International Journal of Mathematics and Mathematical Sciences
Volume 2016 (2016), Article ID 5898356, 10 pages
http://dx.doi.org/10.1155/2016/5898356

Research Article

The Exponentiated Gumbel Type-2 Distribution: Properties and Application

I. E. Okorie,¹ A. C. Akpanta,² and J. Ohakwe³

¹School of Mathematics, University of Manchester, Manchester M13 9PL, UK
²Department of Statistics, Abia State University, PMB 2000, Uturu, Abia State, Nigeria
³Department of Mathematics & Statistics, Faculty of Sciences, Federal University, Otuoke, PMB 126, Yenagoa, Bayelsa, Nigeria

Received 17 May 2016; Accepted 10 July 2016

Academic Editor: Niansheng Tang

Copyright © 2016 I. E. Okorie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce a generalized version of the standard Gumble type-2 distribution. The new lifetime distribution is called the Exponentiated Gumbel (EG) type-2 distribution. The EG type-2 distribution has three nested submodels, namely, the Gumbel type-2 distribution, the Exponentiated Fréchet (EF) distribution, and the Fréchet distribution. Some statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimates was proposed for estimating the model parameters. The usefulness and flexibility of the Exponentiated Gumbel (EG) type-2 distribution were illustrated with a real lifetime data set. Results based on the log-likelihood and information statistics values showed that the EG type-2 distribution provides a better fit to the data than the other competing distributions. Also, the consistency of the parameters of the new distribution was demonstrated through a simulation study. The EG type-2 distribution is therefore recommended for effective modelling of lifetime data.

1. Introduction

The Gumbel distribution, also known as the type-1 extreme value distribution, has received significant research attention, over the years particularly, in extreme value analysis of extreme events. For a review of the recent developments and applications of the Gumbel distribution, see Pinheiro and Ferrari [1]. There is no question that, before now, the Gumbel type-2 distribution is not popularly used in statistical modelling and the reason may not be far from its lack of fits in data modelling. Generally, standard probability distributions are well known for their lack of fits in modelling complex data sets. On this note, users of this distributions across various fields in general and statistics and mathematics in particular have been fantastically motivated to developing sophisticated probability distributions from the existing ones. Exponentiated distributions have been introduced to solve the problem of lack of fits that is commonly encountered when using the standard probability distributions for modelling complex data sets. Results from this advancement have frequently been proven more reasonable than the one based on the standard distributions. Exponentiating distributions are indeed a powerful technique in statistical modelling that offers an effective way of introducing an additional shape parameter to the standard distribution to achieve robustness and flexibility. This method of generalizing probability distributions is traceable to the work of Gupta et al. [2] who introduced the exponentiated exponential (EE) distribution as a generalized form of the standard exponential distribution by simply raising the cumulative density function (cdf) to a positive constant power. Ever since the introduction of the EE distribution, exponentiated distributions have achieved reasonable feats in modelling data sets from various complex phenomena. A good number of standard probability distributions have their corresponding exponentiated versions. Gupta et al. [2] introduced the Exponentiated Weibull distribution as a generalization of the standard Weibull distribution. Nadarajah and Kotz [3] modified the method by Gupta et al. [2] and introduced the Exponentiated Fréchet distribution as a generalization of the standard Fréchet distribution. Using the same method in Nadarajah and Kotz [3], Nadarajah [4] introduced the Exponentiated Gumbel distribution as a generalization of the standard Gumbel distribution. Mudholkar and Srivastava [5] introduced the Exponentiated Weibull family distribution as a generalization of the Weibull family distribution. Ashour and Eltehiwy [6] developed the exponentiated power Lindley distribution generalizing the power Lindley distribution and so forth. Therefore, this paper is aimed at generalizing the standard Gumbel type-2 distribution to a wider class of distribution so as to improve its performance and encourage its applicability, in modelling varieties of complex data sets.

The cumulative density function of the exponentiated family of distributions according to Nadarajah and Kotz [3] is defined by

differentiating (1) with respect to gives the corresponding probability density function as

where and are the vector of parameters and parameter space of the baseline distribution , respectively.

The remaining part of this paper is organized as follows; Section 2 introduces the Gumbel type-2 distribution, its exponentiated version, and special cases (submodels); Section 3 presents some important reliability characteristics of the new distribution and their asymptotic properties; Section 4 presents an explicit derivation of the moments, variance, and moment generating function of the new model; Section 5 presents the th quantile function of the new distribution; Section 6 presents the Rényi’s entropy of the new distribution; Section 7 presents the th order statistics of the new distribution; Section 8 proposes the maximum likelihood estimation method for estimating the parameters of the new distribution; Section 9 presents the application of the new distribution to a real data set and a simulation study; Section 10 is the discussion of results and Section 11 contains the conclusion of the study.

2. Exponentiated Gumbel Type-2 Distribution

Definition 1. According to Gumbel [7–9], a random variable is said to follow the Gumbel type-2 distribution if its cumulative density function (cdf) is given by

while the corresponding probability density function (pdf) is given by

Using (3), we obtain the cdf of the Exponentiated Gumbel (EG) type-2 distribution as

while the corresponding pdf is given by

where and are the shape parameters and is the scale parameter.

Figure 1 shows the plots of the pdf (a) and cdf (b) of the EG type-2 distribution for certain parameter values.

(a)

(b)

Figure 1: Possible shapes of the pdf (a) and cdf (b) of the EG type-2 distribution for fixed parameter values of and selected values of parameter. (solid lines), (dashed lines), (dotted lines), (dot-dashed lines), (long dashed lines), and (two dashed lines).

2.1. Special Cases of the EG Type-2 Distribution

The EG type-2 distribution is developed for the purpose of modelling data sets that arise from complex phenomena. It generalizes some standard distributions; for instance, the EG type-2 distribution reduces to the Gumbel type-2 distribution, Exponentiated Fréchet (EF) distribution, and Fréchet distribution when , , and , respectively.

Theorem 2. If and is distributed according to the EG type-2 distribution then, is distributed according to the exponentiated exponential (EE) distribution due to Gupta et al. [2].

Proof. The transformation of a random variable to a random variable is defined by , where is known as the Jacobian of transformation. Thus, and

Corollary 3. When (8) reduces to the exponential distribution with parameter ; that is, .

3. Some Reliability Properties of the EG Type-2 Distribution

Reliability theory is generally concerned with the estimation of the probability of longevity or failure of a system.

3.1. Reliability Function

Definition 4. The reliability function or the survival function of a random variable is defined by . It could be interpreted as the probability of a system not failing before some specified time , Lee and Wang [10]. The reliability function of the EG type-2 distribution is given by

3.2. Hazard Rate Function

Definition 5. The hazard rate function or the instantaneous failure rate of a random variable is the probability that a system fails given that it has survived up to time and is given by (Lee and Wang [10]). Hence, we define the hazard rate function of the EG type-2 distribution as follows:

Figure 2 shows plots of the reliability function (a) and hazard rate function (b) of the EG type-2 distribution for selected parameter values.

(a)

(b)

Figure 2: Possible shapes of the reliability function (a) and hazard rate function (b) of the EG type-2 distribution for fixed parameter values of and selected values of parameter. (solid lines), (dashed lines), (dotted lines), (dot-dashed lines), (long dashed lines), and (two dashed lines).

3.3. Asymptotics

pdf and cdf of the EG type-2 distribution is unimodal and monotonically increasing, respectively, with increasing values of The reliability function of the EG type-2 distribution is 0 as and 1 as . Also, is a monotonic decreasing function of . For example, when ,

Hence, is strictly a monotonic decreasing function of .

The hazard rate function of the EG type-2 distribution is 0 for both and and its shape appears increasingly upside-down bathtub with decreasing values of .

4. The th Crude Moment of the EG Type-2 Distribution

In probability theory, the moments of a random variable are one of the most important properties of a distribution that could be used to derive other essential properties such as mean, variance, skewness, and kurtosis statistics that describes a probability distribution. The th crude moment of a continuous random variable is defined by ; then the th crude moment of the EG type-2 distribution follows as

Substituting into (13) we have

and thus,

Since can only take values on the positive real line we can introduce the exponential integral defined by (see Chapter 5 of Abramowitz and Stegun [11] and Equation (6.2.6) of Olver et al. [12]):

Substituting in (16) we have

Thus, evaluating (18) at and yields the mean and second crude moment then we can obtain the variance of the EG type-2 distribution as . Denoting by the coefficient of variation (cv), skewness , and kurtosis statistics of the EG type-2 distribution can be obtained by evaluating

respectively.

4.1. The Moment Generating Function of the EG Type-2 Distribution

Generally, the moment generating function (mgf) denoted by of a random variable is defined as

If a random variable is distributed according to the EG Type-2 distribution, then its mgf is given by

5. The th Quantile Function of the EG Type-2 Distribution

The th quantile function of the EG Type-2 distribution is the inverse of (5) and it is obtained as

We can simulate random variables from the EG type-2 distribution through the inversion of the cdf method by simply replacing in (24) with variates. Also, evaluating (24) at gives the median of the distribution.

6. The Rényi Entropy

The Rényi entropy is used to measure uncertainty or variation in a random variable . The Rényi’s entropy measure has been shown to be effective in comparing the tails and shapes of various standard distributions, Song [13]. The Rényi entropy measure for a continuous random variable is given by

Then the Rényi entropy measure for the EG type-2 distribution could be obtained as follows:

Setting in (26) we have

Substituting in (27) we have

using the expression for the function defined in Section 4; we have

and substituting in (29) we have

and thus,

7. The Order Statistics of the EG Type-2 Distribution

The distribution of the th order statistics of a random sample of size is generally given as

Hence, the density of the th order statistics of the EG Type-2 distribution is given by

The density of the th smallest order statistics of the EG Type-2 distribution is given by

The density of the th largest order statistics of the EG Type-2 distribution is given by

8. Parameter Estimation of the EG Type-2 Distribution

In this section, we propose the method of maximum likelihood estimates (MLE) for the estimation of the parameters of the EG type-2 distribution. Suppose a random variable of size has the EG type-2 distribution then, its MLE are obtained as follows:

The likelihood function is given by

with the corresponding log-likelihood function

Taking the partial derivatives of the log-likelihood function with respect to , , and , respectively, and equating to 0 give

Equations (38) can only be solved by some numerical optimization methods such as Newton Raphson’s algorithm to obtain the MLE of , , and .

9. Application

In this section we would fit the EG type-2 distribution to a real and uncensored data set to demonstrate its applicability and flexibility. The goodness of fit of the new distribution would be compared with the three submodels, namely, the Gumbel type-2 distribution, Exponentiated Fréchet distribution, and Fréchet distribution and two other related heavy tail distributions: Weibull distribution , , , and , and log-normal (LN) distribution, , , and . The model comparison would be based on the minimized log-likelihood estimate and the following information statistics: AIC by Akaike [14], AICC by Sugiura [15], CAIC by Bozdogan [16], HQC by Schwarz [17], and BIC by Hannan and Quinn [18]. The model with the smallest minimized log-likelihood and information statistics value is the best. The data set in Table 1 shows the survival times in months of 20 acute myeloid leukemia patients reported in Afify et al. [19].

Table 1: Survival times in months of 20 acute myeloid leukemia patients.

2.226
2.113
3.631
2.473
2.720
2.050
2.061
3.915
0.871
1.548

2.746
1.972
2.265
1.200
2.967
2.808
1.079
2.353
0.726
1.958

9.1. Monte-Carlo Simulation

In this section we present a Monte-Carlo simulation study to investigate the effect of sample size on the maximum likelihood estimates of the parameters of the EG type-2 distribution and further to assess the stability of these parameters. Different sample sizes were drawn from the EG type-2 distribution with parameters , , and using the inverse transformation method with (24) where each sample was replicated 5000 times. Using the simulated random variables we estimate the parameters of the EG type-2 distribution through the method of maximum likelihood estimation and the procedure was repeated 5000 times for each sample size. The mean (parameter estimate) and standard deviation (standard error (se)) of the 5000 parameters each for α, θ, and ϕ for each sample size were computed and the result is presented in Table 3. Furthermore, the corresponding bias and mean square errors (mse) of each of the parameter estimates are tabulated in Table 4. Analogously, Tables 5 and 6 show simulation results for the EG type-2 distribution with parameters , , and .

10. Discussion of Results

From the pdf and cdf plots in Figure 1, the pdf of the EG type-2 distribution is unimodal and increasingly unimodal for increasing values of (shape parameter) while its cdf is monotonic increasing and more monotonically increasing for increasing values of . Also, the plots of the reliability function and hazard rate function in Figure 2 show that is monotonic decreasing and more monotonically decreasing for increasing values of while is upside-down bathtub and becomes more upside-down bathtub for decreasing values of . Results from the model fittings as tabulated in Table 2 indicate that the EG type-2 distribution provides the best fit to the data based on its smallest minimized log-likelihood and information statistics values. Figure 3 depicts the cdf’s of all the estimated distributions in Table 2 superimposed on the empirical cdf of the data, where the cdf of the EG type-2 distribution is closely aligned to the empirical one than the other distributions. From the simulation results in Tables 3, 4, 5, and 6 it is clear that the parameters of the EG type-2 distribution approach the true value as the sample size increases, while the standard error, bias, and the mse decrease down the column with increasing sample size.

Table 2: Results from the survival times data fitting.

Models
Estimates

AIC
AICC
CAIC
HQC
BIC

EG type-2

12.7096
24.12718
52.25436
52.96024
56.24583
52.64312
54.24583

0.2821

G type-2

2.6040
29.08667
62.17334
62.87922
66.16481
62.5621
64.16481

2.0651

EF

0.3928
31.89448
67.78896
68.49484
71.78042
68.17771
69.78042

3.4393

Fréchet

1.7378
35.45610
72.91219
73.13442
74.90793
73.10657
76.90366

LN

0.6971
25.71549
55.43098
56.13687
59.42245
55.81974
57.42245

0.4360

Weibull

2.9237
24.30433
54.60867
56.10867
60.59587
55.1918
54.60013

2.4502

Table 3: Monte-Carlo simulation results of the parameter estimates and standard errors of the EG type-2 distribution with parameters: , , and for different sample sizes.

Sample size

25
292.3201
1.9186
2.5361
2608.3737
1.8417
2.7992

50
16.7755
1.6267
1.8183
600.1862
0.9444
1.1577

75
2.1483
1.5741
1.6634
5.8997
0.6814
0.6847

100
1.8196
1.5463
1.6125
1.5079
0.5516
0.5114

125
1.7586
1.5401
1.5843
1.2131
0.4976
0.4302

150
1.6970
1.5334
1.5684
0.9432
0.4464
0.3834

175
1.6660
1.5332
1.5538
0.7793
0.4072
0.3466

200
1.6269
1.5208
1.5506
0.6975
0.3728
0.3154

225
1.6252
1.5272
1.5386
0.6501
0.3536
0.2952

250
1.6102
1.5251
1.5331
0.5715
0.3268
0.2728

275
1.5909
1.5168
1.5344
0.5333
0.3119
0.2597

300
1.5825
1.5152
1.5320
0.5108
0.3025
0.2511

325
1.5755
1.5152
1.5273
0.4798
0.2870
0.2363

350
1.5730
1.5146
1.5271
0.4624
0.2791
0.2294

375
1.5635
1.5104
1.5255
0.4404
0.2687
0.2205

400
1.5603
1.5129
1.5221
0.4180
0.2558
0.2094

425
1.5517
1.5086
1.5237
0.4034
0.2491
0.2031

450
1.5498
1.5096
1.5213
0.3873
0.2405
0.1966

475
1.5450
1.5079
1.5220
0.3781
0.2361
0.1927

500
1.5433
1.5069
1.5191
0.3620
0.2272
0.1862

Table 4: Monte-Carlo simulation results of the estimators bias and mse of the EG type-2 distribution with parameters: , , and for different sample sizes.

Sample size

25
290.8201
0.4186
1.0361
6887509.5404
3.5666
8.9085

50
15.2755
0.1267
0.3183
360420.7341
0.9078
1.4415

75
0.6483
0.0741
0.1634
35.2238
0.4698
0.4955

100
0.3196
0.0463
0.1125
2.3757
0.3064
0.2742

125
0.2586
0.0401
0.0843
1.5383
0.2492
0.1922

150
0.1970
0.0334
0.0684
0.9283
0.2004
0.1516

175
0.1660
0.0332
0.0538
0.6348
0.1669
0.1230

200
0.1269
0.0208
0.0506
0.5025
0.1394
0.1020

225
0.1252
0.0272
0.0386
0.4383
0.1257
0.0886

250
0.1102
0.0251
0.0331
0.3388
0.1074
0.0755

275
0.0909
0.0168
0.0344
0.2927
0.0976
0.0686

300
0.0825
0.0152
0.0320
0.2677
0.0917
0.0641

325
0.0755
0.0152
0.0273
0.2359
0.0826
0.0566

350
0.0730
0.0146
0.0271
0.2191
0.0781
0.0534

375
0.0635
0.0104
0.0255
0.1980
0.0723
0.0493

400
0.0603
0.0129
0.0221
0.1784
0.0656
0.0443

425
0.0517
0.0086
0.0237
0.1654
0.0621
0.0418

450
0.0498
0.0096
0.0213
0.1524
0.0579
0.0391

475
0.0450
0.0079
0.0220
0.1449
0.0558
0.0376

500
0.0433
0.0069
0.0191
0.1329
0.0517
0.0350

Table 5: Monte-Carlo simulation results of the parameter estimates and standard errors of the EG type-2 distribution with parameters: , , and for different sample sizes.

Sample size

25
957.2346
9.1225
8.1140
4764.4832
171.3331
10.5485

50
118.7143
4.4324
5.8491
1677.2419
1.5587
3.6720

75
23.7643
4.2555
5.3847
804.2310
0.8669
2.1538

100
4.9487
4.1628
5.2897
27.8859
0.6844
1.6550

125
4.3378
4.1331
5.2224
16.5547
0.5964
1.4261

150
3.7404
4.1022
5.1856
3.4572
0.5251
1.2608

175
3.6405
4.0926
5.1459
3.0479
0.4868
1.1427

200
3.5112
4.0785
5.1310
2.1757
0.4479
1.0647

225
3.4261
4.0684
5.1207
2.0217
0.4161
0.9853

250
3.3987
4.0598
5.0902
1.6941
0.3987
0.9206

275
3.3271
4.0538
5.0939
1.5224
0.3699
0.8769

300
3.2975
4.0468
5.0901
1.4489
0.3588
0.8396

325
3.2727
4.0499
5.0834
1.3216
0.3432
0.8059

350
3.2506
4.0458
5.0683
1.2191
0.3264
0.7609

375
3.2390
4.0398
5.0645
1.1950
0.3221
0.7385

400
3.2020
4.0337
5.0716
1.1148
0.3050
0.7162

425
3.2035
4.0379
5.0564
1.0529
0.2926
0.6887

450
3.2039
4.0344
5.0396
1.0356
0.2880
0.6670

475
3.1827
4.0328
5.0502
0.9930
0.2797
0.6472

500
3.1648
4.0334
5.0533
0.9531
0.2724
0.6326

Table 6: Monte-Carlo simulation results of the estimators bias and mse of the EG type-2 distribution with parameters: , , and for different sample sizes.

Sample size

25
954.2346
5.1225
3.1140

29378.3413
120.9573

50
115.7143
0.4324
0.8491

2.6162
14.2029

75
20.7643
0.2555
0.3847

0.8168
4.7864

100
1.9487
0.1628
0.2897
781.3449
0.4949
2.8228

125
1.3378
0.1331
0.2224
275.8201
0.3733
2.0829

150
0.7404
0.1022
0.1856
12.4991
0.2861
1.6240

175
0.6405
0.0926
0.1459
9.6992
0.2455
1.3270

200
0.5112
0.0785
0.1310
4.9943
0.2067
1.1506

225
0.4261
0.0684
0.1207
4.2683
0.1778
0.9854

250
0.3987
0.0598
0.0902
3.0285
0.1626
0.8556

275
0.3271
0.0538
0.0939
2.4246
0.1397
0.7778

300
0.2975
0.0468
0.0901
2.1878
0.1309
0.7130

325
0.2727
0.0499
0.0834
1.8209
0.1202
0.6564

350
0.2506
0.0458
0.0683
1.5489
0.1086
0.5836

375
0.2390
0.0398
0.0645
1.4850
0.1053
0.5496

400
0.2020
0.0337
0.0716
1.2834
0.0941
0.5181

425
0.2035
0.0379
0.0564
1.1500
0.0870
0.4774

450
0.2039
0.0344
0.0396
1.1138
0.0841
0.4465

475
0.1827
0.0328
0.0502
1.0193
0.0793
0.4213

500
0.1648
0.0334
0.0533
0.9355
0.0753
0.4029

Figure 3: cdf plots of the fitted distributions superimposed on the empirical density plot of the survival times data.

11. Conclusion

This paper introduces a new lifetime distribution, the Exponentiated Gumbel (EG) type-2 distribution. The new distribution generalizes the standard Gumbel type-2 distribution and has the following distributions as special cases: Gumbel type-2 distribution, Exponentiated Fréchet distribution, and Fréchet distribution. We have provided explicit mathematical expressions for some of its basic statistical properties such as the probability density function, cumulative density function, th crude moment, variance, coefficient of variation, skewness, kurtosis, moment generating function, and th quantile function and some reliability characteristics like the reliability and hazard rate functions. Estimation of the model parameters was approached through the method of maximum likelihood estimates. The flexibility and applicability of the new lifetime distribution were illustrated with a real data set and the results obtained revealed that the EG type-2 distribution provides the best fit among all the compared related distributions. We recommend the EG type-2 distribution for modelling complex data sets and hope that it would receive significant applications in the future.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

References

E. C. Pinheiro and S. L. P. Ferrari, “A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data,” Journal of Statistical Computation and Simulation, 2015.

R. C. Gupta, P. L. Gupta, and R. D. Gupta, “Modeling failure time data by Lehman alternatives,” Communications in Statistics—Theory and Methods, vol. 27, no. 4, pp. 887–904, 1998.

S. Nadarajah and S. Kotz, “The exponentiated type distributions,” Acta Applicandae Mathematica, vol. 92, no. 2, pp. 97–111, 2006.

S. Nadarajah, “The exponentiated Gumbel distribution with climate application,” Environmetrics, vol. 17, no. 1, pp. 13–23, 2006.

G. S. Mudholkar and D. K. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Transactions on Reliability, vol. 42, no. 2, pp. 299–302, 1993.

S. K. Ashour and M. A. Eltehiwy, “Exponentiated power Lindley distribution,” Journal of Advanced Research, vol. 6, no. 6, pp. 895–905, 2015.

E. Gumbel, “Les valeurs extrêmes des distributions statistiques,” Annales de l'Institut Henri Poincaré, vol. 5, no. 2, pp. 115–158, 1935.

E. J. Gumbel, “The return period of flood flows,” The Annals of Mathematical Statistics, vol. 12, pp. 163–190, 1941.

E. J. Gumbel, Statistical Theory of Extreme Values and Some Practical Applications, vol. 33 of Applied Mathematics, U.S. Department of Commerce, National Bureau of Standards, ASIN B0007DSHG4, Gaithersburg, Md, USA, 1st edition, 1954.

E. T. Lee and J. Wang, Statistical Methods for Survival Data Analysis, vol. 476, John Wiley & Sons, New York, NY, USA, 2003.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series, 1966.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, US Department of Commerce, National Institute of Standards and Technology, 2010.

K.-S. Song, “Rényi information, loglikelihood and an intrinsic distribution measure,” Journal of Statistical Planning and Inference, vol. 93, no. 1-2, pp. 51–69, 2001.

H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Breakthroughs in Statistics, pp. 610–624, Springer, New York, NY, USA, 1992.

N. Sugiura, “Further analysts of the data by akaike's information criterion and the finite corrections: further analysts of the data by akaike's,” Communications in Statistics-Theory and Methods, vol. 7, no. 1, pp. 13–26, 1978.

H. Bozdogan, “Model selection and Akaike's information criterion (AIC): the general theory and its analytical extensions,” Psychometrika, vol. 52, no. 3, pp. 345–370, 1987.

G. Schwarz, “Estimating the dimension of a model,” The Annals of Statistics, vol. 6, no. 2, pp. 461–464, 1978.

E. J. Hannan and B. G. Quinn, “The determination of the order of an autoregression,” Journal of the Royal Statistical Society B: Methodological, vol. 41, no. 2, pp. 190–195, 1979.

A. Z. Afify, Z. M. Nofal, and N. S. Butt, “Transmuted complementary weibull geometric distribution,” Pakistan Journal of Statistics and Operation Research, vol. 10, no. 4, pp. 435–454, 2014.