Exponentiated Transmuted Generalized Rayleigh Distribution : A New Four Parameter Rayleigh Distribution

This paper introduces a new four parameter Rayleigh distribution which generalizes the transmuted generalized Rayleigh distribution introduced by Merovci (2014). The new model is referred to as exponentiated transmuted generalized Rayleigh (ETGR) distribution. Various mathematical properties of the new model including ordinary and incomplete moments, quantile function, generating function and Rényi entropy are derived. We proposed the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. Two real data sets are used to compare the flexibility of the new model versus other models.


Introduction
introduced twelve different forms of cumulative distribution functions for modeling lifetime data.Among those twelve distribution functions, Burr-Type X and Burr-Type XII received the maximum attention.For more detail about those two distributions seeJohnson et al. (1994).Recently, Surles and Padgett (2001) introduced two-parameter Burr Type X distribution and correctly named as the generalized Rayleigh distribution.
The procedure of expanding a family of distributions for added flexibility or to construct covariate models is a well-known technique in the literature.In many applied sciences such as medicine, engineering and finance, amongst others, modeling and analyzing lifetime data are crucial.Several lifetime distributions have been used to model such kinds of data.The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distributions.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.Merovci (2014) introduced transmuted generalized Rayleigh (TGR) distribution.In this article we present a new generalization of the TGR distribution called Exponentiated transmuted generalized Rayleigh (ETGR) distribution.The cumulative distribution function (cdf) of the TGR distribution is given by where ,  > 0, || ≤ 1 and  is a scale parameter,  is a shape parameter and  the transmuted parameter.The corresponding probability density function (pdf) is given by Recently, the   distributions (or exponentiated distributions) have been shown to have a wide domain of applicability, in particular in modeling and analysis of lifetime data.
Definition 1: Let  be an absolutely continuous cdf with support on (, ), where the interval may be unbounded, and let  be a positive real number.The random variable  has an   distributions if its cdf , denoted by, () is given by   ( ) = ( ) = ( ) , > 0, > 0.

G x F x F x x
   which is the  th power of the base line distribution function () and the corresponding pdf of X is given by The class of   distributions contains certain well-known distributions for which their cdf's have closed forms (see, e.g.Gupta and Kundu (1999, 2000, 2001, 2007) and Nadarajah (2011)).Shakil and Ahsanullah (2012) introduced some distributional properties of order statistics and record values from   distributions.
Recently, various generalizations have been introduced based on the above definition.We aim in this paper to define and study the ETGR distribution.The rest of the paper is organized as follows.In Section 2, we define the new distribution and provide some plots for its pdf.Section 3 provides some statistical properties including quantile function, random number generation, moments, generating functions, incomplete moments, mean deviasions and Rényi entropy are derived.In Section 4, the order statistics are discussed.
In Section 5, we present the reliability function (rf), hazard rate function (hrf), reversed hazard rate function (rhrf), cumulative hazard rate function (chrf), moments of the residual life and moments of the reversed residual life.The maximum likelihood estimates (MLEs) for the model parameters and the observed information matrix are provided in Section 6.In Section 7, the ETGR distribution is applied to two real data sets to illustrate its usefulness.Finally, some concluding remarks are given in Section 8.

The ETGR Distribution
The ETGR distribution and its sub-models are presented in this section.The cdf of ETGR (for >0 X ) is given by Using the series expansion The cdf of the ETGR distribution in (3) can be expressed as where where  is a scale parameter representing the characteristic life, ,  and  are shape parameters representing the different patterns of the ETGR distribution and  the transmuted parameter.The corresponding pdf of ( 4) is given by Using the series expansion the pdf in (5) can be expressed in the mixture form as where Plots of the pdf for selected parameter values are given in Figure 1.The ETGR distribution is very flexible model that approaches to different distributions when its parameters are changed.The subject distribution includes as special cases four well known probability distributions as illustrated in corollary 1.

Corollary 1 If
X is a random variable with pdf in ( 5), then we have the following seven cases.

Statistical Properties
The statistical properties of the ETGR distribution including quantile and random number generation, moments, moment generating function, incomplete moments, mean deviasions and Rényi entropy are discussed in this section.

Quantile and Random Number Generation
The quantile function (qf), say , q x of X is the real solution of the following equation ( ) = .
q F x q Then, we can write By putting = 0.5 q in Equation ( 7) gives the median of X .Simulating the ETGR random variable is straightforward.If U is a uniform variate on the unit interval (0,1) , then the random variable = q XX at = qU follows (5).

Moments
The r th moment, denoted by , ' r  of X is given by the following theorem.
Theorem 1 If X is a continuous random variable has the ETGR ( , , , , ), x Proof: where By substituting from Equation (10) into Equation (9), we obtain Therefore, the first and second moments of the ETGR random variable can be obtained by setting = 1,2 r respectively, in Equation ( 8) as follows Then we can get the variance by the relation Based on the above Theorem (1) the coefficient of variation, coefficient of skewness and coefficient of kurtosis of the ETGR ( , , , , ) distribution can be obtained according to the well-known relations.

Corollary 2 Using the relation between the central moments and non-centeral moments,
we can obtain the th n central moment, denoted by , n M of a ETGR random variable as follows

Generating Function
The moment generating function ( mgf ) of the ETGR distribution is given by the following theorem.

Theorem 2 If
X is a continuous random variable has the ETGR ( , , , , ), x By substituting from Equation ( 8) into Equation ( 11), we obtain   X Mt , which completes the proof.The measure of central tendency, measure of dispersion, coefficient of variation, coefficient of skewness and coefficient of kurtosis of X can be obtained according to the above relation in Theorem 2.

Incomplete Moments
The main application of the first incomplete moment refers to the Bonferroni and Lorenz curves.These curves are very useful in economics, reliability, demography, insurance and medicine.The answers to many important questions in economics require more than justknowing the mean of the distribution, but its shape as well.This is obvious not only in the study of econometrics but in other areas as well.The th incomplete moments, denoted by  , Using Equation ( 6) and the lower incomplete gamma function, we obtain Another application of the first incomplete moment is related to the mean residual life and the mean waiting time given by       The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median.The mean deviations about the mean and about the median  is the first incomplete moments that comes from ( 12) by setting =1 s and = M is the median of X .

Rényi and q-Entropies
Entropy refers to the amount of uncertainty associated with a random variable.The Rényi entropy has numerous applications in information theoretic learning, statistics (e.g.classification, distribution identification problems, and statistical inference), computer science (e.g.average case analysis for random databases, pattern recognition, and image matching) and econometrics, see Källberg et al. (2014).The Rényi entropy of a random variable X represents a measure of variation of the uncertainty.The Rényi entropy is defined by Therefore, using Equation ( 6), the Rényi entropy of the random variable X is given by The q-entropy, say () q HX, is defined by The pdf of the j th order statistics for a ETGR distribution is given by where  and the pdf of the smallest order statistics   The joint pdf of where Then the minimum and maximum joint probability density of the ETGR distribution, denoted by  

Reliability Analysis
In this section we introduce the reliability function, the hazard rate function, the cumulative hazard rate function, reversed hazard rate, moments of the residual life and moments of the reversed residual life for the ETGR ( , , , , ). x    

The Reliability, Hazard Rate, Reversed Hazard Rate and Cumulative Hazard Rate Functions
The rf also known as the survival function, which is the probability of an item not failing prior to some time , t is defined by ( ) = 1 ( ).

R x F x 
The rf of the ETGR distribution, say ( , , , , , ), The other characteristic of interest of a random variable is the hrf.The hrf of the ETGR distribution also known as instantaneous failure rate, say ( ), hx is an important quantity characterizing life phenomenon.It can be loosely interpreted as the conditional probability of failure, given it has survived to the time t .The hrf of the ETGR distribution is defined by ( , , , , , ) = ( , , , , , ) / ( , , , , , ) , , , , = .
It is important to note that the units for () hx is the probability of failure per unit of time, or cycles.These failure rates are defined with different choices of parameters.
Plots of the hazard rate function of ETGR for selected parameter values are provided in Figure 2.

Moments of the Residual Life
Several functions are defined related to the residual life.The failure rate function, mean residual life function and the left censored mean function, also called vitality function.It is well known that these three functions uniquely determine () Fx (see Gupta (1975), Kotz  and Shanbhag (1980) where   Another interesting function is the mean residual life function (MRL), defined by  and it represents the expected additional life length for a unit which is alive at age x .The MRL of the ETGR distribution can be obtained by setting =1 k in the above equation.Guess and Proschan (1988) gave an extensive coverage of possible applications of the mean residual life.The MRL has many applications in survival analysis in biomedical sciences, life insurance, maintenance and product quality control, economics and social studies, Demography and product technology (see Lai and Xie (2006)).

Moments of the Reversed Residual Life
The ℎ moments of the reversed residual life, denoted by Then, The k th moments of the reversed residual life of X is given by The mean waiting time (MWT) also known as mean reversed residual life function, defined by  and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in   0, x .The MRRL of the ETGR distribution can be obtained by setting =1 k in the last equation.

Estimation and Inference
The maximum likelihood estimators (MLEs) for the parameters of the ETGR distribution is discussed in this section.Consider the random sample 12 , ,..., n X X X of size n from this distribution with unknown parameter vector  = (, , , )  .Then, the log-likelihood function, say ℓ = ln ℓ(), becomes ℓ = (ln 2 + ln  + ln  + 2 ln ) + ∑ ln   − ∑ (  ) 2 Equation ( 13) can be maximized either directly by using the R (optim function), SAS (PROC NLMIXED), Ox program (sub-routine MaxBFGS) or by solving the nonlinear likelihood equations obtained by differentiating (13).Therefore, the score vector is given by () = .
The maximum likelihood estimator  ̂= ( ̂,  ̂,  ̂,  ̂) of = ( , , , )       is obtained by solving the nonlinear system of equations ( 14) through (17).These equations cannot be solved analytically and statistical software can be used to solve them numerically by means of iterative techniques such as the Newton-Raphson algorithm.For the four parameters ETGR distribution all the second order derivatives exist.
For interval estimation of the model parameters, we require the 44  observed information matrix, whose elements are derived in appendix A,      = for , = , , , .

Applications
In this section we provide two applications of the ETGR distribution to two real data sets.The first data set, strength data, which were originally reported by Badar and Priest (1982) and it represents the strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows.Single fibers were tested under tension at gauge lengths of 10 mm (data set 1) and 20 mm (data set 2), with sample sizes n = 63 and m = 74 respectively.The data are presented below.Several authors analyzed these data sets.Surles andPadgett (1998, 2001), Raqab and Kundu (2005) observed that the generalized Rayleigh distribution works quite well for these strength data.Kundu and Gupta (2006) analyzed these data sets using two-parameter Weibull distribution after subtracting 0.75 from both these data sets.After subtracting 0.75 from all the points of these data sets, Kundu and Gupta (2006) fitted Weibull distribution to both these data sets with equal shape parameters.These two data sets also studied by Rao (2014) to estimation of reliability in multicomponent stressstrength based on generalized.Here I would like to mention that the exact number of (data set 2) is 74 instead of 69, which mentioned in Kundu and Gupta (2009).Here we used these data to compare the proposed ETGR model with TGR, GR and R distributions.
The first data set (gauge lengths of 10 mm) from Kundu and Raqab (2009) where k is the number of parameters and n is the sample size.

Figure 1 :
Figure 1: Plots of the density function for some parameter values we get the transmuted Rayleigh distribution, TR( ,, x  ).

Figure 2 :
Figure 2: Plots of the hrf for some parameter values upper incomplete gamma function.
to construct approximate confidence intervals for the model parameters.Here, ) , ,  and  can be determined as:  th percentile of the standard normal distribution.

Table 1
lists the MLEs of the model parameters for ETGR, TGR, GR and R distributions, the corresponding standard errors are given in parentheses.In this table we shall compare Pak.j.

Table 2
lists the MLEs of the model parameters for ETGR, TGR, GR and R distributions, the corresponding standard errors are given in parentheses.In this table we shall compare our new model with other sub models.

Table 2 : MLEs under the considered models and corresponding
Tables1 and 2compare the ETGR model with the TGR, GR, and Rayleigh distributions.We note that the ETGR model gives the lowest values for the , AIC BIC and CAIC statistics among all fitted models.So, we conclude that the ETGR distribution provides a Pak.j.

stat.oper.res. Vol.XI No.1 2015 pp115-134 134 Appendix A
The elements of the observed information matrix are given by