Properties and Applications of the Modified Kies–Lomax Distribution with Estimation Methods

Alsubie, Abdelaziz

doi:https://doi.org/10.1155/2021/1944864

Journal of Mathematics

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Research Article | Open Access

Volume 2021 | Article ID 1944864 | https://doi.org/10.1155/2021/1944864

Properties and Applications of the Modified Kies–Lomax Distribution with Estimation Methods

Abdelaziz Alsubie¹

Academic Editor: Barbara Martinucci

Received30 Aug 2021

Revised15 Oct 2021

Accepted25 Oct 2021

Published28 Nov 2021

Abstract

The present study introduces a new three-parameter model called the modified Kies–Lomax (MKL) distribution to extend the Lomax distribution and increase its flexibility in modeling real-life data. The MKL distribution, due to its flexibility, provides left-skewed, symmetrical, right-skewed, and reversed-J shaped densities and increasing, unimodal, decreasing, and bathtub hazard rate shapes. The MKF density can be expressed as a linear mixture of Lomax densities. Some basic mathematical properties of the MKF model are derived. Its parameters are estimated via six estimation algorithms. We explore their performances using detailed simulation results, and the partial and overall ranks are provided for the measures of absolute biases, mean square errors, and mean relative errors to determine the best estimation method. The results show that the maximum product of spacings and maximum likelihood approaches are recommended to estimate the MKL parameters. Finally, the flexibility of the MKL distribution is checked using two real datasets, showing that it can provide close fit to both datasets as compared with other competing Lomax models. The three-parameter MKL model outperforms some four-parameter and five-parameter rival models.

1. Introduction

The Lomax distribution has several applications in different applied fields such as biological sciences, income and wealth inequality, engineering, reliability, and actuarial sciences. Chahkandi and Ganjali [1] showed that the Lomax model belongs to decreasing failure rate family. Detailed information about the Lomax distribution can be explored in [2, 3].

The procedure of expanding classical distributions by adding new shape parameters is well-known technique in statistical literature. Recently, various extensions of the Lomax distribution have been constructed using well-known generators to increase its flexibility in modeling various types of data. The new added shape parameters are important to provide skewness and to increase tail weights as well as to enhance the flexibility to model monotonic and nonmonotonic hazard rates particularly if the baseline hazard rate is only monotonic.

Some of the most notable extensions of the Lomax model are the exponentiated-Lomax [4], Marshall–Olkin Lomax [5], Kumaraswamy–Lomax [6], Weibull–Lomax [7], exponentiated half logistic-Lomax [8], and Fréchet Topp–Leone Lomax distributions [9]. On the contrary, some authors studied the estimation of the Lomax distribution using the Bayesian approach. For example, Okasha [10] studied the E-Bayesian estimation for the Lomax distribution based on type-II censored data and Hassan and Zaky [11] studied the entropy Bayesian estimation for the Lomax distribution based on records. For more details about Bayesian inference for probability distributions, the interested reader can refer to [12–19].

Recently, there have been a great interest among statisticians to develop new families or generators of distributions by adding extra shape parameter(s) to the well-known classical distributions. One of the most recent generators is called modified Kies-G (MK-G) family due to Al-Babtain et al. [20]. Consider any baseline cumulative distribution function (CDF) with a vector of parameters , denoted by ; then, the CDF of the MK-G class with additional shape parameter takes the form

The corresponding probability density function (PDF) of (1) reduces to

The hazard rate function (HRF) of the MK-G class has the form

In this paper, a new three-parameter model called the modified Kies–Lomax (MKL) distribution is introduced to extend the Lomax model and improve its flexibility in fitting real-life data in various applied areas. The MKL distribution is generated by replacing the baseline Lomax distribution in the MK-G family. Shafiq et al. [21] adopted the MK-G family to define the MK-Fréchet distribution.

The MKL distribution gives a better fit than some existing rival extensions of the Lomax model which have three, four, and five parameters. The MKL distribution, with three parameters, provides decreasing, unimodal, increasing, bathtub, J shape, and reversed-J shaped hazard functions, as well as right-skewed, symmetrical, left-skewed, and reversed-J shaped densities. Further important aim of the current paper is to explore the estimation of the MKL parameters using classical methods such as maximum likelihood, Anderson–Darling, Cramér–von Mises, least-squares, weighted least-squares, and maximum product of spacings. The performance of these methods are explored via simulation results based on the average values of absolute biases (AVBs), mean square error (MSEs), and mean relative errors of the estimates (MREs). Besides, these measures are ordered using the partial and overall ranks to compare the performances of the proposed estimators and to determine the best estimation method for the MKL parameters. The findings show that the maximum product of spacings and maximum likelihood methods are recommended to estimate the model parameters.

The rest of the article is structured as follows. The MKL distribution is introduced in Section 2. Its basic distributional properties are determined in Section 3. Six estimation approaches are introduced in Section 4. Simulation results for the six estimation methods are given in Section 5. The empirical importance MKL model is checked using two real datasets in Section 6. Some concluding remarks are presented in Section 7.

2. The MKL Distribution

The Lomax distribution with shape parameter, , and scale parameter, , is specified by the CDF

The PDF of the Lomax distribution reduces to

The CDF of the MKL distribution follows, by inserting (4) in (1), as

Its PDF is determined by inserting (4) and (5) in equation (2) as follows:

The survival function (SF) and HRF of the MKL distribution take the forms

Some possible shapes for the PDF and HRF of the MKL distribution are displayed in Figures 1 and 2, respectively. These plots show that the MKL distribution provides right-skewed, symmetrical, left-skewed, and reversed-J shaped densities, as well as decreasing, unimodal, increasing, bathtub, J shape, and reversed-J shaped hazard functions.

3. Mathematical Properties

3.1. Quantile Function

The quantile function (QF) of the MKL distribution is obtained by determining the inverse function of the MKL CDF (6) as

The three quartiles of the MKL distribution can be obtained by setting , and 0.75, respectively, in (9).

Let ; then, the QF can be used directly in generating random data from the MKL distribution as follows:

3.2. Mixture Representation

Here, we derive a useful linear representation for the PDF (7) of the MKL distribution. Al-Babtain et al. [22] derived a linear representation of the MK-G CDF (1) as follows:where is the CDF of the exp-G family with power parameter and the term is given by

Then, the linear representation of the CDF of the MKL distribution takes the formwhere .

By differentiating the previous equation, we obtain a linear representation of the MKL PDF aswhere and is the PDF of Lomax distribution with shape parameter mλ and scale parameter . Hence, the properties of the proposed MKL model, such as moments, follow from those of the Lomax distribution.

3.3. Moments

The moments of the MKL distribution has the form

Then, the moments of the MKL distribution follows, from the moments of Lomax model, as

Setting , and 4, respectively, we obtain the first four-moments of the MKL distribution.

The moment generating function of the MKL distribution follows directly from the previous formula as

The characteristic function of the MKL distribution follows from the last equation by setting .

3.4. Order Statistics

The PDF and CDF of the order statistic of the MKL distribution arewhere is a hypergeometric function.

The joint PDF of the and order d (for ) by

Hence, the joint PDF of two order statistics from the MKL distribution takes the form

3.5. Incomplete Moments

The incomplete moment of the MKL distribution is given bywhere .

The important application of the first incomplete moment is related to the Bonferroni and Lorenz curves defined by and , respectively, where can be evaluated numerically by equation (22) for a given probability . These curves are very useful in economics, demography, insurance, engineering, and medicine. Another application of the first incomplete moment refers to the mean residual life (MRL) and the mean waiting time given by and , respectively.

4. Methods of Estimation

Here, we discuss the estimation of the MKL parameters by different approaches of estimation including the maximum likelihood estimates (MLE), Anderson–Darling estimates (ADE), maximum product of spacings estimates (MPSE), least-squares estimates (LSE), Cramér–von Mises estimates (CVME), and weighted least-squares estimates (WLSE). The parameters of several models have been estimated using classical methods. For example, quasi xgamma-geometric [23], Weibull Marshall–Olkin Lindley [24], heavy-tailed exponential [25], transmuted Burr X exponential [26], and discrete Poisson–Lindley and discrete Lindley [22], among many others.

4.1. Maximum Likelihood Estimation

Let be a random sample of size from the PDF (7); then, the log-likelihood function reduces to

By differentiating equation (22) with respect to , , and , respectively, and equating to zero, we obtain

Solving the previous equations, we obtain estimators of the MKL parameters by the MLEs. The previous equations cannot be solved explicitly; hence, the numerical techniques can be used maximize the log-likelihood function to get the MLEs using several programs such as the R, SAS, Mathcad.

4.2. Ordinary Least-Squares and Weighted Least-Squares Estimators

Let be the order statistics of a random sample of size from the MKL distribution. Hence, we can obtain the LSE of the MKL parameters by minimizing the following equation:

The LSE of the MKL parameters can also be obtained by solving the following equations:where

The WLSE of the MKL parameters are obtained by minimizing the following equation:

Moreover, the WLSE of the MKL parameters are also obtained by solving the following equations:where for are defined in (26)–(28).

4.3. Anderson–Darling Estimation

The ADE of the MKL parameters can be obtained by minimizing

The ADE are also be calculated by solving the following equations:where , for , are defined in (26)–(28).

4.4. Cramér–von Mises Estimation

The CVME of MKL parameters are obtained by minimizing the following equation:or by solving the following equationswhere , for , were defined in (26)–(28).

4.5. Maximum Product of Spacings’ Estimation

The maximum product of spacings (MPS) method is used to estimate the parameters of continuous models as a good alternative to the maximum likelihood method. The uniform spacings of a random sample of size from the MKL distribution is defined bywhere are the uniform spacings, , , and . The MPSE of the MKL parameters can be obtained by maximizingwith respect to , , and . Furthermore, the MPSE of the MKL parameters can also be calculated by solvingwhere , for , were defined in (26)–(28).

5. Simulation Results

Now, we explore and compare the performance of the introduced methods in estimating the MKL parameters based on simulation results. Several sample sizes, , and several parametric values for the parameters, , , and , are considered to generate random samples from the MKL distribution using its QF and to determine the AVBs’, MSEs’, and MREs’ measures using the R program. The AVBs’, MSEs’, and MREs’ measures are calculated using the following equations:where .

Tables 1–6 report detailed simulation results for the six estimation methods including AVBs, MSEs, and MREs for the parameters of the MKL distribution. These tables also present the ordering of the simulation measures, AVBs, MSEs, and MREs, based on partial and overall ranks which are given for each combination and sample size. In conclusion, the estimates, from six estimation methods, of the MKL parameters are quite good and close to their true values, showing small and decreasing AVBs, MSEs, and MREs in all parameter combinations. Moreover, all methods have the consistency property, i.e., the MREs and MSEs decrease as sample size, , increases, for all parameter combinations.

Table 7 illustrates the partial and overall ranks of the six estimators. Table 7 shows that the performance ordering of the estimators are MPSE, MLE, ADE, LSE, WLSE, and CVME, respectively. Based on simulation results and the ranks of the estimators, we conclude that the MPSE outperforms other estimators with an overall score of 39.5 and can confirm the superiority of MPSE and MLE for estimating the parameters of the MKL distribution.

6. Application

In this section, we consider two real-life datasets to show the applicability and flexibility of the new MKL distribution. The first dataset is studied by [9, 27], and it represents successive failures of air conditioning systems for fleet of 13 Boeing 720 jet airplanes. The data are 50, 11, 14, 130, 16, 9, 57, 102, 15, 5, 14, 11, 10, 57, 261, 12, 51, 44, 239, 18, 33, 209, 181, 37, 111, 82, 230, 71, 80, 152, 46, 386, 182, 118, 44, 120, 141, 210, 11, 14, 186, 29, 14, 102, 4, 118, 72, 270, 31, 14, 88, 130, 15, 326, 18, 130, 35, 5, 18, 1, 79, 216, 61, 41, 58, 61, 60, 48, 56, 87, 98, 54, 12, 5, 22, 43, 201, 84, 27, 20, 206, 310, 220, 34, 29, 26, 90, 163, 208, 1, 134, 3, 184, 20, 35, 29, 65, 49, 7, 104, 21, 16, 283, 5, 22, 120, 66, 57, 139, 67, 3, 27, 76, 14, 100, 32, 9, 12, 5, 18, 3, 14, 39, 18, 13, 7, 34, 36, 79, 33, 246, 25, 59, 34, 18, 44, 191, 100, 7, 156, 320, 63, 22, 23, 62, 188, 11, 104, 31, 106, 153, 59, 42, 46, 26, 54, 62, 39, 36, 30, 23, 95, 16, 101, 71, 52, 208, 24, 12, 70, 230, 97, 254, 438, 502, 493, 487, and 603. The second dataset is studied by [8], and it contains 100 observations about waiting times (in minutes) in a bank before the customers receive their services. The data are 0.8, 4.3, 0.8, 1.5, 1.3, 1.8, 2.7, 1.9, 2.1, 1.9, 2.6, 3.1, 3.5, 2.9, 3.2, 3.3, 4, 3.6, 4.2, 4.1, 4.2, 4.4, 4.3, 4.9, 4.4, 7.1, 4.6, 7.1, 4.7, 7.1, 4.7, 6.9, 4.8, 5.0, 4.9, 5.3, 8.9, 5.5, 5.7, 8.8, 5.7, 6.1, 6.2, 9.5, 6.2, 6.3, 6.2, 6.7, 7.1, 10.7, 7.4, 7.7, 7.6, 8, 8.6, 8.2, 8.6, 8.8, 11.0, 8.9, 8.6, 9.6, 11.9, 12.5, 13.0, 13.3, 13.6, 13.7, 33.1, 9.7, 9.8, 11.1, 10.9, 11.0, 17.3, 11.2, 11.5, 12.4, 13.1, 11.2, 20.6, 12.9, 27, 31.6, 23, 13.9, 21.9, 18.9, 19.0, 14.1, 18.2, 15.4, 17.3, 15.4, 18.4, 18.1, 19.9, 21.3, 38.5, and 21.4.

We compare the proposed MKL model with some competing models based on some discrimination measures and criteria such as Akaike information (AIC), consistent Akaike information (CAIC), Bayesian information (BIC), and Hannan–Quinn information (HQIC). The competing distributions include odd Lomax Lomax (OLL) [28], Lomax–Weibull (LW) [29], Lomax-exponential (LE) [29], beta exponentiated-Lomax (BExL) [30], beta-Lomax (BL) [30], exponentiated-Lomax (ExL) [30], Zubair–Lomax (ZL) [31], and Lomax (L) distributions.

The estimates of the model parameters and their SEs (standard errors), using the maximum likelihood, along with discrimination measures are displayed in Tables 8 and 9, respectively, for the two datasets. Furthermore, the estimates of the MKL parameters are obtained using different estimators and listed, for both datasets, in Tables 10 and 11, respectively. These tables also display the values of Anderson–Darling (A), Cramér–von Mises (W), and Kolmogorov–Smirnov (KS) statistic with its value (KS-PV) for both datasets.

The relative histogram with the fitted MKL density along with fitted CDF, SF, and P-P plots for the MKL model is displayed in Figures 3 and 4 for both datasets. These plots support the results in Tables 8 and 9.

It is shown, from Tables 8 and 9, that the MKL distribution has the lowest values for goodness-of-fit criteria for both datasets among all fitted competing models. That is, it provides close fit for the data, and hence, it could be chosen as the best distribution for the two datasets.

From Tables 10 and 11 and based on the values of KS statistic, we conclude that all estimation methods perform well in estimating the parameters of the MKL distribution for the two datasets.

7. Concluding Remarks

In this study, a new three-parameter extension of the Lomax distribution called the modified Kies–Lomax (MKL) distribution has been studied. The MKL distribution provides flexible hazard rate and density functions which can have important forms, depending on its shape parameters, including increasing, unimodal, decreasing, and bathtub hazard rate shapes, and positive-skewed, symmetrical, negative-skewed, and reversed-J shaped densities. Furthermore, its density can be viewed as a linear mixture of Lomax distribution. The MKL parameters have been estimated via six estimation approaches, called, maximum likelihood, least-squares and weighted least-squares, Cramér–von Mises, maximum product of spacings, an Anderson–Darling. The simulation study has been illustrated that maximum product of spacings is the best performing method in terms of partial and overall ranks for absolute biases, mean square errors, and mean relative errors of the estimates. Two real data applications show that all estimators perform well for both datasets. Furthermore, the MKL distribution shows its flexibility in modeling successive failures of air conditioning systems and waiting times’ datasets as compared with some rival Lomax extensions.

Data Availability

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

M. Chahkandi and M. Ganjali, “On some lifetime distributions with decreasing failure rate,” Computational Statistics & Data Analysis, vol. 53, no. 12, pp. 4433–4440, 2009.
View at: Publisher Site | Google Scholar
B. C. Arnold, Pareto Distributions, International Cooperative Publishing House, Fairland, MD, USA, 1983.
N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, Wiley, New York, NY, USA, 2nd edition, 1994.
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.
View at: Publisher Site | Google Scholar
M. E. Ghitany, F. A. Al-Awadhi, and L. A. Alkhalfan, “Marshall-Olkin extended Lomax distribution and its application to censored data,” Communications in Statistics—Theory and Methods, vol. 36, no. 10, pp. 1855–1866, 2007.
View at: Publisher Site | Google Scholar
A. J. Lemonte and G. M. Cordeiro, “An extended Lomax distribution,” Statistics, vol. 47, no. 4, pp. 800–816, 2013.
View at: Publisher Site | Google Scholar
M. H. Tahir, G. M. Cordeiro, M. Mansoor, and M. Zubair, “The Weibull-Lomax distribution: properties and applications,” Hacettepe Journal of Mathematics and Statistics, vol. 44, no. 2, pp. 455–474, 2015.
View at: Google Scholar
A. Afify, H. Yousof, and S. Nadarajah, “The beta transmuted-H family for lifetime data,” Statistics and Its Interface, vol. 10, no. 3, pp. 505–520, 2017.
View at: Publisher Site | Google Scholar
H. Reyad, M. Ç. Korkmaz, A. Z. Afify, G. G. Hamedani, and S. Othman, “The Fréchet Topp Leone-G family of distributions: properties, characterizations and applications,” Annals of Data Science, vol. 8, no. 2, pp. 345–366, 2021.
View at: Publisher Site | Google Scholar
H. M. Okasha, “E-Bayesian estimation for the Lomax distribution based on type-II censored data,” Journal of the Egyptian Mathematical Society, vol. 22, no. 3, pp. 489–495, 2014.
View at: Publisher Site | Google Scholar
A. S. Hassan and A. N. Zaky, “Entropy Bayesian estimation for Lomax distribution based on record,” Thailand Statistician, vol. 19, no. 1, pp. 96–115, 2021.
View at: Google Scholar
T. N. Sindhu, N. Feroze, and M. Aslam, “Preference of prior for Bayesian analysis of the mixed Burr type X distribution under type I censored samples,” Pakistan Journal of Statistics and Operation Research, vol. 10, 2014.
View at: Publisher Site | Google Scholar
T. N. Sindhu, M. Aslam, and A. Shafiq, “Bayesian analysis of two censored shifted Gompertz mixture distributions using informative and noninformative priors,” Pakistan Journal of Statistics and Operation Research, vol. 13, no. 1, pp. 227–243, 2017.
View at: Publisher Site | Google Scholar
T. N. Sindhu, Z. Hussain, and M. Aslam, “On the Bayesian analysis of censored mixture of two Topp-Leone distribution,” Sri Lankan Journal of Applied Statistics, vol. 19, no. 1, 2019.
View at: Publisher Site | Google Scholar
T. N. Sindhu, H. M. Khan, Z. Hussain, and B. Al-Zahrani, “Bayesian inference from the mixture of half-normal distributions under censoring,” Journal of the National Science Foundation of Sri Lanka, vol. 46, no. 4, pp. 587–600, 2018.
View at: Publisher Site | Google Scholar
T. N. Sindhu, M. Riaz, M. Aslam, and Z. Ahmed, “Bayes estimation of Gumbel mixture models with industrial applications,” Transactions of the Institute of Measurement and Control, vol. 38, no. 2, pp. 201–214, 2016.
View at: Publisher Site | Google Scholar
T. N. Sindhu, M. Riaz, M. Aslam, and Z. Ahmed, “A study of cumulative quantity control chart for a mixture of Rayleigh model under a Bayesian framework,” Revista Colombiana de Estadística, vol. 39, no. 2, pp. 185–205, 2016.
View at: Google Scholar
T. N. Sindhu, M. Saleem, and M. Aslam, “Bayesian estimation for Topp Leone distribution under trimmed samples,” Journal of Basic and Applied Scientific Research, vol. 3, no. 10, pp. 347–360, 2013.
View at: Google Scholar
T. N. Sindhu and A. Atangana, “Reliability analysis incorporating exponentiated inverse Weibull distribution and inverse power law,” Quality and Reliability Engineering International, vol. 37, pp. 2399–2422, 2021.
View at: Publisher Site | Google Scholar
A. A. Al-Babtain, M. K. Shakhatreh, M. Nassar, and A. Z. Afify, “A new modified Kies family: properties, estimation under complete and type-II censored samples, and engineering applications,” Mathematics, vol. 8, no. 8, p. 1345, 2020.
View at: Publisher Site | Google Scholar
A. Shafiq, S. A. Lone, T. N. Sindhu, Y. El Khatib, Q. M. Al-Mdallal, and T. Muhammad, “A new modified Kies Fréchet distribution: applications of mortality rate of COVID-19,” Results in Physics, vol. 28, pp. 1–17, 2021.
View at: Publisher Site | Google Scholar
A. A. Al-Babtain, A. M. Gemeay, and A. Z. Afify, “Estimation methods for the discrete Poisson-Lindley and discrete Lindley distributions with actuarial measures and applications in medicine,” Journal of King Saud University—Science, vol. 33, Article ID 101224, 2020.
View at: Publisher Site | Google Scholar
S. Sen, A. Afify, H. Al-Mofleh, and M. Ahsanullah, “The quasi xgamma-geometric distribution with application in medicine,” Filomat, vol. 33, no. 16, pp. 5291–5330, 2019.
View at: Publisher Site | Google Scholar
A. Z. Afify, M. Nassar, G. M. Cordeiro, and D. Kumar, “The Weibull Marshall-Olkin Lindley distribution: properties and estimation,” Journal of Taibah University for Science, vol. 14, no. 1, pp. 192–204, 2020.
View at: Publisher Site | Google Scholar
A. Z. Afify, A. M. Gemeay, and N. A. Ibrahim, “The heavy-tailed exponential distribution: risk measures, estimation, and application to actuarial data,” Mathematics, vol. 8, no. 8, p. 1276, 2020.
View at: Publisher Site | Google Scholar
A. A. Al-Babtain, I. Elbatal, H. Al-Mofleh, A. M. Gemeay, A. Z. Afify, and A. M. Sarg, “The flexible Burr X-G family: properties, inference, and applications in engineering science,” Symmetry, vol. 13, no. 3, p. 474, 2021.
View at: Publisher Site | Google Scholar
M. A. Aldahlan, A. Z. Afify, and A. H. N. Ahmed, “The odd inverse Pareto-G class: properties and applications,” Journal of Nonlinear Science and Applications, vol. 12, no. 5, pp. 278–290, 2019.
View at: Google Scholar
G. M. Cordeiro, A. Z. Afify, E. M. M. Ortega, A. K. Suzuki, and M. E. Mead, “The odd Lomax generator of distributions: properties, estimation and applications,” Journal of Computational and Applied Mathematics, vol. 347, pp. 222–237, 2019.
View at: Publisher Site | Google Scholar
G. M. Cordeiro, E. M. M. Ortega, B. V. Popović, and R. R. Pescim, “The Lomax generator of distributions: properties, minification process and regression model,” Applied Mathematics and Computation, vol. 247, pp. 465–486, 2014.
View at: Publisher Site | Google Scholar
M. Mead, “On five-parameter Lomax distribution:properties and applications,” Pakistan Journal of Statistics and Operation Research, vol. 12, no. 1, pp. 185–199, 2016.
View at: Publisher Site | Google Scholar
R. Bantan, A. S. Hassan, and M. Elsehetry, “Zubair Lomax distribution: properties and estimation based on ranked set sampling,” Computers, Materials & Continua, vol. 65, no. 3, pp. 2169–2187, 2020.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Abdelaziz Alsubie. 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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