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Mathematical Theory and Modeling www.iiste.org

ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)

Vol.4, No.7, 2014

90

Transmuted Inverse Rayleigh Distribution: A Generalization of

the Inverse Rayleigh Distribution.

Afaq Ahmad, S.P Ahmad and A. Ahmed

Department of Statistics, University of Kashmir, Srinagar, India

Abstract:

In this article, we generalize the Inverse Rayleigh distribution using the quadratic rank transmutation map

studied by Shaw et al. (2007) to develop a transmuted inverse Rayleigh distribution. The properties of this

distribution are derived and the estimation of the model parameters is performed by maximum likelihood

method.

Keywords: Inverse Rayleigh Distribution, Transmutation Map, Hazard Rate Function, Reliability Function,

Order Statistics, Parameter Estimation.

1. Introduction:

The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or

distribution. Because of this, considerable effort has been expended in the development of large classes of

standard probability distributions along with relevant statistical methodologies. However, there still remain many

important problems where the real data does not follow any of the classical or standard probability models.

The inverse Rayleigh distribution has many applications in the area of reliability studies. Voda (1972)

mentioned that the distribution of lifetimes of several types of experimental units can be approximated by the

inverse Rayleigh distribution. In this article we use transmutation map approach suggested by Shaw et al. (2007)

to define a new model which generalizes the Inverse Rayleigh model. We will call the generalized distribution

as the transmuted inverse Rayleigh distribution. According to the quadratic rank transmutation map (QRTM),

approach the cumulative distribution (cdf) satisfy the relationship

)()()1()( 2

112 xFxFxF

which on differentiation yields

)(21)()( 112 xFxfxf

where

)(

1xf

and

)(

2xf

are the corresponding probability density function (pdf) associated with F1(x) and

F2(x) respectively and

11

.

We will use the above formulation for a pair of distributions F(x) and G(x) where G(x) is a sub model of F(x).

Therefore, a random variable X is said to have transmuted probability distribution with cdf F(x) if

)1(1,)()()1()( 2

xGxGxF

where G(x) is the cdf of the base distribution. Observe that at

0

, we have the distribution of the base

random variable. Aryal and Tsokos (2009,2011) studied the transmuted extreme distributions. The authors

provided the mathematical characterization of transmuted Gumbel and transmuted Weibull distributions and

their applications to analyze real data sets. Faton Merovci (2013) studied the transmuted Rayleigh distribution,

Ashouret et al (2013). studied the transmuted exponentiated Lomax distribution and discussed some properties

of this family. In the present study we will provide mathematical formulation of the transmuted inverse Rayleigh

distribution and some of its properties.

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ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)

Vol.4, No.7, 2014

91

2. Transmuted Inverse Rayleigh Distribution

A random variable X is said to have a inverse Rayleigh distribution with parameter

>0 if its pdf is given by

)2(0,0,exp

2

),( 23

x

xx

xg

and the corresponding cdf is

)3(0,0,exp),( 2

x

x

xG

Now using (2) and (3) we have the cdf of transmuted inverse Rayleigh distribution

)4(exp1exp),,( 22

xx

xF

Hence, the pdf of transmuted inverse Rayleigh distribution with parameters

and

is

223 exp21exp

2

),,( xxx

xf

Note that the transmuted inverse Rayleigh distribution (TIR) is an extended model to analyze more complex

data. The Rayleigh distribution is clearly a special case for

0

. Figure 1 illustrates some of the possible

shapes of the pdf of a transmuted inverse Rayleigh distribution for selected values of the parameters

and

.

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Vol.4, No.7, 2014

92

Figure 1:The pdf's of various transmuted inverse Rayleigh distributions

x

f(x)

0 1 2 3 4

0.0 0.2 0.4 0.6 0.8 1.0 1.2

lambda= 0.0,theta= 1

lambda= -0.7,theta= 1

lambda= 0.7,theta= 1

lambda= -0.5,theta= 2

lambda= 0.5,theta= 2

Figure 2:The cdf's of various transmuted inverse Rayleigh distributions

x

F(x)

0 1 2 3 4

0.0 0.2 0.4 0.6 0.8 1.0 1.2

lambda= 0.0,theta= 1

lambda= -0.7,theta= 1

lambda= 0.7,theta= 1

lambda= -0.5,theta= 2

lambda= 0.5,theta= 2

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Vol.4, No.7, 2014

93

3. Statistical Properties

This section is devoted to study the statistical properties of the (TIR) distribution specially moments, quantile

function, median, moment generating function.

3.1 Moments: In this subsection we derive the rth moment for the (TIR) distribution.

Theorem 1 The rth moment

r

XE

of a transmuted inverse Rayleigh distributed random variable X is given as

22 21

2

1rr

rr

XE

Especially we have

21)( XE

2

21)1()var(

X

Proof.

0

),,()( dxxfxXE rr

dx

xx

xr

2

02

3exp21exp2

02

3

02

32

exp4exp)1(2 dx

x

xdx

x

xrr

2

1

2

12

2

1

2

2

1

)1( rr

rr

)6(21

2

122

rr r

3.2 Moment Generating Function: In this subsection we derive the moment generating function for the (TIR)

distribution.

Theorem 2. Let X have a transmuted inverse Rayleigh distribution. Then the moment generating function of X

is given by

2

0

221

2

1

!

)( r

r

r

r

Xr

r

t

tM

Proof.

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Vol.4, No.7, 2014

94

0

),,()exp()()( dxxftxeEtM tx

X

dxxf

tx

tx ),,(...

!2)(

1

0

2

00),,(

!

r

r

rdxxfx

r

t

0)(

!

r

r

rXE

r

t

2

0

221

2

1

!

r

r

r

rr

r

t

3.3 Quantiles and Median:

The quantile xq of the transmuted inverse Rayleigh distribution is real solution of the following equation

2

1

2

2

4)1()1(

ln

q

xq

In particular, the median of the distribution is

2

1

2

5.0

2

1()1(

ln

x

4. Random Number Generation and Parameter Estimation

Using the method of inversion we can generate random numbers from the transmuted inversion Rayleigh

distribution as

u

xx

22 exp1exp

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Vol.4, No.7, 2014

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where u ~ U (0, 1). After simplification this yields

)7(

2

4)1()1(

ln

2

1

2

u

x

One can use equation (7) to generate random numbers when the parameters

and

are known. The maximum

likelihood estimates (MLE’s) of the parameters that are inherent within the transmuted inverse Rayleigh

distribution function is given by the following:

Let

n

xxx ,...,,21

be a sample of size n from a transmuted inverse Rayleigh distribution. Then the likelihood

function is given by

n

ii

n

ii

n

ii

n

xx

x

L12

12

1

3exp21exp

2

The log likelihood function is given by

)8(exp21lnlnln2lnln 1 1 1 22

3

n

i

n

i

n

iii

ixx

xnnL

Therefore MLE’s of

and

which maximizes (8) must satisfy the following normal equations

n

i

i

ii

n

ii

x

xx

x

n

L

1

2

22

12

exp21

exp

2

1

ln

n

i

i

i

x

x

L

1

2

2

0

exp21

exp21

ln

The MLE

),(

of

),(

is obtained by solving this nonlinear system of equations. It is usually

more convenient to use nonlinear optimization algorithms such as quasi-Newton algorithm to numerically

maximize the log likelihood function given in (8). Applying the usual large sample approximation, the MLE

can be treated as being approximately bivariate normal with variance-covariance matrix equal to the inverse of

the expected information matrix, i.e.

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1

,0)(

InNn

where

1

I

is the limiting variance-covariance matrix of

. The elements of the 2×2 matrix

I

can be

estimated by

}2,1{,,ln)(

jiLI ji

ij

.

Approximate two sided 100(1-α) % confidence intervals for

and

are, respectively given by

)(

1

112/

Iz

and

)(

1

222/

Iz

where zα is the upper αth quantile of the standard normal distribution. Using R we can easily compute the

Hessian matrix and its inverse and hence the standard errors and asymptotic confidence intervals.

5. Reliability Analysis

The reliability function

)(tR

, which is the probability of an item not failing prior to some time t, is defined by

)(1)( tFtR

. The reliability function of a transmuted inverse Rayleigh distribution is given by

)9(exp1exp1),,( 22

tt

tR

The other characteristic of interest of a random variable is the hazard function defined by

)(1 )(

)( tF

tf

th

.

which is an important quantity characterizing life phenomenon. The hazard rate function for a transmuted

inverse Rayleigh random variable is given

)10(

exp1exp1

exp21exp

2

),,(

22

223

tt

ttt

th

6. Order Statistics

Order statistics make their appearance in many statistical theory and practice. We know that if

)()2()1( .,..,, n

XXX

denotes the order statistics of a random sample

n

XXX ,...,, 21

from a continuous

population with cdf

)(xFX

and pdf

)(xfX

, then the pdf of rth order statistics X(r) is given by

rn

X

r

XXrX xFxFxf

rnr n

xf

)(1)()(

!)(!)1( !

)( 1

)(

For r = 1, 2, . . . ,n.

we have from (2) and (3) the pdf of the rth order inverse Rayleigh random variable X(r) is given by

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rnr

rX xxx

rnr n

xg

223

)( exp1exp

2

)!()!1( !

)(

Therefore, the pdf of the nth order inverse Rayleigh statistic X(n) is given by

)11(exp

2

)( 23

)(

n

nX xx

n

xg

and the pdf of the first order inverse Rayleigh statistic X(1) is given by

)12(exp1exp

2

)(

1

223

)1(

n

Xxxx

n

xg

Note that in particular case of n=2, (11) yields

)13(exp

4

)(

2

23

)2(

xx

xgX

and (12) yields

)14(exp1exp

4

)( 223

)1(

xxx

xgX

Observe that (13) and (14) are special cases of (4) for

11

and

respectively. It has been observed that

a transmuted inverse Rayleigh distribution with

1

is the distribution of

),min( 21 XX

and a transmuted

inverse Rayleigh distribution with

1

is the

),max( 21 XX

. Where X1 and X2 are independent and

identically distributed inverse Rayleigh random variables. Now we provide the distribution of the order statistics

for a transmuted inverse Rayleigh random variable. The pdf of the rth order statistic for a transmuted inverse

Rayleigh distribution is given by

rnr

rX

xxxx

xxx

rnr n

xf

22

1

22

223

)(

exp1exp1exp1exp

exp21exp

2

)()1( !

)(

Therefore, the pdf of the largest order statistic X(n) is given by

1

22223

)( exp1expexp21exp

2

)(

n

nX xxxxx

n

rf

and the pdf of the smallest order statistic X(1) is given by

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Vol.4, No.7, 2014

98

1

22223

)1( exp1exp1exp21exp

2

)(

n

Xxxxxx

n

xf

Note that

0

yields the order statistics of the inverse Rayleigh distribution.

Conclusion

In the present study we have introduced a new generalization of the inverse Rayleigh distribution called the

transmuted inverse Rayleigh distribution. The subject distribution is generated by using the quadratic rank

transmutation map and taking the inverse Rayleigh distribution as the base distribution. Some mathematical

properties along with estimation issues are discussed. The hazard rate function and reliability behavior of

transmuted inverse Rayleigh distribution shows that subject distribution can be used to model reliability data.

References

Aryal, G. R., and Tsokos, C.P. (2009). “On the transmuted extreme value distribution with

application”, Nonlinear Analysis: Theory, Methods and Applications, vol.71, pp. 1401-1407.

Aryal, G. R., and Tsokos, C.P. (2011), “Transmuted Weibull distribution: A generalization of the

Weibull probability distribution”, European Journal of Pure and Applied Mathematics, vol.4, pp. 89-

102.

Ashour, S. and et al. (2013), “Transmuted Exponentiated Lomax distributiom” Austrilian Journal of

Basic and Applied Sciences, vol. 7, pp. 658-667.

Kundu, D., and Raqab, M. Z. (2005), “Generalized Rayleigh distribution: different methods of

estimation”, Computational Statistics and Data Analysis, vol. 49, pp. 187-200.

Merovci, F. (2013), “Transmuted Rayleigh distribution”, Austrian Journal of Statistics, vol. 42, no. 1,

pp. 21-31.

Shaw, W. and Buckley, I. (2007). “The alchemy of probability distributions: beyond Gram-Charlier

expansions, and a skew-kurtotic-normal distribution from a rank transmutation map”, Research Report.

Soliman, A. and et al. (2010), “Estimation and Prediction from the inverse Rayleigh distribution based

on lower record values”, Applied Mathematical Sciences, vol. 4, no. 62, pp. 3057-3066.

Voda, R. (1972), “On the inverse Rayleigh variable”, Rep. Stat. Res. Juse, Vol. 19, no. 4, pp.15-21.