A New Family Of Distributions: Exponential Power-X Family Of Distributions And Its Some Properties

In this study, we have introduced a new family of distributions called Exponential Power-X family of distributions using the method suggested by Alzaatreh et al. (2013).  A special model called Exponential Power-Weibull (EP-W) is examined to illustrate the applicability of this new family of distributions. This new model can be used for skewed data and to model the data having increasing and decreasing  hazard rate. Also, some statistical properties of EP-W model are obtained such as, moments, moment generating function, order statistics. Further, the maximum likelihood estimators (MLE) of unknown parameters are derived. An extensive  Monte-Cario simulation study has been carried out to examine the performance of this estimator  in  terms of mean square error and bias .According to simulation study results, it has been observed that the estimators of model parameters  provide estimation procedures. It has been compared the fits with EP-W model and other models via a real data application .According to real data analysis results, EP-W model is the best fitting model in real data analysis . These situation indicated the applicability of the EP-W model in real life.

1. Introduction:

    In recent years, many new statistical distributions are suggested to modelling lifetime and various dataset. These models are obtained using diverse methods. Some of these studies are listed as follows. Eugene et al. (2002) introduced a family of distributions generated by Beta distributions. Jones (2009), Cordeiro and Castro (2011) introduced the family of distributions generated by the Kumaraswamy distribution. The motivation of this study is method suggested by Alzaatreh et al. (2013). this method can be defined as followes .

  Suppose , is probability density function of  continuous random variable, G(x) is cumutative distribution function (cdf) of any X random variable  and  , is a function that provide  the following properties.

a- 

b-   is a derivable and monotone non-decreasing function.

c- While  , it is   and  while , it is

In this case the family of new distributions is defined as follows

                                                                                 (1)

where  denotes the  cdf of random variable . New distributions obtained using this method are called  as  distributions family. (Alzaatreh et al. 2013)  has introduced  Beta-Exponential-X and Weibull-X  families of distributions. Then, many researchers have found new statistical distributions using this  method. Some of these studies can be listed as follows; (Alzaghal et al. 2013) have suggested Exponentiated T-X Family of distribution, (Tahir et al. 2015) has introduced the odd generalized exponential family of distribution, The logistic-X family of distributions is proposed by (Tahir et al. 2016a) and A New Weibull family of distributions has been generated by  (Tahir et al. 2016b). (Çelik and Guloksuz, 2017) have introduced a new lifetime distribution called “Uniform-Exponential Distribution” using  in (1).

We  introduce  a  new  family  of  distributions  called “Exponential  Power-X family of distributions” using the method  suggested  by  Alzaatreh et al. (2013). This  study  is organized  as  follows. In section 2, exponential power distribution is examined. In section 3, exponential  power-X  family of  distributions  are  introduced. Also a special model of this new family of  distributions  named a EP-W distribution and its statistical properties are examined. Then, in estimation  section, the  unknown  parameters of EP-W distribution  are estimated by maximum likelihood method. In section 7,  the MSE and biases of this estimator are computed  by means of a Monte-Carlo simulation study. In section 8, a real data application is presented to show  whether the real data set can be modelled by EP-W distribution. Finally, concluding remarks are given.

2. Exponential Power Distribution

     EP distribution is introduced by (Smith and Bain, 1975). is used  to  modelling  lifetime data. The cdf,  pdf  and  hazard  function of a X random variable having to  this distribution  with and    parameters are given below.

                                                                                                            (2)

                                                                       (3)

                                                                                                             (4)

, respectively.

3. Exponential Power-X Family of Distribution

In this section, we ıntroduce exponential power T-X family of distribution. This  new family of distribution is obtained by using and  in Eq(1) then . this new family can be  written as follows.

                                           (5)

Where  is pdf of T  random variable having to EP distribution.  denotes cdf of any distribution. .

4. Special Model: Exponential Power-Weibull Distribution

In this part, it is introduced a special model of EP-X family called Exponential Power-Weibull (EP-W) distribution. This model is obtained by taken  in  Eq .(5). Where  is  cdf  of weibull distribution. The cdf and pdf of EP-W distribution  are given by  (6) and (7) respectively. 

.

                                                                                                                 (6)                                                                                                     

                                              (7)

 Here  and

Figure 1. The cdf of the EP-W distribution

Figure 2. The pdf of the EP-W distribution

5. Some Statistical Properties for EP-W distribution
   • Hazard Function

The hazard function of EP-W distribution with  and    is given by

                                                                        (8) 

Figure 3. The hazard function of the EP-W distribution

In order to examine the behavior of the hazard function regarding the EP-W distribution, the derivative of the hazard function is needed. The derivative of the hazard function denoted by is given below:

                                                        (9)

for , >0 Since the hazard function is  is said to be increased .

when taken    for  and for

The hazard regime for EP-W status is bathtub-shaped. Both cases are shown in Figure 3. seen from single graphs.

• Random Numer Generator for EP-W Distribution

To generate random numers from  EP-W distribution with  and   parameters it is  used the   method  of  inversion transformation as

                                                                         (10)                     

Solution of Eq. (9) is given by

                                                                                                    (11)

where, u is a uniform random variable on the unit interval (0,1).

• Moments of EP-W distribution

the  moment  for EP-W distribution with  and  parameters  is given by

                                                                                                               (12)

                            
                                                                                                            (13)                 from the equatıon (13) We  have computed  some statistical properties such as   moment   coefficient  of skewness and  excess kurtosis for  different values of  parameters of  EP-W  distribution  . this  calculations are given in table 1 .

                                                          (14)

                     (15)

Using equations (13), (14) and (15), the moment, variance, skewness and kurtosis coefficients for different parameter values ​​of the EP-W distribution are given in Table 1 Graphs regarding the skewness and kurtosis coefficients are presented in Figure 4.

Table 1.epresents an account r-moment ,varians skewness and kurtosis for different values    of  parameters.

          Figure 4.5. The plots of coefficient of Skewness and Kurtorsis for EP-Wdistribution .

Moment Generating Function  for EP-W distribution

The moment generating function  of random variable X is given by

                                                                                                        (16)                     

• Order Statistics for EP-W distribution

Let  is a iid  random sample from EP-W  distribution with cdf  and pdf given by (6) and (7) respectively. Let  denote the order statistics obtained from this sample. The pdf of   the  order statistic for  is simply as and it is given by;

    (17)                                                                            

where  and  and are cdf and pdf of the EP –W distribution respectively and B(..) is the beta function. By using the binomial series expansion the order statistic function can expressed as follows;

           (18)

6. Parameter Estimation

We consider estimation of unknown parameters of exponential power weibull (EP-W) distribution   are  derived by using  the maximum likelihood based on random sample.

• Maximum Likelihood Estimation

Let  be be a random sample with size n  from   EP-W  distribution.  where  is a  vector parameters. The log-likelihood function is given by;

                                          (19)

By diferentiating partially the log-likelihood function according to  and  parameters, and then  equalizing them to zero we get.

                                         (20)

                                                      (21)    

                                                  (22)

MLE of   and  parameters are obtaind  by simultaneous solutions of the equations (20)-(22) these nonlinear equations can be solved using Newton  Raphson method .  

7. Simulation Study for EP-W distribution

In this part, a Monte-Carlo simulation study based on 10000 replications for different sample sizes such as 25,50,100,200,500 and for different parameter values such as (1.6,0.9,0.7) ,(0.5,0.6.0.4) ,(0.7,0.6.1.2), (0.3,1.3,2)  is performed to see The  performances of  of unknown parameters of  EP-W   distribution in terms of the bias and mean square  error (MSE) . the simulation results are given in table 2.

Table 2. Bias and MSE for  various values of parameters

8. Real Data Analysis

   In this section, A real data application is performed  to examine the fit of the EP-W model in real life and  to compare with other distributions. Real data set was used for these purposes. In order to compare the fits of the distributions, it has been considered the Akaike's Information Criterion (AIC), corrected Akaike's Information Criterion (AICc), the Bayesian Information Criterion (BIC) and -2×log-likelihood value by using these distributions .  for three data sets. These measures are given by

                                                                                                                            (23)

                                                                                                       (24)

                                                                                                                (25)

where k is  number of parameters, n is sample size.  is the value of  log–likelihood function.

The data set for total milk production rates for 107 beef living in the Camauba farm of Brazil Was used. for real data analysis Yousof et.get  (2017), Cordeiro and Brito. (2012 ) Brito  (2009). data set are given in Table3 ;

           Tablo3. Real Data set

The MLE(s) of the unknown parameters  and standart errors for  models are given in Table 4.

                      Tablo 4. Parameter estimaters (standart errors)

Tablo 5. Selection criteria statistics for data set1

Figure 4. Empirical cdf and theoretical cdf.

                      Table6. Real Data set

0.0251    0.0886   0.0891    0.2501      0.3113    0.3451     0.4763     0.5650   0.5671     0.6566 0.6748   0.6751    0.6753     0.7696     0.8375     0.8391     0.8425     0.8645   0.8851     0.9113 0.9120   0.9836    1.0483     1.0596     1.0773     1.1733     1.2570     1.2766   1.2985     1.3211 1.3503   1.3551    1.4595     1.4880     1.5728     1.5733     1.7083     1.7263   1.7460

Tablo 7. Parameter estimaters (standart errors)

Tablo 8. Selection criteria statistics for data set1

                                          Figure 4. Empirical cdf and theoretical cd.