x2
- Í 1 - e 202
x2
__L 2ß
e 202 xkr
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
n
e
k
n
n
n
k
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 (ft, d). 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 ft and d
sample size Parameters
n ß e
MLE
Bias
MSE
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;ß,e) = l2x2ß-1exp(-yy, ß,e > 0
Weighted Rayleigh Distribution (WRD) with PDF given as
xß+1exp (- 222)
f (x;ß,e) = , ß )a ) ; ß,e > 0
eß+222 r( ß +1)
Rayleigh distribution (RD) with PDF given as
x2
f (x; e) = e2exp (-2^); e > 0
Here, several goodness of fit criterion such as -2ll, Akaike Information Criterion (AIC), Bayesian Information 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.
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), -2ll, AIC, BIC, AICC, K-S statistic and P-valuefor Data-set 1.
Model ß Ô -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
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
Table 3: Estimates (standard errors), -2ll, AIC, BIC, AICC,K-S statistic and P-valuefor Data-set 2.
Model SPRD
PRD
WRD
RD
ß Ô -2ll AIC BIC AICC K-S P-value
0.5887 1.6864 162.9906 166.9906 170.5130 167.2906 0.0869 0.901
(0.0736) (0.2041)
0.6198 1.3094 163.2203 167.2203 170.7427 167.5203 0.0903 0.8744
(0.0766) (0.1647)
0.0010 2.2409 181.9592 185.9592 189.4816 186.2592 0.2423 0.0128
(0.3799) (0.2728)
2.2415 181.9277 183.9277 185.6889 184.0252 0.2421 0.0128
(0.1709)
Histogram of data
0 1 2 3 4 5 6 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 likelihood 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
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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