A Compound Class of Unit Burr XII Model: Theory, Estimation, Fuzzy, and Application

Zayed, Mohammad A.; Hassan, Amal S.; Almetwally, Ehab M.; Aboalkhair, Ahmad M.; Al-Nefaie, Abdullah H.; Almongy, Hisham M.

doi:https://doi.org/10.1155/2023/4509889

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Volume 2023 | Article ID 4509889 | https://doi.org/10.1155/2023/4509889

A Compound Class of Unit Burr XII Model: Theory, Estimation, Fuzzy, and Application

Mohammad A. Zayed,^1,2Amal S. Hassan ,³Ehab M. Almetwally,^4,5Ahmad M. Aboalkhair,^1,2Abdullah H. Al-Nefaie,¹and Hisham M. Almongy²

Academic Editor: Punit Gupta

Received12 Aug 2022

Revised24 Sept 2022

Accepted26 Sept 2022

Published26 Apr 2023

Abstract

The current research offers an enhanced three-parameter lifetime model that combines the unit Burr XII distribution with a power series distribution. The novel class of distribution is named the unit Burr XII power series (UBXIIPS). This compounding technique allows for the production of flexible distributions with strong physical meanings in domains such as biology and engineering. The UBXIIPS class includes the unit Burr XII Poisson (UBXIIP) distribution, the unit Burr XII binomial distribution, the unit Burr XII geometric distribution, and the unit Burr XII negative binomial distribution. The statistical properties of the class include formulas for the density and cumulative distribution functions, and limiting behaviour, moments and incomplete moments, entropy measures, and quantile function are provided. For estimating population parameters and fuzzy reliability for the UBXIIP model, maximum likelihood and Bayesian approaches are studied by the Metropolis–Hastings algorithm. For maximum likelihood estimators, the length of asymptotic confidence intervals is specified, whereas, for Bayesian estimators, the length of credible confidence intervals is assigned. A simulation investigation of the UBXIIP model was established to evaluate the performance of suggested estimates. In addition, the UBXIIP distribution is explored using real-world data. The UBXIIP distribution appears to offer some benefits in understanding lifetime data when compared to unit Weibull, beta, Kumaraswamy, Kumaraswamy Kumaraswamy, Marshall-Olkin Kumaraswamy, and Topp–Leone Weibull Lomax distributions.

1. Introduction

In recent years, the development of unit distributions has risen rapidly. These distributions focus on modelling a wide variety of occurrences using data with values ranging from 0 to 1, such as proportions, probabilities, and percentages. The design of parametric, semiparametric, and regression models for the analysis of such data is also in high demand in applied disciplines. The majority of today’s unit distributions are created by appropriately modifying older distributions. Our attention has been drawn to the recently presented unit Burr XII (UBXII) created by Korkmaz and Chesneau [1]. The cumulative distribution function (CDF) and probability density function (PDF) of the UBXII distribution are specified byandwhere and are shape parameters. They investigated some of the UBXII distribution’s characteristics and developed a regression model for it.

Due to theoretical considerations, practical applications, or both, academics have recently grown more interested in the design of novel univariate distributions, which are widely utilised in statistics and related disciplines. Compounding is a useful strategy for creating new distributions by combining certain useful lifetime with truncated discrete distributions. The basic idea behind creating these compounding distributions is that the lifetime of a system with K components and a positive continuous random variable, T_i, that denotes the lifetime of the i^th component, can be represented by a nonnegative random variable, T = min (T₁,…, T_K) if the components are in a series, or T = max (T₁,…, T_K) if the components are in parallel. The continuous random variables T_i are considered to be independent of K in both contexts.

Various compound classes have been provided by mixing continuous distributions with power series (PS) distribution, for example, Weibull-PS and extended Weibull-PS (Morais and Barreto-Souza [2] and Silva et al. [3]), generalized exponential PS (Mahmoudi and Jafari [4]), complementary exponential PS (Flores et al. [5]), the Burr XII-PS (Silva and Corderio [6]), Gompertz PS (Jafari and Tahmasebi [7]), generalized modified Weibull-PS (Bagheri et al. [8]), exponential Pareto PS (Elbatal et al. [9]), exponentiated power Lindley-PS (Alizadeh et al. [10]), generalized inverse Lindley-PS (Alkarni, [11]), Burr–Weibull PS (Oluyede et al. [12]), odd log-logistic PS (Goldoust et al. [13]), new generalized Lindley-Weibull class (Makubate et al. [14]), inverse gamma PS (Rivera et al. [15]), and inverted exponentiated Lomax PS (Hassan et al. [16]) among others.

The unit Burr XII power series (UXIIPS) class is suggested in the current study by mixing the UBXII and PS distributions in a device made up of parallel components. As special examples, this class also includes a number of compound lifetime models. Additionally, it offers us the flexibility to simulate many different behavioural forms of lifetime data using any compound lifetime. There are several hazard rate shapes present in this class of distributions. In addition to the quantile function, some moment measurements, the mean residual life, and uncertainty measures are other distributional features that we offer. The parameters and fuzzy reliability of the UBXII Poisson (UBXIIP) distribution, a specific example from the proposed class, are estimated using Bayesian and non-Bayesian techniques. A numerical simulation experiment is run to assess the accuracy of the estimated values. A real data set is used to explain the utility of the UBXIIP distribution.

The followings are the main physical justifications and significance of the distribution’s UBXII class:(i)To approximate the time to the last failure of a system made up of components linked in parallel.(ii)To construct and generate distributions with symmetric, left-skewed, right-skewed, and U shapes.(iii)To define special models that have a variety of hazard rate functions, including monotonic and nonmonotonic shapes.(iv)To consistently provide better fits than other generated distributions having the same or greater number of parameters.(v)To discuss the Bayesian and non-Bayesian estimators of parameters and fuzzy reliability for one special model from the provided class.(vi)To assess the accuracy of the produced estimators, a numerical simulation investigation is undertaken.(vii)To build heavy-tailed distributions for modelling diverse real data sets used in numerous domains, such as business, environmental research, medicinal studies, demographics, and industrial reliability.(viii)A data study demonstrated the UBXIIP distribution’s superiority over a few other well-known models.

This article is structured as follows: Section 2 provides the PDF of the UBXIIPS class as well as gives associated models. In Section 3, we deduce certain structural features of the UBXIIPS class. The Bayesian and non-Bayesian estimators of the UBXIIP distribution parameters and fuzzy reliability are explained in Section 4. Numerical examination and real data analysis are discussed, respectively, in Sections 5 and 6. Section 7 concludes the article.

2. Construction of the New Class

The new class of UBXIIPS is defined as follows. Given K, supposed that T₁,…, T_K be identically independent distributed random variables having the UBXII distribution (1) with shape parameters and , where K is a discrete random variable having a PS (truncated at zero) distribution. The probability mass function of K is specified bywhere a_k depends only on k, is the scale parameter, and , and denote the first and second derivatives of respectively. We provide certain PS distributions (truncated at zero) defined by (3) in Table 1, including the Poisson, logarithmic, geometric, and binomial distributions.

Let T = max the conditional CDF of is given by

Hence, is the UBXII distribution with parameters and so we obtain

So, the marginal CDF of (5) is given by

A random variable with CDF (6) has UBXIIPS class with parameters shall be denoted by T∼UBXIIPS The PDF of the UBXIIPS class corresponding to (6) is

The survival function and hazard rate function of the UBXIIPS class are represented by

Some special submodels are listed as follows, based on PDF (7) and Table 1:(i)For we obtain the PDF of the UBXIIP distribution as follows:(ii)For we obtain the PDF of the UBXII logarithmic distribution as follows:(iii)For we obtain the PDF of the UBXII geometric distribution as follows:(iv)For we get UBXII binomial distribution as follows:

Figure 1 illustrates graphs of the UBXIIP density for various parameter values. These graphs demonstrate the new distribution’s reliability and modality. The UBXIIP density is unimodal or bell-shaped, as seen in Figure 1. For a given set of parameters, it is left-skewed and reversed J-shaped. The hazard rate function can be decreasing, increasing, bath-tub, and J-shaped.

Proposition 1. The UBXIIPS density function (7) can be explained as an infinite mixture of UBXII densities with parameters

Proof. Using in PDF (7), then it can be reformed as follows:where is the UBXII density function with parameters and defined in (3).

Proposition 2. The UBXII distribution is the limiting distribution of the UBXIIPS class of distributions when

Proof. The limiting distribution of (6), for is determined, using L’Hospital’s rule, as follows:which is CDF of UBXII distribution.

3. General Properties

This section covers some statistical characteristics of the UBXIIPS class. Furthermore, these measures are focused on a one model, specifically the UBXIIP distribution.

3.1. Moments Measures

The moments of a probability distribution are essential tools in any statistical analysis. The r^th moment of T has the UBXIIPS class is presented from (13) as follows:

Using binomial expansion in (15) and let , then of UBXIIPS class has the formwhere and is the gamma function (GF). The r^th central moment () of the UBXIIPS class is given by

Based on (17) and by using well-known relationships, we can determine some measures such as variance (), skewness (), and kurtosis (). In particular, numerical measurements such as , and of the UBXIIP distribution are displayed in Table 2, for various parameter values, including (a) (b) (c) (d) (e) and (f)

As seen from Table 2, as the value of increases, values of increase while the value of decreases expect for set (d). The UBXIIP distribution is negatively skewed and platykurtic.

Furthermore, the r^th incomplete moment of the UBXIIPS class is derived as follows:

Using binomial expansion and let then of UBXIIPS class iswhere is the upper incomplete GF. Additionally, the Lorenz (LZ) and Bonferroni (BI) curves of UBXIIP distribution are derived using the following expressions: and

Moreover, the r^th moment and incomplete moment of the UBXIIP distribution are computed from (16) and (19), respectively, by putting .

Another important function is the mean residual life (MRL) function. The MRL function is defined as an item’s expected life after t years. It is a conditional notion that tells you how long you may expect the object to last. Consequently, we obtain the n^th moment of the residual life of T via the general following formula:

Using binomial series more than one time, then (20) is represented as follows:where is the lower incomplete GF. Put n = 1 in (21), we can calculate the MRL of UBXIIP distribution.

3.2. Quantile Function

The quantile function (QF) is defined as for any CDF. Consequently, the QF of the UBXIIPS class, based on (6), is calculated as follows:

For u = 0.25, u = 0.5, and u = 0.75 in (22), we can determine the first, median, and third quantiles of the UBXIIPS class. More precisely, the QF of the UBXIIP distribution is derived from (22) by letting and as follows:

Equation (23) can be used to generate UBXIIP random variates.

3.3. Uncertainty Measures

We look at certain information measures including Rényi (Ré) entropy and entropy. The entropy quantifies the data’s uncertainty; the higher the entropy number, the greater the data’s uncertainty. The Ré entropy of T has UBXIIPS class is defined by

Expressions for different entropy measures of the UBXIIPS class are obtained. From (7), we deduce as follows:But

Use the relation (Gradshteyn and Ryzhik [17]) in (26) yields

Therefore, (26) will look like this:where, for and Therefore, (25) can be written as

Put (29) in (24), we acquire Ré entropy of the UBXIIPS class as follows:

The entropy of T has UBXIIPS class is defined by

Hence, entropy of UBXIIPS class is obtained as follows:Putting in (30) and (32), we get the Ré entropy and entropy of the UBXIIP distribution.

4. Parameter Estimation of the UBXIIP Model

The parameter and reliability estimators of the UBXIIP distribution based on ML, and Bayesian estimation methods are discussed in this section.

4.1. ML Method

Let T₁,…,T_n be the observed values from the UBXIIP distribution with parameters and The likelihood function, say of the UBXIIP distribution is expressed as:where Then, the log-likelihood function, say , of the UBXIIP distribution is given as follows:

Therefore, the ML equations are given by

Solving the nonlinear equations and numerically using optimization algorism as conjugate-gradient optimization, we get the ML estimators of and

To construct confidence intervals (CIs) for and we need to compute the asymptotic variance-covariance matrix which obtained by inverting the Fisher information matrix , in which elements are negatives of expected values of the second partial derivatives of . The elements of the sample information matrix for the ML method will be

Under some regularity conditions, is approximately normal with mean and covariance matrix . Practically, we estimate by , then

Now, the approximate CIs (ACIs) for and can be obtained as follows:where z_q is the 100 q–th percentile of a standard normal distribution.

4.2. Bayesian Method

Here, we get the Bayesian estimator of the UBXIIP parameters. The Bayesian estimator is regarded as symmetric (squared error loss function (SELF)), which is defined as follows:

Assuming that the prior distribution of and denoted by has an independent gamma distribution. The joint gamma prior density of and can be written as follows:

In Bayesian estimation, the value of the hyperparameter is crucial. If the appropriate prior information for and is provided, , then the joint prior distribution (40) is proportional to the likelihood function (33). As a result, if one does not have previous knowledge of the unknown parameters, it is preferable to employ ML estimation rather than Bayesian estimation because the latter is computationally expensive. We used ML information (estimators and variance) as well as prior information (mean and variance) and solve these functions using the moments method or any iterative approach to obtain the hyperparameter values and , which are referred to as prior values. From (33) and (40), the joint posterior of the UBXIIP distribution with parameters and is

Bayesian estimators may be produced using the Markov chain Monte Carlo (MCMC) technique. The Gibbs sampling, as well as the more generic Metropolis within Gibbs samplers, is important MCMC techniques. The Metropolis–Hastings (MH) algorithm and the Gibbs sampling are two well-known applications of the MCMC approach. The conditional posterior densities of , and are produced as follows:

The Bayesian estimators are obtained based on SELF. It is clear that samples of and can be easily generated by using their conditional posterior distributions which are obtained above based on MH algorithm. Furthermore, using the samples generated from the suggested MH method, one may create the highest posterior density (HPD) credible intervals for , and of the UBXIIP distribution following the MH proposed by Chen and Shao [18].

With two endpoints from the MCMC sample outputs, the lower is 2.5%, and the upper is 97.5% percentiles, respectively, and a 95% HPD interval can be generated. The credible intervals of , and are calculated as follows:(1)Arrange as and , where H is the length of MCMC generated(2)The 95% symmetric credible intervals of , and become , and

4.3. Fuzzy Reliability

Let X be a continuous random variable that represents a system’s failure time. The fuzzy reliability can be calculated using the fuzzy probability formula proposed by Chen and Pham [19]where is a membership function that describes the degree to which each element of a given universe belongs to a fuzzy set. Now, assume that is

For by the computational method of the function of fuzzy numbers, the lifetime can be obtained correspond to a certain value of where proposed by Chen and Pham [19]

Thus, for all values, fuzzy reliability values can be calculated as follows:

The reliability estimator via ML method based on nonfuzzy and fuzzy is as follows:

The reliability estimator via the Bayesian method based on the nonfuzzy and fuzzy parts of the UBXIIP distribution is as follows:

5. Simulation Study

The following algorithm is used to obtain the likelihood and Bayesian estimation of parameters for the UBXIIP model and reliability with tradition and fuzzy with different cuts, and their properties are studied using the mean squared errors (MSE), average absolute bias (AAB), and length of CIs. Our simulation algorithm begins with the creation of all simulation controls. At this stage, we must proceed in the following order: Step 1: Assume various values for the UBXIIP distribution’s parameters vector, where and Step 2: Select the appropriate strength sample size n = 50, and 150. Step 3: Generate the sample random values of UBXIIP distribution by using quantile function in (23). Step 4: Solve differential equations of ML method, to obtain the estimators of the parameters for UBXIIP distribution, we calculate Also, we calculate reliability based on nonfuzzy and fuzzy. Step 5: Estimate hyperparameter values as Step 6: Generate posterior distribution by MCMC techniques of the Bayesian method as , to obtain the estimates of the parameters for UBXIIP distribution, we calculate , and Also, we calculate reliability based on nonfuzzy and fuzzy. Step 7: Compute AAB, MSE, and length of ACI (LACI) for ML estimate and length of credible CI (LCCI) for Bayesian estimates where the level of CIs is 95%. The measures AAB, MSE, and length of CI are defined by: AAB = Mean , MSE = Mean , and length of CI = Upper CI - Lower CI.

The following observations about the behavior of estimates are listed below, as shown in Tables 3–4.(i)The suggested estimates of , and are pretty satisfactory in terms of lowest AAB, MSE, and LACI.(ii)The accuracy of the estimates improves with n.(iii)In terms of AAB, MSE, and LACI, Bayesian MCMC estimates utilizing gamma informative priors outperform frequentist estimates since they include prior knowledge.(iv)The credible CI surpasses the asymptotic CI in terms of the smallest CI length due to the gamma prior knowledge.(v)As decreases, the accuracy of the estimates improves.(vi)As decreases, the accuracy of the estimates improves.(vii)MSE of fuzzy reliability values has smaller values than MSE of nonfuzzy reliability values(viii)As increases, then fuzzy reliability values improve, as well as MSE increases

6. Data Application

This data set is constituted by the total milk production from the first birth of 107 cows of the SINDI race. The data can be found in Cordeiro and Birto [20] and analyzed by Muhammad et al. [21]. Concretely, the data set is 0.4365, 0.4260, 0.5140, 0.6907, 0.7471, 0.2605, 0.6196, 0.8781, 0.4990, 0.6058, 0.6891, 0.5770, 0.5394, 0.1479, 0.2356, 0.6012, 0.1525, 0.5483, 0.6927, 0.7261, 0.3323, 0.0671, 0.2361, 0.4800, 0.5707, 0.7131, 0.5853, 0.6768, 0.5350, 0.4151, 0.6789, 0.4576, 0.3259, 0.2303, 0.7687, 0.4371, 0.3383, 0.6114, 0.3480, 0.4564, 0.7804, 0.3406, 0.4823, 0.5912, 0.5744, 0.5481, 0.1131, 0.7290, 0.0168, 0.5529, 0.4530, 0.3891, 0.4752, 0.3134, 0.3175, 0.1167, 0.6750, 0.5113, 0.5447, 0.4143, 0.5627, 0.5150, 0.0776, 0.3945, 0.4553, 0.4470, 0.5285, 0.5232, 0.6465, 0.0650, 0.8492, 0.8147, 0.3627, 0.3906, 0.4438, 0.4612, 0.3188, 0.2160, 0.6707, 0.6220, 0.5629, 0.4675, 0.6844, 0.3413, 0.4332, 0.0854, 0.3821, 0.4694, 0.3635, 0.4111, 0.5349, 0.3751, 0.1546, 0.4517, 0.2681, 0.4049, 0.5553, 0.5878, 0.4741, 0.3598, 0.7629, 0.5941, 0.6174, 0.6860, 0.0609, 0.6488, 0.2747.

The ML estimate evaluates all of the competing model’s parameters. The Bayesian method is also used to analyze the parameters of the UBXIIP model. The Akaike information criterion (AIC), Bayesian IC (BIC), consistent AIC (CAIC), and Hannan–Quinn IC (HQIC) are all used to compare fitted models, and they all use the estimated log-likelihood as the principal constituent. The Kolmogorov-Smirnov (KSV), Anderson-Darling value (ADV), and Cramér-von Mises value (CvMV) goodness-of-fit statistics are also taken into account. As is customary, the model with the lowest value of these measures better represents the data than the other models. We compute the ML and Bayesian parameter estimators of the UBXIIP model using the standard error (SE). Kumaraswamy (K), beta, Kumaraswamy Kumaraswamy (KK) distribution (El-Sherpieny and Ahmed [22]), Marshall-Olkin Kumaraswamy (MOK) distribution (George and Thobias [23]), unit Weibull (UW) distribution (Mazucheli et al. [24]), and Topp-Leone Weibull-Lomax (TLWL) distribution (Jamal et al. [25]) are used as competing models. Table 5 shows the numerical values of the statistical measures that were considered for each model. The findings reveal that the UBXIIP model fits the data better than the other models.

Table 6 shows different estimates using ML and Bayesian estimation methods. Also, it contains reliability with tradition and fuzzy with different α-cut. We conclude from this table that Bayesian estimates perform better than ML estimates.

Figure 2 gives the plots of the histogram with the fitted PDF, empirical CDF with the fitted CDF, and P-P plots of UBXIIP model. From this figure, we conclude that the UBXIIP distribution is suitable for this data set.

Figure 3 shows convergence plots of MCMC for parameter estimates of the UBXIIP distribution. It illustrates the iterations obtained using the MH algorithm and the Gibbs sampling technique for each parameter. Also, it includes the posterior PDFs of each parameter based on the iterations obtained, to demonstrate the performance of the Bayesian estimates.

The log-likelihood function is maximized by the estimated parameters (Figure 4). The roots were found to always point to the global maximum rather than the local maximum. We were able to corroborate our findings by presenting the log-likelihood function. The estimate is at its maximum location along the curve, as indicated by the blue dot.

We sketched the log-likelihood by contour plot for each parameter for the fraction of total milk production data in Figure 5 by fixing two parameters and modifying the rest. The fraction of total milk production data set performs admirably, as shown in Figure 5, because the three roots of the parameters are global maximums.

7. Concluding Remarks

The unit Burr XII power series class is a four-parameter lifetime distribution that we introduced. The UBXII and PS distributions were combined to produce the proposed class. The hazard rate function of the UBXII distribution can take on a number of forms, including the bathtub and J-shaped. Moments and incomplete moments, mean residual life, information measures, and quantile function are among the statistical features of the UBXII distribution that we have derived. Some new submodels of the class are presented. The classical and Bayesian methods are used to obtain the estimators of parameters as well as reliability estimators via fuzzy and nonfuzzy. Furthermore, the length of ACI and the length of Bayesian credible intervals are determined. For the Bayesian method, we use the MCMC techniques to obtain the proposed estimators. To assess the effectiveness of the provided estimates, a simulation analysis of the UBXIIP model was created. Due to gamma previous information, the credible interval is less than the asymptotic CI in terms of CI length. According to simulation research findings, Bayesian estimates are preferred over non-Bayesian estimates since they have a higher degree of accuracy. Compared to nonfuzzy reliability estimates, the MSE of fuzzy reliability estimates has smaller values. In comparison to the asymptotic CI, the credible interval also has a shorter length. Application to one real data set showed that the UBXIIP distribution performs well than some other distributions. Additionally, results from real data and MCMC plots for parameter estimates of the UBXIIP distribution support the conclusions of the simulation research. The study of stress-strength reliability in light of the ranked set sampling (RSS) technique has recently captured the interest of multiple writers due to its application in a variety of fields. Therefore, in our upcoming work, we want to address the problem of stress-strength estimation for a certain distribution in the class using the RSS method [26–28], and also, for applications of lifetime data [29–31].

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

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University (KFU), Saudi Arabia [GRANT86]. The authors, therefore, gratefully acknowledge KFU for technical and financial support.

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Copyright © 2023 Mohammad A. Zayed 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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