International Journal of Statistics and Systems
ISSN 0973-2675 Volume 12, Number 1 (2017), pp. 119-137
© Research India Publications
http://www.ripublication.com
Weighted Inverse Rayleigh Distribution
Kawsar Fatima and S.P Ahmad
Department of Statistics, University of Kashmir, Srinagar, India. Corresponding author
Abstract
In this paper, we have introduced weighted inverse Rayleigh (WIR)
distribution and investigated its different statistical properties. Expressions for
the Mode and entropy have also been derived. In addition, it also contains
some special cases that are well known. Moreover, we apply the maximum
likelihood method to estimate the parameter , and applications to two real
data sets show the superiority of this new distribution by comparing the fitness
with its special cases.
Keywords: Inverse Rayleigh distribution, weighted distribution, Reliability
Analysis, Entropy, Maximum likelihood estimation, Real life data sets.
1. INTRODUCTION The IR distribution was proposed by Voda (1972). He studies some properties of the
MLE of the scale parameter of inverse Rayleigh distribution which is also being used
in lifetime experiments. If X has IR distribution, its probability density function (pdf)
takes the following form:
0,0;2
)(2
3
xex
xg x
)1.1(
The corresponding cumulative distribution function is
0,0;)(2
xexG x )2.1(
Where 0x the scale parameter 0
The kth moment of the IR distribution is given as the following:
120 Kawsar Fatima and S.P Ahmad
0
2
3.
2)( dxe
xxxE xkk
g
Making the substitution2
1
xy , dydx
x
3
2, so that
2/1
1
yx ,we obtain
0
1)2/1( .)( dyeyxEykk
g
)2/1()( 2/ kxE kkg )3.1(
Then, the expected value of X can be written as:
)2/1()( 2/1 xEg )4.1(
2. WEIGHTED INVERSE RAYLEIGH DISTRIBUTION
The use and application of weighted distributions in research related to reliability,
biomedicine, ecology and many other areas are of great practical importance in
mathematics, probability and statistics. These distributions arise naturally as a result
of observations generated from a stochastic process and recorded with some weight
function. Firstly the model of weighted distributions was introduced by Fisher
(1934). Cox (1962) originally provided the concept of length-biased sampling and
after that Rao (1965) recognized a unifying method that can be used for several
sampling situations and can be displayed by means of the weighted distributions. Cox
(1968) expected mean of the original distribution built on length biased data.
Recently, many researches are applied to length-biased for lifetime distribution, The
Length-Biased Weighted Weibull Distribution; Tanusree Deb Roy et al(2011), The
Length-biased inverse Weibull distribution; Jing Kersey and Broderick O. Oluyede
(2012),The Length biased Beta distribution of first kind; Mir et al (2013),The Length-
biased Exponentiated Inverted Weibull Distribution (2014),The Length-biased
weighted Nakagami Distribution; Sofi Mudasir and S.P Ahmad (2015), The Length-
biased Weighted Lomax Distribution; Afaq et al(2016).
In this study, we propose a new distribution which is a Weighted Inverse Rayleigh
(WIR) distribution. We first provide a general definition of the Weighted Inverse
Rayleigh (WIR) distribution which will subsequently reveal its pdf.
Definition1. If X has a lifetime distribution with pdf )(xg and expected value,
)( kg xE , the pdf of Weighted distribution of X can be defined as:
0,0,)(
)()( xk
xExgxxf k
g
k
)1.2(
Weighted Inverse Rayleigh Distribution 121
Theorem 2.1: - Let X be a random variable of an IR distribution with pdf )(xg .
Then )(
)()( k
g
k
xExgxxf is a pdf of the WIR distribution with scale parameter and
weight parameter k . The notation for X with the WIR distribution is denoted as
),(~ kWIRX . The pdf of X is given by:
0,0,0;)2/1(
2)(
2/3)2/1(
kxexk
xf xkk
)2.2(
Proof: -By definition 1, substitute (1.1) and (1.3) into (2.1), then the pdf for the WIR
distribution can be obtained by:
2
32/
2
)2/1()( x
k
ke
xkxxf
23)2/1(
)2/1(
2)( xk
kex
kxf
Figure 1 illustrates some of the possible shapes of the probability density function of
Weighted inverse Rayleigh distribution for selected values of and k
Figure 1: The probability density function of the WIR distribution for selected values
of and k
122 Kawsar Fatima and S.P Ahmad
We observe from Figure1 that the density function of WIR is positively skewed and
that the curve decreases as the value of increases. So, the shape of the proposed
WIR distribution could be decreasing. Also, we observed that the shape of the
proposed WIR distribution could be unimodal.
Theorem 2.2: - Let X be a random variable of the WIR distribution with parameter
k& . The distribution function of the WIR distribution is written as:
21
,2
1
)(2
kx
k
xF
)3.2(
where dtetxs tx
s
1, is an upper incomplete gamma function.
Proof: - Generally, the distribution function of lifetime distribution is defined as:
dxxfxF
x 0
)()( )4.2(
Substituting (2.3) into (2.4), we obtain:
dxexk
xFx xk
k
0
23)2/1(
)2/1(
2)(
By setting 2x
y , dydx
x
3
2, y
x2
, the above integration becomes:
dyeyk
xFx
y
k
2
12
1
)2/1(
1)(
21
,2
1
)(2
kx
k
xF
The corresponding plots of the WIR distribution function at various values of and kare shown in Figure 2.
Weighted Inverse Rayleigh Distribution 123
Figure 2: The distribution function of the WIR distribution for selected values of
and k .
The distribution curves show the increasing rate.
Theorem 2.3: - Let X be a random variable of the WIR distribution with Parameter
k& . The survival function of the WIR distribution can be written as:
21
,2
1
)(2
kx
k
xS
)5.2(
where
x ts dtetxs0
1, is a lower incomplete gamma function.
Proof: - By definition, the survival function of the random variable X is given by:
)(1)( xFxS .
Using (2.3), the survival function of the WIR distribution can be expressed by:
21
,2
1
1)(2
kx
k
xS
124 Kawsar Fatima and S.P Ahmad
21
,2
12
12
kx
kk
21
,2
12
kx
k
.
Figure 3 illustrates some of the possible shapes of the survival function of Weighted
inverse Rayleigh distribution for selected values of and k
Figure 3: The survival function of the WIR distribution for different values of and
k .
The survival curves show the decreasing rate.
Theorem 2.4: - Let X be a random variable of the WIR distribution with Parameter
k& . The hazard rate of the WIR distribution takes the form:
2
232/1
,2
1
2)(
xk
exxhxkk
)6.2(
Proof: -Let X be a continuous random variable with pdf and survival function, )(xfand )(xS , respectively, then the hazard rate is defined by:
Weighted Inverse Rayleigh Distribution 125
)(
)()(
xSxfxh . )7.2(
Substituting (2.2) and (2.5) into (2.7), we obtain:
)2/1(/,2
1
)2/1(/2)(
2
232/1
kx
kkexxh
xkk
2
232/1
,2
1
2
xk
ex xkk
Figure 4 illustrates some of the possible shapes of the hazard function of Weighted
inverse Rayleigh distribution for selected values of and k
Figure 4: The hazard rate of the WIR distribution for different values of and k .
We can infer from Figure 4 that the shape of the hazard rate is positively skewed, if
the value of increases the hazard rate decreases. We can also say that the hazard rate
shows an inverted bathtub shape or unimodal.
Theorem 2.5: - Let X be a random variable of the WIR distribution with Parameter
k& . The reverse hazard rate of the WIR distribution takes the form:
126 Kawsar Fatima and S.P Ahmad
2
232/1
,2
1
2)(
xk
exxxkk
)8.2(
Proof: -Let X be a continuous random variable with pdf and cdf, )(xf and )(xF ,
respectively, then the reverse hazard rate is defined by:
)(
)()(
xFxfx . )9.2(
Substituting (2.2) and (2.3) into (2.9), we obtain:
)2/1(/,2
1
)2/1(/2)(
2
232/1
kx
kkexx
xkk
2
232/1
,2
1
2
xk
ex xkk
Figure 5 illustrates some of the possible shapes of the Reverse hazard rate function of
weighted inverse Rayleigh distribution for selected values of and k .
Figure 5: The reverse hazard function of the WIR distribution for different values of
and k .
3. SOME SPECIAL CASES OF WEIGHTED INVERSE RAYLEIGH
DISTRIBUTIONS
Weighted Inverse Rayleigh Distribution 127
This section presents some special cases that deduced from equation (2.2) are
Case 1: When 0k , then weighted inverse Rayleigh distribution (2.2) reduces to
inverse Rayleigh distribution (IRD) with probability density function as:
2
3
2)( xe
xxf
)1.3(
Case 2: When 1k , then weighted inverse Rayleigh distribution (2.2) reduces to
length biased inverse Rayleigh distribution (IRD) with probability density function
as:
222/1
)2/1(
2)( xexxf
)2.3(
Case 3: If a random variable is such that XY /1 and kk in Equation (2.2) reduces
to give the weighted Rayleigh distribution (WRD) with probability density function
as:
)12/(
2)(
21)12/(
kexxf
xkk )3.3(
Case 4: If a random variable is such that XY /1 and 1k in Equation (2.2) reduces
to give the length biased Rayleigh distribution (LBRD) with probability density
function as:
)2/1(
4)(
22)2/3(
xexxf
)4.3(
Case 5: If a random variable is such that XY /1 and 0k in Equation (2.2)
reduces to give the Rayleigh distribution (RD) with probability density function as: 2
2)( xxexf )5.3(
4. STATISTICAL PROPERTIES OF THE WIR DISTRIBUTION
This section provides some basic statistical properties of the weighted Inverse
Rayleigh Distribution.
4. 1 The rth Moment of the WIR Distribution
The result of this section gives the rth moment of WIR distribution.
Theorem 4.1: - If )WIR(~ X , then rth moment of a continuous random variable X
is given as follow:
)2/)(1()2/1(
)(2/
krk
XEr
rr
128 Kawsar Fatima and S.P Ahmad
Proof: - Let X is an absolutely continuous non-negative random variable with pdf
)(xf , then rth moment of X can be obtained by:
dxxfxXE rrr
0)()(
From the pdf of the WIR distribution in (2.2), then show that )( rXE can be written
as:
dxexk
xXE xk
krr 23)2/1(
0)2/1(
2)(
Making the substitution,2x
y , dydx
x
3
2, so that
2/1
2/1
yx , we obtain
0
1)2/)(1(2/
.)2/1(
)( dyeyk
XEykr
rr
After some calculations,
)2/)(1()2/1(
)(2/
krk
XEr
rr
)1.4(
Substitute r = 1, 2, in (4.1) we get mean and variance of WIRD
Mean= )2/)1(1()2/1(
)(2/1
kk
XE
)2.4(
)2/)2(1()2/1(
)( 2 kk
XE
)3.4(
Variance =
22/1
)2/)1(1()2/1(
)2/)2(1()2/1(
kk
kk
)4.4(
4.2 Harmonic mean of WIR distribution
The harmonic mean (H) is given as:
dxexkxH
xkk
0
23)2/1(
)2/1(
211
dxxfxX
EH
)(111
0
Weighted Inverse Rayleigh Distribution 129
By setting2
1
xy , we get
)2/)1(1()2/1(
112/1
kkH
)5.4(
4.3 Mode
Consider the density of the WIR distribution given in (2.2), we take the logarithm of
(2.2) as follows:
)2/1(log)log()3()log(2
1)2log()(log2
kx
xkkxf
)6.4(
Differentiating equation (4.6) with respect to x , we obtain
3
2)3()(log
xxk
xxf
)7.4(
Now, set equation (4.7) equal to 0 and solve for x , to get
)3(
20 k
x )8.4(
4.4 Moment generating function
In this sub section we derived the moment generating function of WIR distribution.
Theorem 4.4: - Let X have aWIR distribution. Then moment generating function of
X denoted by )(tM X is given by:
0
2/
)2/)(1()2/1(!
)(r
rr
X krkr
ttM
)9.4(
Proof: -By definition
dxxfeeEtM txtxX
0
)()()(
Using Taylor series
dxxftxtxtM X
0
2
)(!2
)(1)(
1
0 0
( ) ( )!
rr
Xr
tM t x f x dxr
130 Kawsar Fatima and S.P Ahmad
)(!
)(0
r
i
r
X XErttM
0
2/
)2/)(1()2/1(!
)(r
rr
X krkr
ttM
This completes the proof.
4.5 Characteristic function
In this sub section we derived the Characteristic function of WIR distribution.
Theorem 4.5: - Let X have a WIR distribution. Then characteristic function of X
denoted by )(tX is given by:
0
2/
)2/)(1()2/1(!
)()(
r
rr
X krkr
itt )10.4(
Proof: -By definition
dxxfeeEt itxtxiX
1
0
)()()(
Using Taylor series
dxxfitxitxtX
0
2
)(!2
)(1)(
dxxfxritt r
r
r
X
00
)(!
)((
)(!
)()(
0
r
r
r
X XEritt
0
2/
)2/)(1()2/1(!
)()(
r
rr
X krkr
itt
This completes the proof.
5. SHANNON’S ENTROPY OF WEIGHTED INVERSE RAYLEIGH
DISTRIBUTION
For deriving entropy of the weighted Inverse Rayleigh distribution, we need the
following definition that more details of this can be found in Thomas J.A. et.al.
(1991).
Weighted Inverse Rayleigh Distribution 131
The oblivious generalizations of the definition of entropy for a probability density
function f defined on the real line as:
dxxfxfxfExH )()(log)]([log)(0
Provided the integral exists.
Theorem 5.1: - Let ),...,,( 21 nxxxx be n positive identical independently distributed
random samples drawn from a population having weighted Inverse Rayleigh density
function as
23)2/1(
)2/1(
2)( xk
kex
kxf
Then Shannon’s entropy of weighted Inverse Rayleigh distribution is
)2/1()2/1()log(2
)3(
2
)2/1(log)(
)2/1(kkkkxH k
Proof: -Shannon’s entropy is defined as
)]([log)( xfExH
23)2/1(
)2/1(
2log)( xk
kex
kExH
2
2/1 1)log()3(
)2/1(
2log)(
xExEk
kxH
k
)1.5(
Now,
dxxfxxE )()log()log(0
dxexk
xxE xkk
232/1
0 )2/1(
2)log()log(
By setting 0,,,0,,2
,32
yxyxas
yxdx
xdy
xy
dyeyyk
xE ykk
0
2/
2/1
2/12/
log)2/1(
)log(
After solving the above expression, we get
21)log(
2
1)log(
kxE )2.5(
132 Kawsar Fatima and S.P Ahmad
Also,
dxexkxx
E xkk
232/1
0 22 )2/1(
211
By setting 0,,,0,,2
,32
yxyxas
yxdx
xdy
xy
dyeykx
E yk
0
1)2/2(
2 )2/1(
11
After solving the above expression, we get
)2/1(12
kx
E
)3.5(
Substitute the values of equation (5.2) and (5.3) in equation (5.1), we get
)2/1()2/1()log(2
)3(
2
)2/1(log)(
)2/1(kkkkxH k
)4.5(
The above relation (5.4) indicates the Shannon’s entropy of Weighted Inverse
Rayleigh distribution.
6. ESTIMATION OF PARAMETERS
In a view to estimating the parameter of theWIR distribution, we make use of the
method of Maximum Likelihood Estimation (MLE). Let ),...,,( 21 nxxxx be a
random sample having probability density function (2.2), and then the likelihood
function is given by
2
1
3)2/1(
)2/1(
2)( ix
n
i
kin
knnex
kxL
)1.6(
By taking logarithm of (6.1), we find the log likelihood function as
n
i i
n
ii
xxkknknnxL
12
1
ln)3()2/1(logln)2/1(2ln)(ln
)2.6(
Differentiating equation (6.2) with respect to and equate to zero, we get
01)2/1()(ln
1
2
n
iix
knxL
n
iin
ii
xTTkn
x
kn1
2
1
2
where;)2/1()2/1(
)3.6(
Weighted Inverse Rayleigh Distribution 133
7. APPLICATION
In this section, we illustrate the usefulness of the weighted Inverse Rayleigh
distribution. We fit this distribution to two data sets and compare the results with its
special cases that areInverse Rayleigh distribution, Length Biased Inverse Rayleigh
distribution,weighted Rayleigh distribution, Length Biased Rayleigh distribution, and
Rayleigh distribution. The first real data set represents the 72 exceedances for the
years 1958–1984 (rounded to one decimal place) of flood peaks (in m3/s) of the
Wheaton River near Carcross in Yukon Territory, Canada. The data are as follows:
1.7, 2.2, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.9, 13.0, 12.0, 9.3,1.4, 18.7, 8.5, 25.5, 11.6,
14.1, 22.1, 1.1, 2.5, 14.4, 1.7, 37.6,0.6, 2.2, 39.0, 0.3, 15.0, 11.0, 7.3 , 22.9, 1.7, 0.1,
1.1, 0.6, 9.0, 1.7, 7.0, 20.1, 0.4, 2.8 ,14.1, 9.9 ,10.4 ,10.7 ,30.0, 3.6,5.6, 30.8, 13.3, 4.2,
25.5, 3.4, 11.9 ,21.5, 27.6 ,36.4 ,2.7 ,64.0,1.5, 2.5, 27.4, 1.0, 27.1, 20.2, 16.8, 5.3, 9.7,
27.5, 2.5 and 7.0. Recently, Merovci and Puka (2014) studied these data using the
Transmuted Pareto (TP) distribution.The second data set is regarding remission times
(in months) of a random sample of 128 bladder cancer patients given in Lee and
Wang (2003). The data set is given as follows : 0.08, 2.09, 2.73, 3.48, 4.87, 6.94,
8.66, 13.11, 23.63, 0.20, 2.22, 3.52, 4.98, 6.99, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06,
7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54,
3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15,
2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96,
36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 15.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63,
17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87,
11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98,
19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28,
2.02, 3.36, 6.93, 8.65, 12.63 and 22.69.
We fit the WIR distribution to above two data sets and compare the fitness with its
special cases such as the IR, LBIR, WR, LBR and Rayleigh distributions. The
required numerical evaluations are carried out using the Package of R software. The
MLEs of the parameters with standard errors in parentheses and the corresponding
log-likelihood values, AIC, AICC and BIC are displayed in Table 1and 2.
134 Kawsar Fatima and S.P Ahmad
Table 1: MLEs (S.E in parentheses) for Wheaton river flood data
Distribution Parameter Estimates
-2Log L AIC AICC BIC
k
Weighted
Inverse
Rayleigh(WIRD)
1.64725
(0.04472)
0.09136
(0.02813)
575.2203 579.2203 579.3942 583.7736
Length Biased
Inverse Rayleigh
(LBIRD)
_ 0.25899
(0.04316)
669.7216 671.7216 671.7787 673.9983
Inverse Rayleigh
(IRD)
_ 0.51799
(0.06105)
915.6792 917.6792 917.7363 919.9559
Weighted
Rayleigh (WRD)
0.51008
(0.34756)
0.00421
(0.00067)
664.1989 668.1989 664.3728 672.7522
Length Biased
Rayleigh(LBRD)
_ 0.00503
(0.00046)
682.7753 684.7753 684.807 687.052
Rayleigh(RD) _ 0.00335
(0.00036)
605.6757 607.6757 607.7074 609.9524
Table 2: MLEs (S.E in parentheses) for bladder cancer data
Distribution Parameter Estimates
-2Log L AIC AICC BIC
k
Weighted
Inverse Rayleigh
(WIRD)
1.61586
(0.03672)
0.11869
(0.02649)
975.4733 979.4733 979.5693 985.1774
Length Biased
Inverse Rayleigh
(LBIRD)
_ 0.30899
(0.03862)
1111.258 1113.258 1113.29 1118.11
Inverse Rayleigh
(IRD)
_ 0.61798
(0.05462)
1497.54 1499.54 1499.572 1502.392
Weighted
Rayleigh (WRD)
0.51433
(0.27344)
0.00657
(0.00085)
1082.764 1086.764 1086.86 1092.468
Weighted Inverse Rayleigh Distribution 135
Length Biased
Rayleigh(LBRD)
_ 0.00782
(0.00055)
1101.261 1103.261 1103.293 1106.113
Rayleigh (RD) _ 0.00521
(0.00044)
992.2544 994.2544 994.2861 997.1064
In order to compare the six distribution models, we consider the criteria like AIC
(Akaike information criterion), AICC (corrected Akaike information criterion) and
BIC (Bayesian information criterion). The better distribution corresponds to lesser
AIC, AICC and BIC values.
Lp log22AIC )1(
)1(2AICAICC
pnpp
and Lnp log2logBIC
where p is the number of parameters in the statistical model, n is the sample
size and - 2logL is the maximized value of the log-likelihood function under the
considered model. From Table 1 and 2, it is obvious that the Weighted Inverse
Rayleigh distribution have the lesser AIC, AICC and BIC values as compared to other
sub-models. So we can conclude that the WIR distribution provides better fit than
Inverse Rayleigh distribution, Length Biased Inverse Rayleigh distribution,weighted
Rayleigh distribution, Length Biased Rayleigh distribution, and Rayleigh distribution
Figure 6: Plots of the fitted WIR, IR, LBIR, WR, LBR and Rayleigh distributions for
data sets 1 and 2.
136 Kawsar Fatima and S.P Ahmad
7. CONCLUSION
In this paper, we have introduced Weighted Inverse Rayleigh (WIR) distribution,
which acts as a generalization to so many distributions viz. IRD, LBIRD, WRD,
LBRD and RD. After introducing WIRD, we investigated its different mathematical
properties. Two real data sets have been considered in order to make comparison
between special cases of WIRD in terms of fitting. After the fitting of WIRD and its
special cases to the data sets considered, the results are given in Table 1and 2. It is
evident from the Table 1 and 2 that, WIRD possesses minimum values of AIC, AICC
and BIC on its fitting, to two real life data sets. Thereforewe can conclude that the
WIRD will be treated as a best fitted distribution to the data sets as compared to its
other special cases.
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