Abstract
In this study, it has been aimed to introduce a new statistical distribution called Exponential Power  Chen by using the method suggested by Alzaatreh et al. (2013). Some statistical properties such as moments, coefficies of skewness and kurtosis, random number generator for Exponential Power Chen (EPCH) distribution are obtained. Moreover, the maximum likelihood estimators (MLEs) for unknown parameters of EPCH distribution have been derived and a Monte Carlo simulation study based on mean square errors and biases of this estimators for various sample sizes have been performed. Finally, an application using real data set has been presented for this new distribution.
Highlights
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 MonteCario 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 EPCh distribution in real life
Full Text
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 betapareto 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.
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 BetaExponentialX distribution, WeibullPareto distribution and WeibullX families of distributions. Alzaghal et al. [4] suggested Exponentiated TX Family of distributions, Tahir et al. [22,23,24] introduced the odd generalized exponential family of distributions, The logisticX family of distributions and A new Weibull family of distributions. Çelik and Guloksuz [8] suggested a new lifetime distribution called as UniformExponential 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.
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)
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 EPCh 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 EPCh distribution for different parameter values
Figure 2. pdf plots of EPCh distribution for different parameter values
Figure 3. hf plots of EPCh distribution for different parameter values
Figure 4. ihf function plots of EPCh distribution for different parameter values
4.1. Random Numer Generator for EPCh 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 EPCh 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 EPCh distribution, the moment, variance, skewness and kurtosis coefficients are given in Table 1.
Table1. moment, variance skewness and kurtosis values for EPCh distribution






Skewness 
Kurtosis 

0.4065 0.2414 0.1357 
0.3187 0.1221 0.0413 
0.3163 0.0812 0.0177 
0.3591 0.0632 0.0086 
0.1535 0.0638 0.0229 
1.0300 1.2960 1.5390 
3.3540 4.2560 5.2950 

0.2424 0.4661 0.6704 
0.0786 0.2424 0.4661 
0.0295 0.1349 0.3327 
0.0122 0.0786 0.2424 
0.0198 0.0251 0.0167 
0.3006 0.4042 0.9544 
2.3390 2.5590 3.8510 

0.3180 0.3859 0.4661 
0.2033 0.2019 0.2424 
0.1675 0.1223 0.1349 
0.1596 0.0813 0.0786 
0.1022 0.0529 0.0251 
1.1610 0.2814 0.4042 
3.7670 2.2520 2.5590 

0.2768 0.4591 0.7158 
0.1573 0.3976 0.8854 
0.1164 0.4318 1.3263 
0.1003 0.5337 2.2171 
0.0807 0.1868 0.3730 
1.2310 0.9616 0.6951 
4.0110 3.1610 2.5440 
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 momentgenerating function (mgf) of a random variable X having EPCh distribution, , is obtained as follows.
(20)
4.4. Order Statistics for EPCh 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 EPCh 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 nonnegative 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 nonnegative 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 nonnegative 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




(1.5,0.2,0.5,1.5) 
1.4529 
3.5196 
3.8974 
(1.5,0.2,0.5,2) 
1.0402 
2.3229 
3.7419 
(1.5,0.2,0.5,2.5) 
0.7466 
3.1971 
3.6839 
(1.5,0.2,0.5,2.5) 
0.5238 
3.1419 
3.6825 
(1.5,0.9,0.2,0.5) 
5.1339 
6.0212 
6.4235 
(1.5,1.5,0.2,0.5) 
2.4975 
3.8020 
4.0010 
(1.5, 2,0.2,0.5) 
2.9014 
2.7051 
2.4862 
(1.5, 2,0.2,0.5) 
2.7345 
2.4192 
2.1165 
(0.2,0.5,0.3,0.01) 
26.8366 
27.9178 
28.5566 
(0.5,0.5,0.3,0.01) 
56.9885 
58.4539 
59.342 
(1.5,0.5,0.3,0.01) 
123.8780 
125.7015 
126.8180 
(2,0.5,0.3,0.01) 
148.8599 
150.7784 
151.9081 
(1.5,2,0.3,0.01) 
165.5861 
164.6076 
163.6312 
(1.5,2,0.5,0.01) 
20.2103 
19.2162 
18.2262 
(1.5,2,0.7,0.01) 
7.8161 
6.8215 
5.8369 
(1.5,2,0.9,0.01) 
4.4221 
3.4286 
2.4650 
(2,1.5,1.5,0.01) 
1.8048 
0.8286 
0.0752 
(1.5,0.6,0.5,0.01) 
15.5495 
15.2069 
14.6775 
(1.5,0.6,0.6,0.01) 
9.1948 
8.6193 
8.0192 
(1.5,0.6,0.7,0.01) 
5.9921 
5.5846 
4.9763 
(1.5,0.6,0.9,0.01) 
3.1885 
2.9301 
2.4320 
(0.2,0.5,0.2,0.5) 
0.4386 
0.5408 
0.5999 
(0.5,0.5,0.2,0.5) 
2.0837 
3.7894 
4.3722 
(1.5,0.5,0.2,0.5) 
12.5983 
14.3185 
15.8991 
(2,0.5,0.2,0.5) 
19.8341 
22.1319 
23.7242 
(1.5,0.4,0.15,0.2) 
433.4017 
463.4297 
483.7846 
(1.5,0.4,0.2,0.2) 
72.1829 
78.2397 
82.4504 
(1.5,0.4,0.3,0.2) 
13.0904 
14.1164 
15.0439 
(1.5,0.4,0.4,0.2) 
5.61672 
5.8228 
6.33779 
(2.5,0.5,0.2,0.1) 
256.8916 
266.7358 
273.5476 
(2.5,0.8,0.2,0.1) 
207.7318 
212.4193 
215.5393 
(2.5,0.9,0.2,0.1) 
204.6623 
208.2875 
210.7467 
(2.5,1.5,0.2,0.1) 
214.6724 
215.3445 
215.6490 


Figure 7. Plots of the mean remaining life function for EPCh 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
Parameters 
Renyi entropy values 






0.1726 0.3104 0.3356 
1.0210 0.6808 1.0690 
1.1790 1.2710 1.3010 

0.7903 0.5325 0.4510 
0.6042 0.3378 0.6247 
0.5136 0.5624 0.8571 

2.2360 0.5671 0.3104 
1.7100 0.5974 0.6808 
1.5470 0.8232 1.2710 

0.8002 0.1984 0.2906 
0.9500 0.9898 1.1210 
1.6660 1.2550 1.3110 
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 EPCh distribution are given in Table 4.
Table 4. Plots of Shannon entropy for different parameter values of EPCh distribution

Shannon Entropy 

1.021 0.6806 1.069 

0.6042 0.3378 0.6247 

1.71 0.5974 0.6808 

0.95 0.9898 1.121 
Let be a random sample with size n taken from distribution. The loglikelihood function is given as follows.
(30)
where . Derivatives according to unknown parameters of the loglikelihood 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.
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
Parameters 







n 
bias 
Mse 
bias 
mse 
Bias 
mse 
bias 
Mse 
(0.5,1.4,0.2,0.5) 
100 
0.0222 
0.0438 
0.0134 
0.1736 
0.0063 
0.0139 
0.0329 
0.0169 
150 
0.0147 
0.0132 
0.0103 
0.0725 
0.0032 
0.0034 
0.0209 
0.0106 

200 
0.0077 
0.0061 
0.0086 
0.0432 
0.0025 
0.0019 
0.0139 
0.0073 

300 
0.0019 
0.0037 
0.0069 
0.0268 
0.0026 
0.0011 
0.0077 
0.0044 

500 
0.0009 
0.0018 
0.0096 
0.0150 
0.0019 
0.0005 
0.0043 
0.0022 

(0.2,0.8,0.6,0.5) 
100 
0.0080 
0.0543 
0.0261 
0.1965 
0.1574 
2.0267 
0.0320 
0.0374 
150 
0.0108 
0.0030 
0.0221 
0.0386 
0.0491 
0.2309 
0.0277 
0.0235 

200 
0.0061 
0.0027 
0.0164 
0.0260 
0.0388 
0.0817 
0.0171 
0.0179 

300 
0.0033 
0.0009 
0.0111 
0.0148 
0.0226 
0.0312 
0.0099 
0.0114 

500 
0.0013 
0.0006 
0.0135 
0.0093 
0.0102 
0.0153 
0.0078 
0.0069 

(0.3,0.9,0.6,0.4) 
100 
0.0100 
0.7033 
0.0799 
2.1006 
0.2131 
4.6283 
0.0540 
0.0378 
150 
0.0067 
0.1823 
0.0269 
0.5555 
0.0844 
0.6790 
0.0320 
0.0237 

200 
0.0074 
0.0518 
0.0198 
0.1855 
0.0462 
0.1454 
0.0239 
0.0174 

300 
0.0061 
0.0024 
0.0108 
0.0182 
0.0286 
0.0450 
0.0131 
0.0110 

500 
0.0023 
0.0011 
0.0123 
0.0094 
0.0126 
0.0198 
0.0095 
0.0064 

(0.2,1.5,0.4,0.2) 
100 
0.0120 
0.0082 
0.0798 
0.7168 
0.0289 
0.1880 
0.0076 
0.0024 
150 
0.0087 
0.0021 
0.0611 
0.1454 
0.0116 
0.0141 
0.0055 
0.0016 

200 
0.0055 
0.0015 
0.0442 
0.1099 
0.0099 
0.0098 
0.0033 
0.0012 

300 
0.0025 
0.0010 
0.0380 
0.0721 
0.0055 
0.0056 
0.0025 
0.0007 

500 
0.0009 
0.0006 
0.0375 
0.0466 
0.0017 
0.0031 
0.0020 
0.0004 

(0.3, 2,0.5,0.9) 
100 
0.0213 
0.0100 
0.0666 
0.3942 
0.0515 
0.1520 
0.0317 
0.0644 
150 
0.0127 
0.0048 
0.0597 
0.2636 
0.0250 
0.0355 
0.0215 
0.0400 

200 
0.0096 
0.0033 
0.0563 
0.2045 
0.0175 
0.0215 
0.0173 
0.0303 

300 
0.0041 
0.0022 
0.0480 
0.1337 
0.0105 
0.0114 
0.0105 
0.0189 

500 
0.0009 
0.0013 
0.0467 
0.0812 
0.0050 
0.0062 
0.0083 
0.0113 
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×loglikelihood 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
Distribution 
MLE 
EPCh 
, , 
Exponentiated Exponential 
, 
Weibull 
, 
Exponential power 
, 
Table 9. Selective criteria statistics for Real data set 1
Dağılım 
2LogL 
AIC 
BIC 
KS 
pvalue 
EPCh 
93.6475 
101.6475 
105.6304 
0.1383 
0.8359 
Exponentiated Exponential 
109.2411 
113.2411 
115.2325 
0.2493 
0.1663 
Weibull 
109.5036 
133.5036 
115.4950 
0.2205 
0.2853 
Exponential power 
102.9713 
106.9713 
108.9628 
0.2049 
0.3706 
Table 4.10. Parameter estimates (standard errors) for Data set 2
Distribution 
MLE 
EPCh 
, , 
Exponential power 
, 
Chen 
, 
Table 11. Selective criteria statistics for Real data set 2
Dağılım 
2LogL 
AIC 
AICc 
KS 
pvalue 
EPCh 
290.2455 
298.2455 
298.6265 
0.0743 
0.5776 
Exponential power 
300.3445 
304.3445 
304.4566 
0.1158 
0.1044 
Chen 
295.3769 
299.3769 
299.4891 
0.1158 
0.1044 
Figure 8. Goodness of fit plots for data set 1 Figure 9. Goodnes of fit plots for data set 2
REFERENCES