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Transmuted Inverse Rayleigh Distribution: A Generalization of the Inverse Rayleigh Distribution

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Transmuted Inverse Rayleigh Distribution: A Generalization of the Inverse Rayleigh Distribution

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