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

&e 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. &e 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. &e 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. &e 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. &e three-parameter MKL model outperforms some four-parameter and five-parameter rival models.


Introduction
e 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]. e 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. e 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.
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 υ, denoted by G(x; υ); then, the CDF of the MK-G class with additional shape parameter a takes 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. e 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.
e MKL distribution gives a better fit than some existing rival extensions of the Lomax model which have three, four, and five parameters. e 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. e 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. e findings show that the maximum product of spacings and maximum likelihood methods are recommended to estimate the model parameters.
e rest of the article is structured as follows. e 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. e empirical importance MKL model is checked using two real datasets in Section 6. Some concluding remarks are presented in Section 7.

(6)
Its PDF is determined by inserting (4) and (5) in equation (2) as follows: e 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.
ese 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.

Quantile Function.
e quantile function (QF) of the MKL distribution is obtained by determining the inverse function of the MKL CDF (6) as  Journal of Mathematics e three quartiles of the MKL distribution can be obtained by setting p � 0.25, 0.5, and 0.75, respectively, in (9).
Let p ∼ U(0, 1); then, the QF can be used directly in generating random data from the MKL distribution as follows: 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 H (j+1)a+k (x) is the CDF of the exp-G family with power parameter (j + 1)a + k and the term δ j,k is given by en, the linear representation of the CDF of the MKL distribution takes the form By differentiating the previous equation, we obtain a linear representation of the MKL PDF as where Ψ j,k,m � (a(− 1) j+m+1 /j![(j + 1)a + k]) (j + 1)a + k k (j + 1)a + k m and h m (x) is the PDF of Lomax distribution with shape parameter mλ and scale parameter β. Hence, the properties of the proposed MKL model, such as moments, follow from those of the Lomax distribution.

Moments.
e rth moments of the MKL distribution has the form en, the rth moments of the MKL distribution follows, from the moments of Lomax model, as Setting r � 1, 2, 3, and 4, respectively, we obtain the first four-moments of the MKL distribution. e moment generating function of the MKL distribution follows directly from the previous formula as e characteristic function of the MKL distribution follows from the last equation by setting t � it.

Order Statistics.
e PDF and CDF of the ith order statistic of the MKL distribution are 4 Journal of Mathematics e joint PDF of the rth and ith order d (for 1 ≤ r < i ≤ n) by Hence, the joint PDF of two order statistics from the MKL distribution takes the form

Incomplete Moments.
e sth incomplete moment of the MKL distribution is given by where B z (a, b) � z 0 t a− 1 (1 − t) b− 1 dt. e important application of the first incomplete moment is related to the Bonferroni and Lorenz curves defined by L(p) � (Ψ 1 (t)/μ 1 ′ ) and B(p) � (Ψ 1 (x p )/(pμ 1 ′ )), respectively, where x p can be evaluated numerically by equation (22) for a given probability p. ese 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 m 1 (t) � ([1 − Ψ 1 (t)]/(S(t) − t)) and M 1 (t) � ((t − Ψ 1 (t))/F(t)), respectively.

Maximum Likelihood Estimation
. Let x 1 , x 2 , . . . , x n be a random sample of size n from the PDF (7); then, the loglikelihood function reduces to By differentiating equation (22) with respect to a, λ, and β, respectively, and equating to zero, we obtain Journal of Mathematics Solving the previous equations, we obtain estimators of the MKL parameters by the MLEs. e 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.

Ordinary Least-Squares and Weighted Least-Squares
Estimators. Let x 1: n , x 2: n , . . . , x 2: n be the order statistics of a random sample of size n from the MKL distribution. Hence, we can obtain the LSE of the MKL parameters by minimizing the following equation: e LSE of the MKL parameters can also be obtained by solving the following equations: where e 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 ϑ k (x i:n ) for k � 1, 2, 3 are defined in (26)-(28).

Anderson-Darling Estimation.
e ADE of the MKL parameters can be obtained by minimizing (2i − 1) log F x i:n + log S x i:n . (31) e ADE are also be calculated by solving the following equations: 6 Journal of Mathematics where ϑ k (x i:n ), for k � 1, 2, 3, are defined in (26)-(28).

Cramér-von Mises
Estimation. e CVME of MKL parameters are obtained by minimizing the following equation: or by solving the following equations where ϑ k (x i:n ), for k � 1, 2, 3, were defined in (26)-(28).

Maximum Product of Spacings' Estimation.
e maximum product of spacings (MPS) method is used to estimate the parameters of continuous models as a good alternative to the maximum likelihood method. e uniform spacings of a random sample of size n from the MKL distribution is defined by where D i are the uniform spacings, F(x 0 ) � 0, F(x n+1 � 1), and n+1 i�1 D i � 1. e MPSE of the MKL parameters can be obtained by maximizing with respect to a, λ, and β. Furthermore, the MPSE of the MKL parameters can also be calculated by solving where ϑ k (x i:n ), for k � 1, 2, 3, were defined in (26)-(28).

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, n � 20, 70, 150, { }, are considered to generate N � 10, 000 random samples from the MKL distribution using its QF and to determine the AVBs', MSEs', and MREs' measures using the R program. e AVBs', MSEs', and MREs' measures are calculated using the following equations: where φ � (a, λ, β) ′ . Tables 1-6 report detailed simulation results for the six estimation methods including AVBs, MSEs, and MREs for the parameters of the MKL distribution. ese 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, n, 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.

Journal of Mathematics 9
Kolmogorov-Smirnov (KS) statistic with its p value (KS-PV) for both datasets. e 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. ese 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            Figure 4: e fitted MKL PDF, CDF, SF, and P-P plots for waiting times in a bank data. 16 Journal of Mathematics both datasets among all fitted competing models. at 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.

Concluding Remarks
In this study, a new three-parameter extension of the Lomax distribution called the modified Kies-Lomax (MKL) distribution has been studied. e 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. e 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. e 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
e data used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.