- 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.
- 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.
- 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. .
- 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
- Some Statistical Properties for EP-W distribution
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.
|
|
|
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SK
|
KU
|
|
(1.5,3,0.8)
|
1.4900
|
3.7800
|
12.3500
|
47.4000
|
1.5590
|
1.0610
|
3.8310
|
|
(1.5,3,1.5)
|
0.9220
|
1.0710
|
1.4180
|
2.0500
|
0.2210
|
0.2150
|
2.3660
|
|
(1.5,3,2)
|
0.8260
|
0.7950
|
0.8390
|
0.9460
|
0.1130
|
-0.0790
|
2.3560
|
|
(0.3,0.6,0.5)
|
0.6383
|
1.0400
|
2.5117
|
7.7343
|
0.6351
|
2.0670
|
8.4110
|
|
(0.3,0.6,1)
|
1.1927
|
2.1277
|
4.6455
|
11.5560
|
0.7052
|
0.7185
|
2.9830
|
|
(0.3,0.6,2)
|
1.8479
|
3.9757
|
9.3853
|
23.6414
|
0.5609
|
-0.0799
|
2.3560
|
|
(1,2,1.5)
|
1.0550
|
1.4040
|
2.1280
|
3.5210
|
0.2909
|
0.2152
|
2.3662
|
|
(1.5,2,1.5)
|
0.7036
|
0.6240
|
0.6304
|
0.6954
|
0.1289
|
0.2152
|
2.3663
|
|
(2.6,2,1.5)
|
0.4059
|
0.2077
|
0.1211
|
0.0770
|
0.0429
|
0.2152
|
2.3665
|
|
(0.5,0.5,0.6)
|
0.3385
|
0.2448
|
0.4469
|
0.3064
|
0.1302
|
1.6150
|
5.9573
|
|
(1.5,0.5,0.6)
|
0.1128
|
0.0272
|
0.0091
|
0.0038
|
0.0145
|
1.6152
|
5.9574
|
|
(3,0.5,0.6)
|
0.0564
|
0.0068
|
0.0011
|
0.0002
|
0.0036
|
1.6160
|
5.9580
|
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)
- 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 .
- 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
|
Parameters
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n
|
|
yan
|
mse
|
yan
|
mse
|
yan
|
mse
|
|
(1.6,0.9,0.7)
|
25
|
MLE
|
0.0211
|
0.0118
|
0.0826
|
0.0685
|
-0.0205
|
0.0071
|
|
50
|
MLE
|
0.0207
|
0.0057
|
0.0416
|
0.0294
|
-0.0041
|
0.0028
|
|
100
|
MLE
|
0.0221
|
0.0029
|
0.0191
|
0.0128
|
0.0041
|
0.0012
|
|
200
|
MLE
|
0.0204
|
0.0012
|
0.0090
|
0.0062
|
0.0076
|
0.0006
|
|
500
|
MLE
|
0.0176
|
0.0008
|
0.0050
|
0.0024
|
0.0081
|
0.0003
|
|
(0.5,0.6.0.4)
|
25
|
MLE
|
0.0126
|
0.0018
|
0.0408
|
0.0148
|
-0.0213
|
0.0046
|
|
50
|
MLE
|
0.0109
|
0.0010
|
0.0190
|
0.0063
|
-0.0088
|
0.0021
|
|
100
|
MLE
|
0.0091
|
0.0005
|
0.0089
|
0.0027
|
-0.0022
|
0.0009
|
|
200
|
MLE
|
0.0085
|
0.0003
|
0.0044
|
0.0013
|
0.0010
|
0.0005
|
|
500
|
MLE
|
0.0083
|
0.0002
|
0.0020
|
0.0005
|
0.0023
|
0.0002
|
|
(0.7,0.6.1.2)
|
25
|
MLE
|
-0.0012
|
0.0079
|
0.0402
|
0.0149
|
-0.0257
|
0.0118
|
|
50
|
MLE
|
0.0068
|
0.0040
|
0.0190
|
0.0061
|
-0.0083
|
0.0051
|
|
100
|
MLE
|
0.0083
|
0.0020
|
0.0110
|
0.0028
|
0.0037
|
0.0022
|
|
200
|
MLE
|
0.0105
|
0.0011
|
0.0051
|
0.0013
|
0.0072
|
0.0011
|
|
500
|
MLE
|
0.0111
|
0.0006
|
0.0016
|
0.0005
|
0.0099
|
0.0005
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|
(0.3, 1.3, 2)
|
25
|
MLE
|
-0.0011
|
0.0013
|
0.0883
|
0.0710
|
-0.0324
|
0.0243
|
|
50
|
MLE
|
0.0013
|
0.0007
|
0.0401
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0.0292
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-0.0025
|
0.0091
|
|
100
|
MLE
|
0.0024
|
0.0004
|
0..0205
|
0.0132
|
0.0120
|
0.0041
|
|
200
|
MLE
|
0.0025
|
0.0002
|
0.0190
|
0.0065
|
0.0218
|
0.0025
|
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500
|
MLE
|
0.0029
|
0.0001
|
0.0046
|
0.0024
|
0.0241
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0.0014
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- 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
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|
0,4365
|
0,6012
|
0,6789
|
0,5481
|
0,5627
|
0,4612
|
0,5349
|
0,6488
|
|
|
|
0,4260
|
0,1525
|
0,4576
|
0,1131
|
0,515
|
0,3188
|
0,3751
|
0,2747
|
|
|
|
0,5140
|
0,5483
|
0,3259
|
0,729
|
0,0776
|
0,216
|
0,1546
|
|
|
|
|
0,6907
|
0,6927
|
0,2303
|
0,0168
|
0,3945
|
0,6707
|
0,4517
|
|
|
|
|
0,7471
|
0,7261
|
0,7687
|
0,5529
|
0,4553
|
0,6220
|
0,2681
|
|
|
|
|
0,2605
|
0,3323
|
0,4371
|
0,4530
|
0,4470
|
0,5629
|
0,4049
|
|
|
|
|
0,6196
|
0,0671
|
0,3383
|
0,3891
|
0,5285
|
0,4675
|
0,5553
|
|
|
|
|
0,8781
|
0,2361
|
0,6114
|
0,4752
|
0,5232
|
0,6844
|
0,5878
|
|
|
|
|
0,4990
|
0,4800
|
0,348
|
0,3134
|
0,6465
|
0,3413
|
0,4741
|
|
|
|
|
0,6058
|
0,5707
|
0,4564
|
0,3175
|
0,065
|
0,4332
|
0,3598
|
|
|
|
|
0,6891
|
0,7131
|
0,7804
|
0,1167
|
0,8492
|
0,0854
|
0,7629
|
|
|
|
|
0,5770
|
0,5853
|
0,3406
|
0,6750
|
0,8147
|
0,3821
|
0,5941
|
|
|
|
|
0,5394
|
0,6768
|
0,4823
|
0,5113
|
0,3627
|
0,4694
|
0,6174
|
|
|
|
|
0,1479
|
0,5350
|
0,5912
|
0,5447
|
0,3906
|
0,3635
|
0,6860
|
|
|
|
|
0,2356
|
0,4151
|
0,5744
|
0,4143
|
0,4438
|
0,4111
|
0,0609
|
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
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|
|
|
|
|
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|
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|
|
The MLE(s) of the unknown parameters and standart errors for models are given in Table 4.
Tablo 4. Parameter estimaters (standart errors)
|
Distribution
|
MLE Estimaters
|
|
EP-Weibull
|
, ,
|
|
Exponentiated
exponential
|
,
|
|
Weibull
|
,
|
|
Exponential
|
|
Tablo 5. Selection criteria statistics for data set1
|
Distribution
|
-2LogL
|
AIC
|
AICc
|
BIC
|
K-S
|
p-values
|
|
EP-W
|
-54,2257
|
-48,2257
|
-41,7597
|
-40,2072
|
0,066602
|
0,729563
|
|
Üstell Exponentiated
exponential
|
-10,0775
|
-6,0775
|
-5,96212
|
-0,73184
|
0,147648
|
0,018834
|
|
Weibull
|
-42,695
|
-38,695
|
-38,5796
|
-33,3494
|
0,083244
|
0,448642
|
|
Exponential
|
51,90155
|
53,90155
|
53,93964
|
56,57438
|
0,319267
|
6,72E-10
|
|
|
|
|
|
|
|
|
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)
|
|
MLE Estimaters
|
|
EP-Weibull
|
, ,
|
|
Exponentiated
exponential
|
,
|
|
weibull
|
,
|
|
Exponential
|
|
|
Transmuted exponentiated Exponential
|
, ,
|
Tablo 8. Selection criteria statistics for data set1
|
Dağılım
|
-2LogL
|
AIC
|
AICc
|
BIC
|
K-S
|
p-değeri
|
|
EP-W
|
49.0556
|
53.0556
|
53.3889
|
56.3827
|
0.0882
|
0.9217
|
|
Exponentiated
exponential
|
63.0761
|
67.0761
|
67.4094
|
70.4032
|
0.1705
|
0.2071
|
|
Weibull
|
54.9506
|
58.9506
|
59.2839
|
62.277
|
0.1174
|
0.6552
|
|
exponential
|
73.3353
|
75.3353
|
75.4434
|
76.9988
|
0.2716
|
0.0063
|
|
Transmuted exponentiated Exponential
|
59.5441
|
65.5441
|
66.2298
|
70.5348
|
0.1449
|
0.3863
|
|
|
|
|
|
|
|
|
Figure 4. Empirical cdf and theoretical cd.