Exponential Power-Chen Distribution and Its Some Properties

In this paper, we have introduced a new lifetime distribution named as Exponential Power Chen (EPCh) distribution by using method of Alzaatreh et al. [2]. Some statistical properties of EPCh distribution such as moments, moment generating function, order statistics, mean remaining life, Renyi and Shannon entropies are obtained. Furthermore, shapes of pdf, cdf, hf and ihf for this distribution are examined. From these shapes, it is concluded that EPCh distribution can be used to model the data having increasing, decreasing and bathtube shaped hazard rates. Further, the maximum likelihood estimators (MLE) for unknown parameters of EPCh distribution are derived. An Monte-Cario simulation study has been carried out to examine the performance of this estimators in terms of mean square error and bias. To illustrate the applicability of this new distribution, EPCh distribution for two real data sets are compared  with some known distributions  such  as Exponentiated  exponential, Weibull and Chen distribution using some goodness of fit measures. According to real data analysis results obtained from both data sets, EPCh distribution has the best fitting among compared distributions. This demonstrates the applicability of the EP-Ch distribution in real life

A data set can show fitting to many distributions. It is important for statistical inference to determine statistical distribution that best fits to a data set. In recent years, many new statistical distributions have been suggested to modeling real data sets. These new distributions have better fit than current distributions for some real data sets. In literature, many methods have been developed to obtain new continuous distributions. Eugene et al. [9] introduced a family of distributions generated by beta distributions. Cordeiro and Castro [6] introduced the family of distributions generated by the Kumaraswamy distribution. Nadarajah and Kotz [15,16] suggested The beta gumbel and the beta exponential distributions.  Akinsete et.al. [1] introduced The beta-pareto distribution. Alzaatreh et al. [2] introduced the new class of distributions by extend method of Eugene et al. [8]. The motivation of this study is method suggested by Alzaatreh et al. [2]. This method can be defined as follows. Suppose  and  is probability density function (pdf) and cumulative distribution function (cdf) of a continuous  random variable, respectively. Let G(x) is cumulative distribution function (cdf) of any random variable X  and   is a function that has the following properties.

1.  
2. is a differantiable and monotone non-decreasing function.
3. When , and while , .

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

                                                                                                         (1)

New distributions obtained by using this method are called as  distributions family. (Alzaatreh et al. [2]). Recently, many researchers have found new statistical distributions using this  method. Some of these studies can be included as follows: Alzaatreh et al. [2,3]  introduced Beta-Exponential-X distribution, Weibull-Pareto distribution and Weibull-X families of distributions.  Alzaghal et al. [4] suggested Exponentiated T-X Family of distributions, Tahir et al. [22,23,24] introduced the odd generalized exponential family of distributions, The logistic-X family of distributions and A new Weibull family of distributions. Çelik and Guloksuz [8]  suggested a new lifetime distribution called as Uniform-Exponential Distribution.

The main purpose of this study is to introduce a new statiscal distribution with four parameters by using method of Alzaatreh et.al. [2] and this new distribution is called as Exponential Power Chen (EPCh) distribution. The rest of this paper is organized as follows. In section 2, information is given about Exponential power and Chen distributions. In section 3, EPCh distribution with parameters  have been introduced. In section 4, the some statistical properties such as hazard function, random number generator, moment generating function, moments, variance, skewness and kurtosis coefficients, renyi and shannon enropies for this new distribution  are presented. In section 5, maximum likelihood (ML) estimators for parameters of  EPCh distribution are obtained. In section 6, a simulation study to see  the performances of this estimators in terms of mean square errors (MSEs) and biases is performed. In section 7, a real data analysis is presented. Finally, conclusion is given in section 8.

1. Exponential Power (EP) and Chen Distributions

EP distribution introduced by Smith and Bain [21] is used  to  modeling  lifetime data. The cdf,  pdf  and  hazard  function (hf) of a random variable X having EP distribution  with   parameters can be written in order as follows :

                                                                                                                 (2)

                                                                           (3)

                                                                                                                   (4)

Another distribution used to model  lifetime data is Chen distribution suggested by Chen [7]. The cdf,  pdf  and  hf of  a random variable X having Chen distribution  with   parameters are given, respectively, by

                                                                                                         (5)

                                                                                   (6)

                                                                                                                          (7)

3. Exponential Power-Chen Distribution

This new distribution is obtained by using method of Alzaatreh et al. [2]. Suppose that  in Eq. (1.1) is defined as follows:

                                                                                                             (8)

where  is defined in Eq. (5). If it is used pdf of EP distribution defined in Eq. (3) instead of  and , a new distribution called as EP-Ch distribution with parameters is obtained. Cdf, pdf, hf, inverse hazard functions (ihf) and rf of  distribution with are given as follows.

                                    (9)

                  (10)

                              (11)

                                         (12)

                                                        (13)

The plots of df, pdf, hf and ihf for the various parameter values ​​of the distribution are given in the following order: Figure 1, Figure 2, Figure 3 and Figure 4.

Figure 1. Df plots of EP-Ch distribution for different parameter values

Figure 2. pdf plots of EP-Ch distribution for different parameter values

Figure 3. hf plots of EP-Ch distribution for different parameter values

Figure 4. ihf function plots of EP-Ch distribution for different parameter values

4. Some Statistical Properties for EP-Ch distribution

4.1. Random Numer Generator for EP-Ch Distribution

The  method  of  inversion transformation has been used to generate random numbers from  distribution as following

                   (14)

Where u is defined on the unit interval (0,1). When u = 0.5 in Eq (14) the median for EPCh distribution is obtained. In this case, the median can be written as follows;

                                 (15)

4.2.Moments for EP-Ch distribution

 The  moment of a random variable X having EPCh distribution  with  parameter is obtained as follows;

                                                   (16)

Where y can be written as  follows:

From the equation (16), as a result of  transformation,  moment is obtained as follows.

                                                                                 (17)

By using the equation (17), the coefficients of skewness (CS) and kurtosis (CK) can be computed using the following formulas;

                                                                       (18)

                                  (19)

For different parameter values ​​of EP-Ch distribution, the  moment, variance, skewness and kurtosis coefficients are given in Table 1.

Table1.  moment, variance skewness and kurtosis values for EP-Ch distribution

The plots of coefficients of skewness and kurtosis are given in Figure 5 and Figure 6.

Figure 5. The plots of coefficient of Skewness for EPCh distribution

                       Figure 6. The plots of coefficient of Kurtosis for EPCh distribution

4.3.Moment Generating Function

The moment-generating function (mgf) of a random variable X having EP-Ch  distribution, , is obtained as follows.

                                                                   (20)

4.4. Order Statistics for EP-Ch Distribution

Let  be a random sample taken from  distribution. Let  indicate the order statistics obtained from this sample. The pdf of the  order statistic for  is shown as  and it is given by;

                    (21)

Where , B(..) is the beta function.  and  are cdf and pdf of the EPCh distribution, respectively.

4.5. Mean Remaining Life

The mean remaining life function, , defined as the expected value of the remaining lifetime after a fixed time t for a continuous random variable T with a life function, , is stated as follows:

                                                 (22)

Guess and Prosehan [11]. The other formula for  is obtained by the help of Tonelli's theorem [20,26] and is given as follows:

                                                                                     (23)                  

From equation (22), the mean remaining life function for the EP-Ch distribution is given by

                                                                      (24)

Where k can be written as follows:

As a result of applying the transformation  in the Eq. (4.11),  is obtained as follows.

                                                                                                (25)

where .  According to Bryson and Siddique [5] and Ghitany et al.[10], if hazard function of a  non-negative  continuous  random  variable  is decreasing (increasing), then mean remaining life function of is increasing (decreasing). The values of  for t = 1,2,3  and  the different parameter values ​​of EPCh distribution are given in Table 2 and its plot is given by Figure 7.

Lemma 1: Let X be  a  non-negative  continuous  random  variable with  HR function  h(x)

and  mean  residual  life  function  μ(x)  If  h(x)  is  decreasing  (increasing),  then  μ(x)  is

increasing (decreasing) (see Bryson and Siddique, 1969; Ghitany et al., 2011)

Lemma 1: Let X be  a  non-negative  continuous  random  variable with  HR function  h(x)

and  mean  residual  life  function  μ(x)  If  h(x)  is  decreasing  (increasing),  then  μ(x)  is

increasing (decreasing) (see Bryson and Siddique, 1969; Ghitany et al., 2011).

Table 2. values ​​for t = 1,2,3  and  the different parameter values ​​of EPCh distribution

Using Lemma 1 and Lemma 2, we note that:

a  Since h(x) is decreasing for

< 1, then μ(x) is increasing by Lemma 1

             Figure 7. Plots of the mean remaining life function for EP-Ch distribution                  

4.5. Measures of Uncertainty for EPCh distribution

In this section, Renyi entropy (R'enyi [18] ) and Shannon entropy (Shannon [20] ) are presented for EPCh distribution. A larger entropy value indicates a higher level of uncertainty in the data.

4.5.1. R´enyi Entropy and Shannon Entropy

R´enyi entropy (R´enyi,[18])  is an extension of Shannon entropy and has been used in many fields such as physics, engineering, and economics. The Rényi entropy for any distribution is defined as follows:

                                     (26)

The Rényi entropy for EPCh  distribution is given by

                           (27)

Where k can be written as  follows: 

.

Renyi entropy values for various parameter values of  distribution is given in Table 3.

Table 3. R´enyi entropy for some selected parameter values ​​of EPCh distribution

Figure 8. Plot of the R´enyi entropy is concave for different values of .

As seen from Figure7, Renyi entropy for EPCh distribution is a concave monotonically decreasing function. At the large values, ​​ the Rényi entropy is small.

R´enyi entropy tends to Shannon entropy for [17]. The Shannon entropy is described as follows:

                                                                                   (28)

The Shannon entropy for the EPCh distribution is obtained as follows.                                (29)

where                           

the Shannon  entropy values for different parameter values ​​of the EP-Ch distribution are given in Table 4. 

Table 4. Plots of Shannon entropy for different parameter values ​​of EP-Ch distribution

6. Maximum Likelihood Estimation

Let  be a random sample with size n taken from  distribution.  The log-likelihood function is given as follows.

                             (30)

where . Derivatives according to unknown parameters of the log-likelihood function are as follows:

                                                                 (31)

                     (32)                               

        (33)

                                                     (34)

MLEs of  parameters are obtained by the simultaneous solutions of the equations (31) - (34). These nonlinear equations can be solved using iterative methods.        

6. Simulation Study

In this section, a simulation study based on 5000 replications to investigate the performances of MLEs of the unknown parameters in terms of bias and mean squared error (MSE) for  distribution for different sample sizes n =100,150,200,300,500 and for different parameter values such as  (0.5,1.4,0.2,0.5), (0.2,0.8,0.6,0.5), (0.3,0.9,0.6,0.4), (0.2,1.5,0.4,0.2) and (0.3, 2,0.5,0.9) is performed. The simulation results are given in Table 5.

Table 5. Bias and MSE for  various values of parameters

7. Real Data Analysis

In this  section, two real data analysis are considered to illustrate that the EPCh  distribution can be better than known distributions  such  as Exponentiated  exponential, Weibull and Chen distribution. For this aim, EPCh distribution are compared  with above distributions using goodness of fit measures such as the Akaike's  Information  Criterion (AIC), corrected Akaike's  Information Criterion  (AICc), the Bayesian Information Criterion (BIC) and -2×log-likelihood value.  These measures are given by

                                                                                                              (35)

                                                                                                         (36)

                                                                                                                 (37)

where k is a number of parameters, n is sample size and  is the value of  log–likelihood function. The first data set which shows failure times of components is the real data set taken from book of Murthy et al [14] are given in Table 6.            

           Table6.  Real Data set based on failure times (Data Set 1)

0.0014    0.0623    1.3826    2.0130   2.5274    2.8221   3.1544    4.9835   5.5462   5.8196    5.8714    7.4710    7.5080   7.6667    8.6122   9.0442    9.1153   9.6477  10.1547  10.7582 

The second data set which states graft survival times in months of 148 renal transplant patients was obtained by Henderson and Milner [13] and was included in the book of Hand et al. [12].

Table 7. Real Data set based on surviving times (Data Set 2)

0.0035, 0.0068, 0.01, 0.0101, 0.0167, 0.0168, 0.0197, 0.0213, 0.0233, 0.0234, 0.0508, 0.0508, 0.0533, 0.0633, 0.0767, 0.0768, 0.077, 0.1066, 0.1267, 0.13, 0.1639, 0.1803, 0.1867, 0.218, 0.2967, 0.3328, 0.37, 0.3803, 0.4867, 0.6233, 0.6367, 0.66, 0.66, 0.718, 0.78, 0.7933, 0.7967, 0.8016, 0.83, 0.841, 0.91, 0.9233, 1.0541, 1.0607, 1.0633, 1.0667, 1.1067, 1.2213, 1.2508, 1.2533, 1.38, 1.4267, 1.4475, 1.45, 1.5213, 1.5333, 1.5525, 1.5533, 1.5541, 1.5934, 1.62, 1.63, 1.6344, 1.66, 1.7033, 1.7067, 1.7475, 1.7667, 1.77, 1.7967, 1.8115, 1.8115, 1.8933, 1.8934, 1.9508, 1.9733, 2.018, 2.09, 2.1167, 2.1233, 2.21, 2.2148, 2.2267, 2.25, 2.2533, 2.3738, 2.4082, 2.418, 2.4705, 2.5213, 2.5705, 3.1934, 3.218, 3.2367, 3.2705, 3.3148, 3.3567, 3.4836, 3.4869, 3.6213, 3.941, 3.9433, 4.0001, 4.1733, 4.1734, 4.2311, 4.2869, 4.3279, 4.3902, 4.4267.

The MLE(s) and their standard errors for the unknown parameters of above the distributions are given in Table 8 for data set 1 and Table 10 for data set 2. Goodness of fit measures for these data sets are shown in Table 9 for data set 1 and Table 11 for data set 2. Plots of empirical and  theoritical distribution functions of random variables having compared distributions are given by Figure 8 for data set 1 and Figure 9 for data set 2.

Table 8. Parameter estimates (standard errors) for Data set 1

Table 9. Selective criteria statistics for Real data set 1

Table 4.10. Parameter estimates (standard errors) for Data set 2

Table 11. Selective criteria statistics for Real data set 2

Figure 8. Goodness of fit plots  for data set 1            Figure 9. Goodnes of fit plots for data set 2