Weighted Inverse Rayleigh Distribution · biased Weighted Lomax Distribution; Afaq et al(2016). In this study, we propose a new distribution which is a Weighted Inverse Rayleigh (WIR)

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:

mailto:[email protected]

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


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.


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


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.


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:


)(

)()(

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.


k& . The reverse hazard rate of the WIR distribution takes the form:


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


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


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


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


)(!

)(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).


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(


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(


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.


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


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.


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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