The Extended Exponential Weibull Distribution: Properties, Inference, and Applications to Real-Life Data

S. Mastor, Adam Braima; Ngesa, Oscar; Mung’atu, Joseph; Alfaer, Nada M.; Afify, Ahmed Z.

doi:https://doi.org/10.1155/2022/4068842

Complexity

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4068842 | https://doi.org/10.1155/2022/4068842

The Extended Exponential Weibull Distribution: Properties, Inference, and Applications to Real-Life Data

Adam Braima S. Mastor,¹Oscar Ngesa,²Joseph Mung’atu,³Nada M. Alfaer,⁴and Ahmed Z. Afify⁵

Academic Editor: M. De Aguiar

Received21 Jun 2022

Accepted21 Aug 2022

Published23 Sept 2022

Abstract

A novel version of the exponential Weibull distribution known as the extended exponential Weibull (ExEW) distribution is developed and examined using the Lehmann alternative II (LAII) generating technique. The new distributions basic mathematical properties are derived. The maximum likelihood estimation (MLE) technique is used to estimate the unknown parameters of the proposed distribution. The estimators’ performance is further assessed using the Monte Carlo simulation technique. Eventually, two real-world data sets are utilized to show the applicability of the new distribution.

1. Introduction

Analysis of lifetime data has seen applications in various fields, including health, business, engineering, finance, etc. [1, 2]. The main objective of such analysis is usually to model the distribution time to an event and/or the determinants of time to event of interest.

For modeling lifetime data, a variety of probability models are available, including log-logistic, beta, gamma, Weibull, exponential, and others. Furthermore, in many cases, these traditional models are inappropriate for modeling lifetime data [3, 4] necessitating the use of updated versions of current distributions [5]. New models bring up new possibilities for theoretical and practical researchers to solve real-world issues since they suit asymmetric and complicated random occurrences so well [6].

The Weibull distribution has been used to cope with several challenges in a wide range of survival data and to model lifespan data. The Weibull distribution, with its negatively and positively skewed density forms, is the primary option when modeling monotone hazard rates [7].

The parameters of this distribution’s tremendous flexibility allow for a range of techniques, all of which have the same key property.

The hazard rate is a monotone function that can be decreasing, constant, or growing [8]. The Weibull distribution is inappropriate for survival data with a nonmonotone failure rate function.

As a result, scientists explored for extensions and modifications of the Weibull distribution. Mudholkar and Srivastava [9] developed a three-parameter model by exponentiating the Weibull distribution, known as the exponentiated Weibull distribution. Silva et al. [10] proposed the beta modified Weibull distribution. Bourguignon et al. [11] introduced the Weibull-G family. Pinho et al. [12] developed the gamma-exponentiated Weibull distributions (GEW). Xie et al. [13] developed a three-parameter modified Weibull extension.

Lee et al. [14] introduced the beta Weibull model, which can be incorporated into data sets with nonmonotone and monotone hazard rate functions (hrf) and has the exponential, exponentiated Weibull, and exponentiated exponential models as submodels, among others. Carrasco et al. [15] defined and investigated a novel four-parameter modification of the Weibull distribution that can simulate bathtub-shaped failure rate forms.

Cordeiro et al. [16] developed the exponential Weibull distribution. Bander et al. [17] proposed the (P-A-L) extended Weibull distribution, which is a novel generalization of the three-parameter extended Weibull distribution. Almheidat et al. [18] suggested a generalization of the Weibull distribution, described by four parameters that specify the shape and scale properties. Famoye et al. [19] developed the Weibull-normal distribution and found that the Weibull-normal distribution can be unimodal or bimodal. Aldahlan [20] introduced a novel model called the inverse Weibull inverse exponential (IWIE) distribution. Hassan and Abd-Allah [21] proposed the exponentiated Weibull Lomax, a novel five-parameter model derived from the exponentiated Weibull-generated family.

Additional important generalized forms of the Weibull model are introduced by Korkmaz et al. [22–24], Abouelmagd et al. [25–27], Cordeiro et al. [28], Bhatti et al. [29], Nasir et al. [30], Alizadeh et al. [31], Afify et al. [32, 33], Hussein et al. [34], Mead et al. [35] and Nassar et al. [36].

In recent years, the technique of parameter induction has attracted a lot of attention, for example, Tahir and Nadarajah [37] reviewed the most common G families introduced in the last decade using the parameter induction technique. Tahir and Cordeiro [38] presented a survey for probability families formulated by parameter induction techniques, and they developed some new G families. Ahmad et al. [39] reviewed and presented a brief survey of recent advances in distribution theory with a focus on the parameter induction technique. Recently, Muse et al. [40] presented a survey of the log-logistic (LL) distribution and its generalizations by focusing on the new LL distributions formulated from the parameter induction technique.

The addition of one or more extra shape parameters to the parent distribution makes it more flexible, which is especially useful when analyzing tail features.

Based on the above discussion, this study proposes a new probability distribution called the extended exponential Weibull (ExEW) distribution, as a modification of the exponential Weibull distribution using the Lehmann-type II approach.

There are two main methods for developing the exponentiated family (EF) of distributions in the literature. These techniques are the Lehmann alternative I (LAI) technique, which has gotten a lot of attention, and the Lehmann alternative II (LAII) technique, which has received less attention.

The method of LAII aids in the derivation and understanding of its many features. According to Nadarajah and Tahir [37], the LAII technique is defined as follows:

If is the cumulative distribution function (cdf) and is the survival function (sf) of the existing distribution, then, by taking one minus the power of , the cdf of the LAII family or the exponential family (EF) follows as

According to (1), the probability density function (pdf) reduces to

The sf is defined as follows:

The hazard rate function (HRF) is

The reverse hazard function is

The cumulative hazard function H(t) is

The following can be used to briefly outline the article’s motivations: (i) using the LAII parameter induction approach, a tractable extension of the Weibull distribution will be introduced, which provides increasing, decreasing, and constant hazard rate forms. (ii) Createing a modified Weibull distribution with a more adaptable kurtosis compared to the standard Weibull model. (iii) Extending the parent Weibull distribution to one that is extended so that its density function can display symmetrical, asymmetrical, unimodal, J, and reversed-J shapes. (iv) Creating a more comprehensive model that can be used to represent different types of data in the fields of engineering, medicine, actuarial science, and other applied fields. This fact is demonstrated by modeling two real-life data sets from the engineering and medical disciplines, demonstrating its superiority as compared to other competing distributions. (v) lastly, our motivation stems from the desire to demonstrate how the inclusion of a single parameter may increase the application and tractability of the parent distribution. The remainder of the research develops and talks about each of those issues.

The remainder of the article is structured as follows: Section 2 discusses the basic lifetime functions of the proposed distribution and its submodels. Section 3 provides some of the mathematical properties of the ExEW distribution. The estimation of the ExEW parameters is investigated in Section 4. An extensive Monte Carlo simulation study is presented in Section 5. Section 6 presents two real-life data applications of the proposed distribution. In Section 7, concluding remarks are presented.

2. The ExEW Distribution

The ExEW distribution generalizes the exponential Weibull distribution. It is formulated by using the LAII approach.

Let , then the cdf of the ExEW distribution can be defined by applying (1) as follows:where , , and are shape parameters and is the scale parameter.

2.1. Submodels

The proposed ExEW distribution has some submodels that are often utilized in parametric survival modeling. Its submodels include the exponential (E), Weibull (W), exponentiated exponential (EE) [41], and the exponential Weibull (EW) distributions [16]. These submodels are listed in Table 1.

2.2. Probabilistic Functions for the ExEW Distribution

In this section, the pdf, hrf, and sf of the ExEW are presented. In addition to the above probabilistic functions, cumulative hrf (chrf) and reverse hrf (rhrf) are also formulated.(1)The pdf corresponding to (7) takes the form Figure 1 illustrates pdf shapes of the ExEW distribution for various choices of the parameters. The pdf of the ExEW distribution can be symmetrical, asymmetrical, unimodal, J, and reversed-J shapes.(2)The sf corresponding to (7) is as follows:(3)The hrf of the proposed distribution is expressed as Figure 2 shows the hrf which is clearly decreasing, increasing, constant, J-shaped, and reverse J-shaped.(4)The rhrf is written as follows:(5)The chrf is obtained as

3. Mathematical Properties

In this part, the ExEW distribution’s mathematical properties are discussed.

3.1. The Quantile Function

The quantile function (qf) is the reverse of the cdf and is significant in quantitative and statistical data analysis. A probability distribution can be described using either the qf or the cdf [42].

Let , for , then the qf of the ExEW distribution is given by reversing (7), thus for x, then the qf will be the solution of the equation,

3.2. Residual and Reverse Residual Life Functions

In reliability analysis and risk management, residual life has a wide range of applications. The residual life of the ExEW r.v. is

Furthermore, the reverse residual life of the ExEW distribution can be calculated as follows:

3.3. Moments

Moments can be used to analyze some of a distribution’s most important characteristics and properties, such as dispersion, tendency, kurtosis, and skewness.

The moment of an r.v. is

Proof. An r.v. X with pdf f(x), the moment, is written as follows:(8) is substituted for equation (17) and the result isApplying binomial expansion and further simplification, (18) becomesBut,By using McLaurin’s series expansionSubstituting (21) in (19), we haveLetWe getHence,Then,which is as follows:Substituting (27) in equation (22), we obtainAfter simplification, (28) becomesTable 2 reports the values of the first five moments, standard deviation (SD), coefficient of variation (CV), skewness (CS), and kurtosis (CK) of the ExEW model for various parameter values.
From Table 2, the ExEW distribution is quantitatively versatile in terms of mean and variance. As evidenced by its values, CS can be right skewed, almost symmetrical, or somewhat left-skewed. The CK values indicate whether the ExEW distribution is leptokurtic, platykurtic, or mesokurtic. All of these features point to the ExEW distribution’s versatility, which makes it an ideal choice for modeling.

3.4. Moment Generating Function (mgf)

The mgf of the ExEW distribution is written as follows:

Using the results from Subsection 3.3, we can get the closed-form for the mgf. Then, the mgf of ExEW distribution reduces to

4. Maximum Likelihood Estimation

Maximum likelihood (ML) is used to estimate the unknown parameters of the ExEW distribution using a full sample.

If denote a random sample from the ExEW distribution with an unknown parameter vector , then the ML function follows as

Then, the log-likelihood function reduces to

The parameter estimates are produced by performing a partial derivative of with respect to each parameter, as follows:

The unknown parameters can be calculated via resetting the aforementioned equations to zero and calculating them all at once. These equations can also be numerically solved using statistical software (for example, the adequacy model package in R software) or an iterative technique such as the Newton–Raphson algorithm.

Because of the predicted information matrix is too complicated to set confidence intervals for the parameters, the observed information matrix is used.

The following is how the information matrix is obtained:whereas the regularity criteria are satisfied and the parameters are within the interior of the parameter space but not on the boundary; converges in distribution to , where is the predicted. When is substituted by the observed information matrix assessed at , the asymptotic behavior remains true. To construct two-sided 95% confidence interval for model parameters, we use the asymptotic multivariate normal distribution , where is the significance level.

5. Simulation Study

In this part, a comprehensive numerical inspection using Monte Carlo simulations is achieved to evaluate the capability of the ML estimates (MLEs) for the ExEW model. The absolute biases (AB), root mean square errors (RMSEs), and coverage probability (CP) are calculated for different small and large samples and parameter settings to evaluate the performance of MLEs. To produce random samples from the ExEW, the qf (13) is employed. With , and 200, the simulation experiments are repeated times. For set I, , , set II: , , set III: , , and set IV: , .

The following equations are used to calculate the AB, RMSE, and CP for the estimates.where .

The CP is the proportional of times the confidence interval, which is given by

The AB and RMSE values of the parameters , and for various sample sizes are shown in Tables 3 and 4. The visual comparisons of these results are shown in Figures 3–10. The findings show that the RMSE decreases as the sample size grows until it hits zero. Furthermore, the AB decreases as sample size grows. As a result, the MLEs and their asymptotic features can be used to build confidence ranges even for tiny sample numbers. Additionally, the confidence intervals’ CPs are quite close to the nominal 95 percent level.

6. Applications to Real-Life Data

To illustrate the applicability of the ExEW distribution, we analyze and compare the fitting of the ExEW distribution with other competing models by using two real-life data sets. The ExEW distribution is compared to submodels such as the W, EE [41], and EW distributions [16], and other common lifetime distributions including the log-logistic (LL), beta Weibull (BW) [14], beta extended Weibull (BEW) [43], modified beta Weibull (MBW) [44], and tan-log-logistic (TanLL) distributions [45].

The competing models’ pdfs are as follows:(1)The pdf of the W distribution is(2)The pdf of EE distribution is(3)The pdf of EW distribution is(4)The pdf of the LL distribution is(5)The pdf of BW distribution is(6)The pdf of BEW distribution is(7)The pdf of MBW is(8)The pdf of TanLL distribution is

To specify which statistical distribution best fits the two data, a variety of analytical measures are applied, such as the Bayesian information criterion (BIC), the Akaike information criterion (AIC), consistent AIC (CAIC), and the Hannan–Quinn information criterion (HQIC). Moreover, goodness-of-fit measures like the log-likelihood are also adopted.

6.1. Likelihood Ratio Tests

The ExEW distribution has some submodels, including the W, EE, and EW distributions. As a result, the likelihood ratio test (LRT) is used to evaluate the following hypotheses:(1), and , this means that the sample comes from the W distribution. , and , this means that the sample comes from the ExEW distribution.(2), this indicates that the sample size comes from the EE distribution. , this indicates that the sample size comes from the ExEW distribution.(3), this means that the sample comes from the EW distribution. , this means that the sample comes from the ExEW distribution.

The LRT is written aswhere denotes the constrained MLEs for the null hypothesis , whereas denotes the unconstrained MLEs for the alternative hypothesis . The LRT follows the chi-square distribution with degrees of freedom when the null hypothesis is true.

The null hypothesis is rejected if the p value is less than 5%.

6.2. Application to Airplane Windshield Data

The airplane windshield data consists of 84 observations. Ramos et al. [46] recently studied the data set.. The data set is reported in Table 5, and its descriptive statistical analysis is shown in Table 6 which indicates that the skewness coefficient has a positive value, the data is right skewed. Due to the kurtosis having negative value, the data are platykurtic.

Figure 11 illustrates the TTT transform plot with a concavity pattern, indicating that the data has an increasing hazard rate shape. This confirms that the hrf in Figure 2 is appropriate for analyzing this data.

The MLEs of the parameters of the fitted models, as well as the corresponding standard errors, are shown in Table 7. At the 5% significance level, all of the ExEW parameters are significant. The ExEW model fits the airplane windshield data better than its submodels and other rival distributions. Table 8 shows that the ExEW has the highest log-likelihood and the lowest CAIC, HQIC, BIC, and AIC values as compared to the other models. Although the ExEW model provides the greatest fit to the data, the W distribution is a suitable option since its fit values are more similar to the ExEW model. Table 9.

Figure 12 illustrates the fitted density shapes for competitive models, demonstrating that the ExEW distribution fits aircraft windshields better.

6.3. Application to COVID-19 Fatality Rate Data

The second set of data about the COVID-19 fatality rate from Mexico contains 108 days, and it collected between March 4 and July 20, 2020. It is available at https://covid19.who.int. The data are recently studied by Almongy et al. [47]. Table 10 lists the data observations and Table 11 shows the descriptive statistical analysis of the data. Because the skewness coefficient has a positive value, the data are right skewed. The data are platykurtic since the kurtosis is smaller than three. Figure 13 displays the TTT plot with a concavity shape, indicating that the data have an increasing failure rate. This demonstrates that the ExEW distribution is appropriate for analyzing this type of data.

Table 12 shows the MLEs of the parameters of the fitted models, together with the standard errors in brackets. At the 5% significance level, all of the ExEW parameters are significant. Table 13 provides the analytical measures of competing distributions and shows that the ExEW provides the best fit to the data. Table 14) shows the LRT tests for the proposed model and its submodels.

The fitted densities for competing models are depicted in Figure 14, demonstrating that the ExEW distribution fits the COVID-19 mortality rate data better.

7. Conclusion

The mathematical and statistical properties of the ExEW are proposed and described in this study. The ExEW distribution includes several known sub models as special cases. Some mathematical features of the new model are derived. The ExEW parameters are estimated via the maximum likelihood method, and the estimators’ behaviour is evaluated via Monte Carlo simulations. Based on goodness-of-fit statistics and analytical performance measurements, the ExEW model fits two real-world data sets better than its submodels and other typical parametric survival models. As a consequence, we conclude that the ExEW distribution is the most fitting model among the distributions studied and is a good contender for modeling lifetime events.

There are various possible future extensions to this study. The presence of explanatory factors and long-term survival, for example, is common in practice. In addition, a survival regression model that works for both whole and incomplete (truncation, censored) data might be useful. Hence, our approach can be investigated more in various settings.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank the Pan African University, Institute for Basic Sciences, Technology, and Innovation (PAUSTI), Nairobi, Kenya. This study was funded by the Taif University Researchers Supporting Project (TURSP-2020/316), Taif University, Taif, Saudi Arabia.

References

Z. Ahmad, E. Mahmoudi, G. G. Hamedani, and O. Kharazmi, “New methods to define heavy-tailed distributions with applications to insurance data,” Journal of Taibah University for Science, vol. 14, no. 1, pp. 359–382, 2020.
View at: Publisher Site | Google Scholar
S. Nasiru, P. N. Mwita, and O. Ngesa, “Exponentiated generalized power series family of distributions,” Annals of Data Science, vol. 6, no. 3, pp. 463–489, 2019.
View at: Publisher Site | Google Scholar
S. Nasiru, P. N. Mwita, and O. Ngesa, “Exponentiated generalized exponential Dagum distribution,” Journal of King Saud University Science, vol. 31, no. 3, pp. 362–371, 2019.
View at: Publisher Site | Google Scholar
A. H. Muse, S. Mwalili, O. Ngesa, S. J. Almalki, and G. A. Abd-Elmougod, “Bayesian and classical inference for the generalized log-logistic distribution with applications to survival data,” Computational Intelligence and Neuroscience, vol. 2021, Article ID 5820435, 24 pages, 2021.
View at: Publisher Site | Google Scholar
Y. L. Tung, Z. Ahmad, O. Kharazmi, C. B. Ampadu, E. H. Hafez, and S. A. Mubarak, “On a new modification of the Weibull model with classical and Bayesian analysis,” Complexity, vol. 2021, Article ID 5574112, 19 pages, 2021.
View at: Publisher Site | Google Scholar
O. S. Balogun, M. Z. Iqbal, M. Z. Arshad, A. Z. Afify, and P. E. Oguntunde, “A new generalization of Lehmann type-II distribution: theory, simulation, and applications to survival and failure rate data,” Scientific African, vol. 12, 2021, e00790.
View at: Publisher Site | Google Scholar
W. He, Z. Ahmad, A. Z. Afify, and H. Goual, “The arcsine exponentiated-x family: validation and insurance application,” Complexity, vol. 2020, Article ID 8394815, 18 pages, 2020.
View at: Publisher Site | Google Scholar
T. V. F. SantanaSantana, E. M. M. OrtegaOrtega, and G. M. Cordeiro, “Generalized beta Weibull linear model: estimation, diagnostic tools and residual analysis,” Journal of Statistical Theory and Practice, vol. 13, no. 1, pp. 16–23, 2019.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
G. O. Silva, E. M. M. OrtegaOrtega, and G. M. Cordeiro, “The beta modified Weibull distribution,” Lifetime Data Analysis, vol. 16, no. 3, pp. 409–430, 2010.
View at: Publisher Site | Google Scholar
M. Bourguignon, R. B. Silva, and G. M. Cordeiro, “The Weibull-G family of probability distributions,” Journal of Data Science, vol. 12, no. 1, pp. 53–68, 2021.
View at: Publisher Site | Google Scholar
L. Pinho, G. Cordeiro, and J. Nobre, “The gamma-exponentiated Weibull distribution,” Journal of Statistical Theory and Applications, vol. 11, no. 4, pp. 379–395, 2012.
View at: Google Scholar
M. Xie, Y. Tang, and T. N. Goh, “A modified weibull extension with bathtub-shaped failure rate function,” Reliability Engineering & System Safety, vol. 76, no. 3, pp. 279–285, 2002.
View at: Publisher Site | Google Scholar
C. Lee, F. Famoye, and O. Olumolade, “Beta-Weibull distribution: some properties and applications to censored data,” Journal of Modern Applied Statistical Methods, vol. 6, no. 1, pp. 173–186, 2007.
View at: Publisher Site | Google Scholar
J. M. Carrasco, E. M. Ortega, and G. M. Cordeiro, “A generalized modified Weibull distribution for lifetime modeling,” Computational Statistics & Data Analysis, vol. 53, no. 2, pp. 450–462, 2008.
View at: Publisher Site | Google Scholar
G. M. Cordeiro, E. M. Ortega, and A. J. Lemonte, “The exponential-Weibull lifetime distribution,” Journal of Statistical Computation and Simulation, vol. 84, no. 12, pp. 2592–2606, 2014.
View at: Publisher Site | Google Scholar
A.-Z. Bander, P. R. D. Marinho, A. Fattah, and G. M. Cordei̇re, “The (PAL) extended Weibull distribution: a new generalization of the weibull distribution,” Hacettepe journal of mathematics and statistics, vol. 45, no. 3, pp. 851–868, 2016.
View at: Google Scholar
M. Almheidat, C. Lee, and F. Famoye, “A generalization of the Weibull distribution with applications,” Journal of Modern Applied Statistical Methods, vol. 15, no. 2, pp. 788–820, 2016.
View at: Publisher Site | Google Scholar
F. Famoye, E. Akarawak, and M. Ekum, “Weibull-normal distribution and its applications,” Journal of Statistical Theory and Applications, vol. 17, no. 4, pp. 719–727, 2018.
View at: Publisher Site | Google Scholar
M. A. Aldahlan, “The inverse Weibull inverse exponential distribution with application,” International Journal of Contemporary Mathematical Sciences, vol. 14, no. 1, pp. 17–30, 2019.
View at: Publisher Site | Google Scholar
A. S. Hassan and M. Abd-Allah, “Exponentiated Weibull-Lomax distribution: properties and estimation,” Journal of Data Science, vol. 16, no. 2, pp. 277–298, 2021.
View at: Publisher Site | Google Scholar
M. Ç. Korkmaz, E. Altun, H. M. Yousof, and G. G. Hamedani, “The hjorth’s IDB generator of distributions: properties, characterizations, regression modeling and applications,” Journal of Statistical Theory and Applications, vol. 19, no. 1, p. 59, 2020.
View at: Publisher Site | Google Scholar
M. C. Korkmaz and A. I. Genc, “A lifetime distribution based on a transformation of a two-sided power variate,” Journal of Statistical Theory and Applications, vol. 14, no. 3, pp. 265–280, 2015.
View at: Publisher Site | Google Scholar
M. Ç. Korkmaz, H. M. Yousof, M. Alizadeh, and G. Hamedani, “The Topp-Leone generalized odd log-logistic family of distributions: properties, characterizations and applications,” Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, vol. 68, no. 2, pp. 1506–1527, 2019.
View at: Publisher Site | Google Scholar
T. H. M. Abouelmagd, M. S. Hamed, L. Handique et al., “A new class of distributions based on the zero truncated Poisson distribution with properties and applications,” The Journal of Nonlinear Science and Applications, vol. 12, no. 03, pp. 152–164, 2018.
View at: Publisher Site | Google Scholar
T. H. M. Abouelmagd, M. S. Hamed, G. G. Hamedani et al., “The zero truncated Poisson Burr X family of distributions with properties, characterizations, applications, and validation test,” The Journal of Nonlinear Science and Applications, vol. 12, no. 05, pp. 314–336, 2019.
View at: Publisher Site | Google Scholar
T. H. M. Abouelmagd, S. Al-mualim, M. Elgarhy, A. Z. Afify, and M. Ahmad, “Properties of the four-parameter Weibull distribution and its applications,” Pak. J. Statist, vol. 33, no. 6, pp. 449–466, 2017.
View at: Google Scholar
G. M. Cordeiro, E. Altun, M. C. Korkmaz, R. R. Pescim, A. Z. Afify, and H. M. Yousof, “The xgamma family: censored regression modelling and applications,” Revstat-Statistical Journal, vol. 18, no. 5, pp. 593–612, 2020.
View at: Google Scholar
F. A. Bhatti, G. G. Hamedani, M. C. Korkmaz, G. M. Cordeiro, H. M. Yousof, and M. Ahmad, “On Burr III Marshal Olkin family: development, properties, characterizations and applications,” Journal of Statistical Distributions and Applications, vol. 6, no. 1, pp. 12–21, 2019.
View at: Publisher Site | Google Scholar
M. A. Nasir, M. C. Korkmaz, F. Jamal, and H. M. Yousof, “On a new Weibull Burr XII distribution for lifetime data,” Sohag Journal of Mathematics, vol. 5, no. 2, pp. 47–56, 2018.
View at: Publisher Site | Google Scholar
M. Alizadeh, M. . Ç. Korkmaz, J. A. Almamy, and A. Ahmed, “Another odd log-logistic logarithmic class of continuous distributions,” Istatistikçiler Dergisi: Istatistik ve Aktüerya, vol. 11, no. 2, pp. 55–72, 2018.
View at: Google Scholar
A. Z. Afify, H. Al-Mofleh, H. M. Aljohani, and G. M. Cordeiro, “The Marshall-Olkin-Weibull-H family: estimation, simulations, and applications to COVID-19 data,” Journal of King Saud University Science, vol. 34, no. 5, 2022, 102115.
View at: Publisher Site | Google Scholar
A. Z. Afify, A. I. Shawky, M. Shawky, and M. Nassar, “A new inverse Weibull distribution: properties, classical and bayesian estimation with applications,” Kuwait Journal of Science, vol. 48, no. 3, 2021.
View at: Publisher Site | Google Scholar
E. A. Hussein, H. M. Aljohani, A. Z. Aljohani, and A. Z. Afify, “The extended Weibull-Fréchet distribution: properties, inference, and applications in medicine and engineering,” AIMS Mathematics, vol. 7, no. 1, pp. 225–246, 2021.
View at: Publisher Site | Google Scholar
M. E. Mead, A. Afify, and N. S. Butt, “The modified kumaraswamy Weibull distribution: properties and applications in reliability and engineering sciences,” Pakistan Journal of Statistics and Operation Research, pp. 433–446, 2020.
View at: Publisher Site | Google Scholar
M. Nassar, A. Z. Afify, S. Dey, and D. Kumar, “A new extension of Weibull distribution: properties and different methods of estimation,” Journal of Computational and Applied Mathematics, vol. 336, pp. 439–457, 2018.
View at: Publisher Site | Google Scholar
M. H. Tahir and S. Nadarajah, “Parameter induction in continuous univariate distributions: well-established G families,” Anais da Academia Brasileira de Ciências, vol. 87, no. 2, pp. 539–568, 2015.
View at: Publisher Site | Google Scholar
M. H. Tahir and G. M. Cordeiro, “Compounding of distributions: a survey and new generalized classes,” Journal of Statistical Distributions and Applications, vol. 3, no. 1, pp. 13–35, 2016.
View at: Publisher Site | Google Scholar
Z. Ahmad, G. G. Hamedani, and N. S. Butt, “Recent developments in distribution theory: a brief survey and some new generalized classes of distributions,” Pakistan Journal of Statistics and Operation Research, pp. 87–110, 2019.
View at: Publisher Site | Google Scholar
A. H. Muse, S. M. Mwalili, and O. Ngesa, “On the log-logistic distribution and its generalizations: a survey,” International Journal of Statistics and Probability, vol. 10, no. 3, p. 93, 2021.
View at: Publisher Site | Google Scholar
S. Nadarajah, The Exponentiated Exponential Distribution: A Survey, Springer, USA, 2011.
N. Midhu, P. Sankaran, and N. Unnikrishnan NairUnnikrishnan Nair, “A class of distributions with the linear mean residual quantile function and it’s generalizations,” Statistical Methodology, vol. 15, pp. 1–24, 2013.
View at: Publisher Site | Google Scholar
G. M. Cordeiro, E. Ortega, and G. Silva, “The beta extended Weibull family,” Journal of Probability and Statistical Science, vol. 10, no. 10, pp. 15–40, 2012.
View at: Google Scholar
S. Nadarajah, M. Teimouri, and S. H. Shih, “Modified beta distributions,” Sankhya B, vol. 76, no. 1, pp. 19–48, 2014.
View at: Publisher Site | Google Scholar
A. H. Muse, A. H. Tolba, E. Fayad, O. A. Abu Ali, M. Nagy, and M. Yusuf, “Modelling the covid-19 mortality rate with a new versatile modification of the log-logistic distribution,” Computational Intelligence and Neuroscience, vol. 2021, Article ID 8640794, 14 pages, 2021.
View at: Publisher Site | Google Scholar
M. W. A. Ramos, P. R. D. Marinho, R. V. da Silva, and G. M. Cordeiro, “The exponentiated Lomax Poisson distribution with an application to lifetime data,” Advances and Applications in Statistics, vol. 34, no. 2, p. 107, 2013.
View at: Google Scholar
H. M. Almongy, E. M. Almetwally, H. M. Aljohani, A. S. Alghamdi, and E. Hafez, “A new extended Rayleigh distribution with applications of covid-19 data,” Results in Physics, vol. 23, Article ID 104012, 2021.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Adam Braima S. Mastor 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.

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