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: 2 2 xg )( x xe 0,0; )1.1( x3
The corresponding cumulative distribution function is 2 x xexG 0,0;)( )2.1( Where x 0 the scale parameter 0 The kth moment of the IR distribution is given as the following: 120 Kawsar Fatima and S.P Ahmad
2 k k 2 x g )( xxE e dx. 0 x3 1 1 2 Making the substitution y , dydx , so that x 2/1 ,we obtain x 2 x3 y
y k )( k 1)2/1( eyxE dy. g 0
kk 2/ g xE k )2/1()( )3.1(
Then, the expected value of X can be written as: 2/1 g xE )2/1()( )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 has a lifetime distribution with pdf xg )( and expected value, k g xE )( , the pdf of Weighted distribution of X can be defined as: k xgx )( )( 0,0, )1.2( xf k xk g xE )(
Weighted Inverse Rayleigh Distribution 121
Theorem 2.1: - Let X be a random variable of an IR distribution with pdf xg )( . k xgx )( Then )( is a pdf of the WIR distribution with scale parameter and xf k g xE )( weight parameter k . The notation for with the WIR distribution is denoted as kWIRX ),(~ . The pdf of is given by: )2/1( 2 k 2 xf )( /3 xk xex k 0,0,0; )2.2( k )2/1(
Proof: -By definition 1, substitute (1.1) and (1.3) into (2.1), then the pdf for the WIR distribution can be obtained by: k x 2 2 xf )( e x k 2/ )2/1( xk 3
k )2/1( 2 2 xf )( k3ex x k )2/1(
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: k 1 , 2 x 2 xF )( )3.2( k 1 2 where , s1etxs t dt is an upper incomplete gamma function. x
Proof: - Generally, the distribution function of lifetime distribution is defined as: x xfxF )()( dx )4.2( 0
Substituting (2.3) into (2.4), we obtain:
k )2/1( x 2 2 xF )( k3ex x dx 0 k )2/1(
2 By setting y , dydx , y , the above integration becomes: x 2 x3 x2
k 1 1 1 2 y xF )( ey dy k )2/1( 2 x
k 1 , 2 x 2 xF )( k 1 2
The corresponding plots of the WIR distribution function at various values of and k are 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: k 1 , 2 x 2 xS )( )5.2( k 1 2 x where , 1etxs ts dt is a lower incomplete gamma function. 0
Proof: - By definition, the survival function of the random variable X is given by: xFxS )(1)( .
Using (2.3), the survival function of the WIR distribution can be expressed by: k 1 , 2 x 2 xS 1)( k `1 2 124 Kawsar Fatima and S.P Ahmad
kk 1 1 , 2 2 2 x k 1 2 k 1 , 2 2 x . k 1 2
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 2 kk 32/1 ex x xh )( )6.2( k 1 , 2 x 2
Proof: -Let be a continuous random variable with pdf and survival function, xf )( and xS )( , respectively, then the hazard rate is defined by: Weighted Inverse Rayleigh Distribution 125
xf )( xh )( . )7.2( xS )(
Substituting (2.2) and (2.5) into (2.7), we obtain: 2 2 kk 32/1 x kex )2/1(/ xh )( k 1 k )2/1(/, 2 x 2 2 2 kk 32/1 ex x k 1 , 2 2 x 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 2 kk 32/1 ex x x)( )8.2( k 1 , 2 x 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: xf )( x)( . )9.2( xF )( Substituting (2.2) and (2.3) into (2.9), we obtain: 2 2 kk 32/1 x kex )2/1(/ x)( k 1 k )2/1(/, 2 x 2 2 2 kk 32/1 ex x k 1 , 2 2 x
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 k 0 , then weighted inverse Rayleigh distribution (2.2) reduces to inverse Rayleigh distribution (IRD) with probability density function as: 2 2 xf )( e x )1.3( x3 Case 2: When k 1, then weighted inverse Rayleigh distribution (2.2) reduces to length biased inverse Rayleigh distribution (IRD) with probability density function as: 2/1 2 2 xf )( 2ex x )2.3( )2/1(
Case 3: If a random variable is such that /1 XY and kk in Equation (2.2) reduces to give the weighted Rayleigh distribution (WRD) with probability density function as: 2 2 1)12/( ex xkk xf )( )3.3( k )12/(
Case 4: If a random variable is such that and k 1 in Equation (2.2) reduces to give the length biased Rayleigh distribution (LBRD) with probability density function as: 2 4 2)2/3( ex x xf )( )4.3( )2/1(
Case 5: If a random variable is such that /1 XY and k 0 in Equation (2.2) reduces to give the Rayleigh distribution (RD) with probability density function as: 2 xf 2)( xe x )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.
th Theorem 4.1: - If X )WIR(~ , then r moment of a continuous random variable X is given as follow: r 2/ XE r )( kr )2/)(1( r k )2/1( 128 Kawsar Fatima and S.P Ahmad
Proof: - Let X is an absolutely continuous non-negative random variable with pdf th xf )( , then r moment of X can be obtained by:
r r xfxXE )()( dx r 0 From the pdf of the WIR distribution in (2.2), then show that XE r )( can be written as: k )2/1( 2 2 )( xXE krr 3ex x dx k )2/1( 0 2/1 2 Making the substitution, , , so that , we obtain y 3 dydx x 2/1 x 2 x y r 2/ y XE r )( y kr 1)2/)(1( e dy. k )2/1( 0
After some calculations, r 2/ XE r )( kr )2/)(1( )1.4( r k )2/1( Substitute r = 1, 2, in (4.1) we get mean and variance of WIRD 2/1 Mean= XE )( k )2/)1(1( )2.4( k )2/1( XE 2 )( k )2/)2(1( )3.4( k )2/1( 2 2/1 Variance = k )2/)2(1( k )2/)1(1( )4.4( k )2/1( k )2/1(
4.2 Harmonic mean of WIR distribution The harmonic mean (H) is given as: 111 E xf )( dx H 0 xX
k )2/1( 211 2 k3ex x dx kxH )2/1( 0 Weighted Inverse Rayleigh Distribution 129
1 By setting y , we get x 2
1 1 k )2/)1(1( )5.4( H k )2/1( 2/1
4.3 Mode Consider the density of the WIR distribution given in (2.2), we take the logarithm of (2.2) as follows: k xf 1)2log()(log xk k )2/1(log)log()3()log( )6.4( 2 x2 Differentiating equation (4.6) with respect to x , we obtain
xf k 2)3()(log )7.4( x x x3 Now, set equation (4.7) equal to 0 and solve for , to get 2 x0 )8.4( k)3(
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 X tM )( is given by: t rr 2/ tM )( kr )2/)(1( )9.4( X )2/1(! r0 kr Proof: -By definition tx tx )()()( X xfeeEtM dx 0 Using Taylor series tx)( 2 1)( )( X tM tx xf dx 0 !2
r 1 t r MX () t x f() x dx r0 r! 0 130 Kawsar Fatima and S.P Ahmad
r t r X tM )( XE )( i0 r! t rr 2/ X tM )( kr )2/)(1( r0 kr )2/1(! 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 X t)( is given by: it)( rr 2/ X t)( kr )2/)(1( .4( 10) r0 kr )2/1(! Proof: -By definition 1 itx itx )()()( X xfeeEt dx 0 Using Taylor series itx)( 2 1)( )( X t itx xf dx 0 !2 it)( r ( r )( X t xfx dx r0 r! 0 r it)( r X t)( XE )( r0 r! it)( rr 2/ X t)( kr )2/)(1( r0 kr )2/1(!
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: ([log)( xfExH )] xfxf )()(log dx 0 Provided the integral exists.
Theorem 5.1: - Let ,( 21 ,..., xxxx n ) be n positive identical independently distributed random samples drawn from a population having weighted Inverse Rayleigh density function as k )2/1( 2 2 xf )( k3ex x k )2/1( Then Shannon’s entropy of weighted Inverse Rayleigh distribution is )2/1( kk )3( xH log)( k k )2/1()2/1()log( 2 k )2/1( 2
Proof: -Shannon’s entropy is defined as ([log)( xfExH )] k )2/1( 2 3 2 ExH log)( k ex x k )2/1( 2 k 2/1 1 log)( )log()3( )1.5( xH ExEk 2 k )2/1( x Now, xE xfx )()log()log( dx 0 k 2/1 2 2 xE x)log()log( k3ex x dx 0 k )2/1( 2 By setting , 0,,,0,, y 2 dy 3 dx x as yxyx x x y
2/ 2/1 k 2/ k xE )log( log ey dy )2/1( 0 2/1 k y y
After solving the above expression, we get 1 k xE )log( 1)log( )2.5( 2 2 132 Kawsar Fatima and S.P Ahmad
Also, k 2/1 211 2 E k3ex x dx 2 0 xx 2 k )2/1( 2 By setting , 0,,,0,, y 2 dy 3 dx x as yxyx x x y 11 E 1)2/2( ey yk dy x2 k )2/1( 0
After solving the above expression, we get k )2/1(1 E )3.5( x2 Substitute the values of equation (5.2) and (5.3) in equation (5.1), we get )2/1( kk )3( xH log)( k k )2/1()2/1()log( )4.5( 2 k )2/1( 2 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 ,..., xxxx n ) be a random sample having probability density function (2.2), and then the likelihood function is given by 2 knn )2/1( n 2 )( k3 xi )1.6( xL n i ex k )2/1( i1 By taking logarithm of (6.1), we find the log likelihood function as
n n ln )( ln knnxL )2/1(2 ln )3()2/1(log ln xkkn )2.6( i 2 i1 i1 xi Differentiating equation (6.2) with respect to and equate to zero, we get ln xL )( kn 1)2/1( 0 n 2 xi i1 n kn kn )2/1()2/1( 2 ˆ where; n xT i )3.6( 2 T i1 xi i1 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 1.64725 0.09136 575.2203 579.2203 579.3942 583.7736 Inverse (0.04472) (0.02813) Rayleigh(WIRD) Length Biased _ 0.25899 669.7216 671.7216 671.7787 673.9983 Inverse Rayleigh (0.04316) (LBIRD)
Inverse Rayleigh _ 0.51799 915.6792 917.6792 917.7363 919.9559 (IRD) (0.06105)
Weighted 0.51008 0.00421 664.1989 668.1989 664.3728 672.7522 Rayleigh (WRD) (0.34756) (0.00067) Length Biased _ 0.00503 682.7753 684.7753 684.807 687.052 Rayleigh(LBRD) (0.00046)
Rayleigh(RD) _ 0.00335 605.6757 607.6757 607.7074 609.9524 (0.00036)
Table 2: MLEs (S.E in parentheses) for bladder cancer data
Distribution Parameter Estimates -2Log L AIC AICC BIC
k
Weighted 1.61586 0.11869 975.4733 979.4733 979.5693 985.1774 Inverse Rayleigh (0.03672) (0.02649) (WIRD) Length Biased _ 0.30899 1111.258 1113.258 1113.29 1118.11 Inverse Rayleigh (0.03862) (LBIRD)
Inverse Rayleigh _ 0.61798 1497.54 1499.54 1499.572 1502.392 (IRD) (0.05462)
Weighted 0.51433 0.00657 1082.764 1086.764 1086.86 1092.468 Rayleigh (WRD) (0.27344) (0.00085) Weighted Inverse Rayleigh Distribution 135
Length Biased _ 0.00782 1101.261 1103.261 1103.293 1106.113 Rayleigh(LBRD) (0.00055)
Rayleigh (RD) _ 0.00521 992.2544 994.2544 994.2861 997.1064 (0.00044)
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.
pp )1(2 log22AIC Lp AICC AIC and log2logBIC Lnp pn )1(
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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