A New Two-Parameter Burr-Hatke Distribution: Properties and Bayesian and Non-Bayesian Inference with Applications

Department of Statistics, Mathematics and Insurance, Benha University, Benha 13511, Egypt Department of Mathematics & Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt


Introduction
Survival and reliability analysis is an important area of statistics and it has various applications in several applied sciences such as engineering, economics, demography, medicine, actuarial science, and life testing. Different lifetime distributions have been introduced in the statistical literature to provide greater flexibility in modeling data in these applied sciences.
One of the important features of generalized distributions is their capability for providing superior fit for various life-time data encountered in the applied fields. Hence, the statisticians have been interested in constructing new families of distributions to model such data. Some recent notable families are the following: the exponential T-X [1], transmuted Burr-X [2], Marshall-Olkin Burr-III [3], Marshall-Olkin Burr [4], and log-logistic tan [5] families.
On the other hand, there are some useful techniques to add an additional parameter to extend and enhance the flexibility of the classical distributions such as the inverse-power (IP) transformation. Let X and Y be two random variables. e inverse transformation, say X � Y − 1 , or the IP transformation, say X � Y − (1/η) , has been adopted by many authors to construct generalized inverted distributions. For example, the generalized inverse gamma [6], the inverse Lindley with two parameters [7], the inverse Lindley [8], the inverse-power Maxwell [9], the inverse-power Lindley [10], and inverse-power Lomax [11].
In this paper, we are motivated to propose a more flexible version of the Burr-Hatke (BH) distribution to increase its flexibility in modeling real-life data. e BH model provides only a decreasing hazard rate (HR) shape; hence, its use will be limited to modeling the data that exhibits only increasing failure rate. e proposed distribution is called the inverse-power Burr-Hatke (IPBH) distribution. e IPBH model can accommodate right-skewed shape, symmetrical shape, reversed J shape, and left-skewed shape densities. Its HR can be an increasing shape, a unimodal shape, or a decreasing shape.
e IPBH provides more accuracy and flexibility in fitting engineering and medicine data. e IPBH distribution was constructed using the inverse-power (IP) transformation.
Isaic-Maniu and Voda [12] proposed the BH distribution with shape parameter α. Its cumulative distribution function (CDF) has the following form: Its probability density function (PDF) takes the following form: We also considered ten various classical and Bayesian methods for estimating the IPBH parameters and provided detailed numerical simulations to explore their performances based on the mean square errors (MSE), mean relative estimates (MRE), and absolute biases (BIAS). e classical estimators proposed included the maximum product of spacing estimators (MPSE), Anderson-Darling estimators (ADE), Cramér-von Mises estimators (CVME), least-squares estimators (LSE), maximum likelihood estimators (MLE), right-tail Anderson-Darling estimators (RTADE), and weighted least-squares estimators (WLSEs).
e Bayesian estimators of the IPBH parameters have been obtained under symmetric and asymmetric loss functions, namely, the square errors (SE), general entropy (GE), and linear exponential (LN) loss functions. We have compared the estimation methods by conducting extensive simulations study to explore their performances and to determine the best method of estimation, based on partial and overall ranks, which gives accurate estimates for the IPBH parameters.
is article is outlined in the following eight sections. e IPBH distribution is defined in Section 2. Some of its properties are discussed in Section 3. In Section 4, seven classical approaches of estimation are explored. e Bayesian estimators of the IPBH parameters under three loss functions are discussed in Section 5. In Section 6, the performances of classical and Bayesian approaches of estimation are explored via simulations. e applicability and flexibility of the IPBH distribution are illustrated in Section 7 using two real-life datasets. Some useful conclusions are presented in Section 8.

The IPBH Distribution
By applying the IP transformation to the BH CDF (1), the CDF of the IPBH distribution follows (for x > 0) as e corresponding PDF of the IPBH distribution reduces to where η and α are shape parameters. e inverse BH (IBH) distribution follows simply as a special case by replacing η � 1 in equation (4). e survival function (SF) and HR function of the IPBH distribution take the following forms, respectively: Possible shapes of the density and HR functions of the IPBH distribution are displayed in Figures 1 and 2, respectively.

Mathematical Properties
In this section, some distributional properties are addressed.

Quantile Function.
e quantile function (QF) of the IPBH distribution is derived from the CDF (3) as where W[·] is Lambert function. e three quartiles of the IPBH distribution follow directly from (6) with p � 0.25, 0.5, and 0.75.
Assuming that p∼uniform (0, 1), the QF (6) can be applied to generate random datasets of size n from the IPBH distribution by the following formula: e shapes of SK and KU of the IPBH model for several values of α and η are displayed in Figure 3.

Moments.
e rth moments of the IPBH distribution have the following forms:

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e first four moments of the IPBH distribution follow from the above formula with r � 1, 2, 3, and 4. e moment generating function of the IPBH distribution takes the following form:

Incomplete Moments.
e rth incomplete moment (ICM) of IPBH distribution follows as (for kη < r) where D � (η + ηk − r) and B z (a, b) � z 0 t a− 1 (1 − t) b− 1 dt. e first ICM can be used to calculate the Bonferroni and Lorenz curves that are, respectively, defined by can be determined numerically using equation (7) for a certain probability p. e two curves have their importance in insurance, economics, medicine, demography, and engineering. e first ICM is also adopted to calculate the mean residual life (MRL) and mean waiting time that are derived as m 1 (t) � ([1 − Ψ 1 (t)]/(S(t) − t)) and M 1 (t) � ((t − Ψ 1 (t))/F(t)), respectively.

Order Statistics.
e density function of the ith order statistic (OS) of the IPBH distribution takes the following form: e associated CDF reduces to is a hypergeometric function. e PDFs and CDFs of the minimum OS, (Y n ), and maximum OS, (T n ), can be obtained simply from the last two formulae with i � 1 and i � n, respectively. e limiting distributions of (Y n ) and (T n ) are expressed by eorem 2.1.1 in [17].

Classical Inference
In this section, different classical estimation methods of the IPBH parameters are discussed.

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Solving the three previous equations using the statistical software such as Maple, R, SAS, or Mathematica gives the MLE of the IPBH parameters.

Least Squares and Weighted Least Squares.
Consider the order statistics of a random sample, say x 1:n , x 2:n , . . . , x n:n , from the IPBH distribution.
en, the LSE of the IPBH parameters follow by minimizing: We also can obtain the LSE by solving the formula where e WLSE of the IPBH parameters are calculated by minimizing: e WLSE are also calculated by solving the following formula: where ψ p (x l:n ), p � 1, 2, are specified by (19) and (20).

Anderson-Darling and Right Tail Anderson-Darling.
e ADE of the IPBH parameters are obtained by minimizing: (2l − 1) log F x l:n + log S x l:n .
(23) ese estimators can be determined by the derivation of the following equation: where ψ p (x l:n ), p � 1, 2, are specified by (19) and (20). e RADE of the IPBH parameters can be determined by minimizing: and they also can be determined by solving where ψ p (x l:n ), p � 1, 2, are specified by (19) and (20).

Cramér-von Mises.
e CVME of the IPBH parameters are derived by minimizing: e CVME can also be derived by solving the following formula: where ψ p (x l:n ), p � 1, 2, are specified by (19) and (20).

Maximum Product of Spacings.
e maximum product of spacings (MPS) approach is a useful alternative to the ML approach. e uniform spacings, say D l , of a random sample from the IPBH distribution are defined by where n+1 l�1 D l � 1, F(x 0 ) � 0, and F(x n+1 ) � 1. e MPSE of the IPBH parameters are obtained by maximizing: Moreover, the MPSE can be determined using 6 Journal of Mathematics where ψ p (x l:n ), p � 1, 2, 3, are specified by (19) and (20).

Bayesian Estimation
In this section, we estimate the parameters of the IPBH distribution from complete sample by the Bayes estimators (BE) using symmetric and asymmetric loss functions. Now, we adopted the SE, GE, and LN loss functions to obtain the parameters estimates. We also consider that α and η are independent. We adopted two independent gamma priors for the two parameters α and η. e two independent gamma priors have the forms en, the joint PDF prior of α and η takes the form Hence, the posterior function reduces to According to the SE loss function, the BE for B � B(Θ), where π * (Θ) is as in equation (34). e BE under the LN loss function has the form such that E Θ [exp(− cΘ)] exists. e BE Θ GE under GE loss function is such that E Θ [Θ − q ] exists. In fact, the integrals in equations (35)-(37) cannot be found analytically. Hence, the Markov chain Monte Carlo (MCMC) technique is adopted to approximate these integrals. Moreover, we use the Metropolis-Hastings algorithm as an example of the MCMC technique to obtain the estimates.

Simulation Results
is section is devoted to determining the performance and behavior of several estimation approaches in estimating the IPBH parameters based on detailed simulation results. For this purpose, several sample sizes, n � 20, 50, 100, 200, 500 { }, and several values of the parameters α and η, α � 0.5, 0.75, 1.5 and η � 0.5, 1.5, are considered. We generated N � 5000 random samples from the IPBH distribution using its QF (6). e compared estimators are checked in terms of their average absolute biases (BIAS), average mean square errors (MSE), and average mean relative errors of the estimates (MRE) which are obtained, for all parameter values and sample sizes, using the R program. e BIAS, MSE, and MRE can be determined by the three following equations: where θ � (α, η) ′ .
Tables 1-4 report the simulation results including BIAS, MSE, and MRE of the IPBH parameters using the ten estimation approaches. Moreover, Tables 1-4 report the rank of each one of the ten estimators among all the estimators in each row by the superscript indicators, and the partial sum of the ranks for each column, say Ranks, in a certain sample size. From the tabulated results, it is observed that the ten estimation methods show the property of consistency for all studied cases. Table 5 displays the partial and overall rank of these estimators. Form the results in Table 5, we can conclude that the Bayesian method outperforms all other classical methods under the three loss functions, with respective overall scores of 29, 46, and 50 for the SE, GE, and LN loss functions, respectively. erefore, we confirm the superiority of the Bayesian approach for the IPBH distribution.

Two Real-Life Applications
is section is devoted to analyzing two real-life datasets to explore the importance and flexibility of the IPBH distribution as compared with some of its other competing distributions.

Dataset I.
is dataset consists of 63 observations which are generated to simulate the strengths of glass fibers [18]. e 63 observations of the dataset are as follows:

Dataset II.
is dataset represents the relief times of 20 patients who are receiving an analgesic [19]. e 20 relief times are as follows: 1.     Here, we show empirically that the IPBH distribution can provide a more adequate fit than ten competing distributions, namely, the BH, W, E, ILL, F, G, IWL, IL, IP, and INM distributions. We adopted some discrimination or information criterions (IC) such as minus Tables 6 and 7 report estimates of the parameters, by the maximum likelihood approach, standard errors (SEs), and the nine discrimination measures for the two datasets, respectively.
e figures in these tables show that the new IPBH model provides a close fit to both modeled datasets among other competing distributions. e fitted curves for the PDF, CDF, SF, and P-P plots of the IPBH distribution are depicted in Figures 4 and 5 for the two datasets, respectively. e values of discrimination measures in Tables 6 and 7 show great improvement in fitting using the IPBH model Table 5: Partial and overall ranks of the ten estimation methods for the IPBH distribution. 6 9 4 8 10 7 1 3 2 e ten estimation approaches are also adopted to estimate the IPBH parameters from the two datasets. Tables 8  and 9 report the estimates of α and η along with the values of − ℓ, A, W, K-S, and K-S p value for both datasets, respectively. e proposed estimation approaches show a similar    very well performance in estimating α and η of the IPBH distribution. A visual comparison shows the close performance of these estimators as shown in Figure 6 which represents the P-P plots of the IPBH distribution for the ten methods.

Conclusions
In this article, we introduce a more flexible extension of the Burr-Hatke distribution called inverse-power Burr-Hatke (IPBH) distribution that provides more accuracy and flexibility in fitting engineering and medicine data. e new model was generated based on the inverse-power transformation technique. e hazard rate function of the IPBH distribution exhibits an increasing shape, a decreasing shape, or an upside-down bathtub shape. e IPBH model can accommodate right-skewed shape, symmetrical shape, reversed J shape, and left-skewed shape densities. Some of its basic mathematical properties are derived. e two parameters of the IPBH distribution are estimated using ten classical and Bayesian estimation approaches. e behavior and performance of these estimators are explored using simulation results. We also determined the best estimation approach using partial and overall ranks for all estimators. As expected, the Bayesian method outperforms other classical methods under the different loss functions. e flexibility and practical importance of the IPBH distribution are explored empirically using two real-life datasets. It is shown that the IPBH distribution has a superior fit compared to the Burr-Hatke distribution and other competing models.
For some possible directions for future studies, the IPBH model can be modified with "polynomial variable transfer" to introduce new model with several free parameters which makes it attractive for analysis and approximation of specific data from different areas such as growth theory, test theory, biostatistics, and computer viruses propagation. Furthermore, different approximation problems related to the "saturation" in Hausdorff sense can be explored for the new model along with some numerical examples using CAS Mathematica to validate the results. More details about these directions can be explored in [20,21].
Moreover, the T-X family may be applied to define the new inverse-power Burr-Hatke-G family of distributions. Several properties of this new family may be established, its special sub-models may be explored, and their applications in different applied fields may also be addressed.

Data Availability
is work is mainly a methodological development and has been applied on secondary data, but, if required, data will be provided.  Figure 6: P-P plots of the IPBH distribution using several estimation approaches for dataset I (a) and dataset II (b).

Conflicts of Interest
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