I.    CONCLUSIONS

Rayleigh distribution generalized by TL family This new distribution named as TLR It is a positive skew distribution. Some statistical properties of the distribution has been found such as r-th moment around zero, mode, quantile and moment generating functions. The parameters of distribution estimated by Bayesian technique under Sq and E linex loss functions The results of estimation applied in simulated and real data sets, The important conclusion in simulated data set that the Elinex are best estimators when the mean of shape parameter of E linex function less than zero and scale parameter of TLR less than 2.5. Also the estimators under Sq  is the best estimator when µ>0 except the case of scale parameter of TLR equal to 2.5 In real data set it is concluded that the Elinex estimator when the location parameter µ equal -2 is the best.

1. Introduction

Probability distributions have received agreat attention in order to achieve flexibility in modeling the studied phenomena.Well- known probability distributions lacks flexibility in modeling convoluted data for complex phenomena, which has made the increasing interest of many researchers to generalize known probability distributions to become more flexible in application than the known distributions.

On this basis, various generalizations were made for the distributions, including those based on the Beta distribution or simpler probability distributions than a Beta distribution and their random variables defined within the interval (0,1). Eugene et al (2002) presented  Beta distribution family using the cumulative distribution function (c.d.f) of a random variable w  Beta(a,b) which is defined as : 

        dw                                                                                                                (1)

if (x) is replaced by the cumulative  function G (x) of any basic distribution, the family of the generalized

   dw                                                                                                             (2)                               

where is the B(a,b) is the complete Beta function.

Famoy et al (2005) used this generalization on many basic distributions, such as the Weibull distribution and Hanook et al (2013) generalized inverse Weibull distribution using the beta generalization.

Suggestions for  probability distributions are simpler than the Beta distribution are defined within  closed interval [0,1] such as the kumaraswamy distribution and the Topp-Leone distribution. Cordiero and Castro (2011) introduced  the kumaraswamy distributions family  as  the cumulative function of this family which  takes the following form

Nadarajah and Kotz (2013)  introduced Topp-Leone distribution, which is one of the simplest probability distributions that  statisticians used as a alternative to the Beta distribution, as its cumulative  function takes the following form

    Where  0 < x < 1, α > 0                                                                                                (3) 

Shomrani et al (2016) used the c.d.f. defined in eq. (3) above to present a new generalization as an alternative for  the Beta generalization called the Topp - Leone family of distributions, as the cumulative function for this generalization takes the following form

 -∞ < x < ∞,  α > 0                                                                                     (4)

Shomrani et al (2016) gave an example of using the generalization defined in equation (4) on the exponential distribution. Al - Saiary and Bakoban (2020) used this generalization on the generalized inverted exponential distribution of the baseline distribution and the distribution properties and  estimations were studied according to maximum likelihood.

Rayleigh distribution is one of the probability distributions that belong's to the distributions of survival times and has wide applications in survival analysis, reliability theory, communications and engineering. This distribution has a linear increasing hazard function with time. Therefore it was also used in modeling the survival times of patients with incurable diseases whose degree of  risk increases over time. to avoid  restricting the use of this distribution various  generalization  have been made on  it on  this basis. In this research, the Topp-Leone generalization was used on this distribution and its statistical properties with estimation of its parameters in Bayesian technique  under expected linex (E.linex) and Square quadratic loss functions were studied.

The research is divided into six parts, the first part dealt with the introduction, the second part dealt with the  Topp - Leone Rayleigh distribution,  the third part discusses  the properties of the distribution, and the fourth part includes estimating the parameters of the distribution in the Bayesian style, and the fifth part dealt with the practical side, as the theoretical results were applied to the data of the time when an injury which  occurred to dialysis patients by months.

I.    TOPP-LEONE RAYLEIGH  DISTRIBUTION

If the basic distribution of the random variable x is a Rayleigh distribution with the scale parameter  θ  where the  c.d.f. and the p.d.f.  are as follows:

1-        x>0  , θ> 0                                                                                                                    (5)

      x,θ >0                                                                                                                        (6) 

Using eq. (no. 5 , 6) in eq.(no.4), we get the c.d.f. of the Topp-Leone Rayleigh Distribution (TLR) as follows:

      x , α , θ > 0                                                                                                         (7)

By taking the first derivative of both sides of the eq. (no.7 ) with respect to x, the p.d.f.  of x is

                                                                               (8)

This distribution  is symbolized as an acronym   x ~  T LR  (α ,θ)

The survival and hazard functions of the TLR distribution are

                                                                                                                        (9)

                                                                                                              (10)

f(x),F(x), s(x), h(x)  have  been  plotted when

α=(1.5,2.5) and θ =(0.5,1,1.5,2.5)

Fig. 1. Plotting f (x),F(x), s (x), h (x) of the TLR distribution at different values of its parameters.

We notice from Figure (1) above that the skewed distribution is positive, and the degree of skewness increases with the increase in the value of the shape parameter ( ). As for the hazard function, it takes the  (J) shape and its intensity increases.

II.   PROPERTIES OF DISTRIBUTION

In this section, some properties of the distribution will be presented which are moments, mode, median, skewness, kurtosis , and  quantile function as  follows

• r-order moment about zero

The r- th moment about zero for the random variable x is

Ε                                                                                     (11)

Using the binomial expansion over the expression and  making some mathematical simplifications, we get

Εx^r=                                                                                                               (12)                                    

• Mode

The mode is the solution of the derivative eq. (no.8) above with respect to x

 =0                                                                                                                        (13)

• The median and the quantile function

The median and the quantile function are found by solving the equation             and the inverse function F(x) respectively  to get                                  

    where   0<q< 1                                                                                                          (14)

• Moment Generating Function

The moment generating function of  x  is:

µx(t)=Ε

µ_x(t) = 4αθ                                                                                     (15)

• Kurtosis and skewness

The skewness and  Kurtosis are   found  mathematically  using the moments about zero by solving  the following formulas, respectively (sarhan and zaindin . 2000).

Skewness=                                                                                                                               (16)

Kurtosis=                                                                                                                                     (17)

III. BAYESIAN ESTIMATION

In this section, the two parameters  of the TLR distribution are estimated when the two parameters, using the in formative prior, and assuming that these parameters  are independent, then the common initial distribution is

We choose the prior distribution for α and θ, which are

α Gamma( )  and    θ

By providing  n of the  sample observations, the likelihood  function is as follows

                                                                                     (18)

And joint  posterior distribution for α and θ  is

                                                              (19)

Where as       

We notice from the common posterior distribution defined in eq. (no.19) above that it is difficult to find the posterior marginal distribution for each parameter. Therefore, Laplace approximation was used. Azevedo, F.A.et al (1994)

The approximate posterior – distribution  of   is as follows

                                                                                                                         (20) 

It is the kernel of N2( ) where  represents ( ) the posterior mode and W=-

Under the Sq loss function, the  approximate Bayes estimator  of the  parameter vector  is  and the quadratic  risk function is tr

As for estimating the parameters under the linex  loss function, we need to find the approximate posterior marginal distribution for each parameter and describe them

Where  represents the element in position (1,1) of the matrix    and  represent the element in position (2,2) of the matrix  

Under the linex loss function, the estimates for α and θ are as follows

                                                                                                                                           (21)          

We find an important problem in the Bayesian estimation using the linex loss function, which is to choose a value for the parameter b, where the negative value of b provides a greater weight under the estimate, and when a large positive quantity, this will provide a greater weight for the above estimate. In order to make a balance in the weights between above and below estimation, Nassar,M.et al.(2022) suggested that the shape parameter should be a random variable, and in this case the balance between above and below estimation will be achieved at all possible values of the random variable They suggested three distributions, namely the generalized logistic distribution of type I, the normal  and the standard Gumbel. In this paper, we choose a probability distribution for the shape parameter b described by   and the probability density function b is

                                                                                                                                      (22)

Using the above distribution, the Elinex estimate is above

                                                                                                                           (23) 

                                                                                                                          (24)            

Using the distribution defined in eq.(no.22) in eq. (no.23,24) we get

                                                                                                                                  (25)

                                                                                                                                  (26)

We also need the expected value of the linear exponential loss function    and it is found as follows

                                  (27)

                          (28)

= - µ       ]                                                      (29)

Using the Maclaurian series on      up to the first order and making simplifications and making integrations, we get

µ                      

                                                                                                                                                               (30)

In the same way, the risk function for θ is

 -µ                

            (31)

IV. SIMULATION STUDY

In this section in order to investigate and compare the performance of different Bayesian estimators using the E linex and Sq loss functions.

The data generated from TLR with sample sizes (n=25,50,100) and ( α=1.5) (θ=0.5,2.5) The values of  hyperparameters for the priors of  α,θ are ( Also we consider the values of shape parameter of linex loss function are (µ=-2, -1, 0.5) Bayesian estimators under E linex   and Sq loss function for parameter of TLR and MSE of estimators has been evaluated and put the results in Tables (1-3) below.

TABLE 1. Bayesian estimators for α and θ under sq and Elinex loss functions at α=1.5, µ= -2 and MSE

TABLE 2. Bayesian estimators for α and θ under sq and Elinex loss functions at α=1.5, µ= --1 and MSE

It is seen from tables (1,2) above The E linex estimators is better than Sq estimators except for the case of θ=2.5 and all sample sizes

TABLE 3. Bayesian estimators for α and θ under sq and Elinex loss functions at α=1.5, µ= 0.5 and MSE

It is seen from tables (3) above that Bayesian  estimators under Sq   is better E linex  Except for the case of θ=2.5, n=25

V.   REAL DATA ANALYSIS

In this section the data set represent the time of infection of  kidney dialysis patients in month taken from Klein and Moesch erger (2006)The goodness of fit kolmogrov smirnov (K.S.)and Chi-square goodness of fits statistic is used to fit real data set we put  the  hypothesis H0:Data follow TLR  r.s  H1: :Data follow Doesnt TLR

The results of test statistics in table (4) below

TABLE 4. K.S. and Chi-square tests of fitting data by TLR distribution

From table (4), it is concluded the null  hypothesis  is accepted Bayesian estimators for parameters of TLR under Sq and the E linex which defined in eq. (no.30) three different values for the mean of shape parameter(b) of linex loss function are (µ=-2, -1,0.5 ) The  results are in table (5) below

TABLE 5. Bayesian estimators under linex and Sq loss function and Bayesian risks for time of infection data in months

From table above the best estimators for parameters of TLR are

  ,  with