Abstract
Using a class of claim distributions, we introduce the Weibull claim distribution, which is a new extension of the Weibull distribution with three parameters. The maximum likelihood estimation method is used to estimate the three unknown parameters, and the asymptotic confidence intervals and bootstrap confidence intervals are constructed. In addition, we obtained the Bayesian estimates of the unknown parameters of the Weibull claim distribution under the squared error and linear exponential function (LINEX) and the general entropy loss function. Since the Bayes estimators cannot be obtained in closed form, we compute the approximate Bayes estimates via the Markov Chain Monte Carlo (MCMC) procedure. By analyzing the two data sets, the applicability and capabilities of the Weibull claim model are illustrated. The fatigue life of a particular type of Kevlar epoxy strand subjected to a fixed continuous load at a pressure level of 90% until the strand fails data set was analyzed.
1. Introduction
The use of statistical distributions to model life phenomena has attracted considerable research interest. Recent articles have demonstrated the potential of statistical distributions in modelling life data. The Weibull distribution with two parameters is a well-known model that can be effectively used for data modelling in lifetime analysis. The Weibull distribution was introduced by Frechet [1] and first applied by Rosin and Rammler [2] to describe the distribution of a particle size. The Weibull distribution has many applications in most fields. Let X be a random variable (R.V.) that follows the two-parameter Weibull distribution , then its cumulative distribution function (CDF), denoted by , is given by
The corresponding probability density function (PDF), survival function (SF), and hazard rate function (HRF) of the Weibull R.V. are given, respectively, byand
In this article, we focus on modelling a new three-parameter modification of the Weibull distribution, the Weibull claim distribution. In 1997, Marshall and Olkin [3] developed a new family by adding a shape parameter to the basic distribution, which many researchers used to find and study new distributions. The Weibull claim distribution is introduced using the class of claim distributions introduced by Ahmad et al. [4] based on the Marshall and Olkin mechanism. The CDF and the PDF of a class of claim distributions are, respectively, given byand
In the field of Big Data science and other related fields, the best possible description of real-world phenomena is an important research topic (see [5–8] for details). Recent studies have shown the potential of statistical models in various areas of applied science. The aim of this paper is to deepen this research area of distribution theory and propose a new statistical model that provides a better fit to survival time data.
The structure of this paper is as follows. Section 2 introduces the Weibull claim distribution. The bivariate extension of the Weibull claim distribution is discussed in Section 3. In Section 4, we first discuss the likelihood estimation of the Weibull claim distribution and the asymptotic and bootstrap confidence intervals based on the observed Fisher information matrix. We then discuss the Bayesian estimation of the unknown parameters under the squared error, LINEX, and the general entropy loss function and perform a Monte Carlo simulation study. In Section 5, simulations and two real data sets are performed to evaluate the efficiency of the proposed model. Section 6 provides a brief conclusion.
2. The Weibull Claim Distribution
The CDF and the PDF of the Weibull claim distribution can be constructed directly by substituting the formulas (1)–(3) into (4) and (5), respectively. Now, X is a Weibull claim R.V. if its CDF is given by
The corresponding PDF is given by
The main motivations for using the Weibull claim in practice are as follows:(1)A very simple and convenient way to modify the Weibull distribution.(2)The improvement of the functions and flexibility of the Weibull distribution.(3)The introduction of an extended version of the Weibull distribution with a closed form of the distribution function.(4)The best fit to data in many sciences and fields.(5)Another important motivation for the proposed approach is the introduction of new distributions by adding only one additional parameter instead of two or more parameters.
3. Bivariate Weibull Claim Distribution
Morgenstern [9] introduced the copula model. Copula models are used to represent the common CDF of the two marginal univariate distributions. If is the CDF of and and is the dependence measure between and , the joint CDF defined by the copula model, denoted by , is given by
Conway [10] introduced the joint CDF and PDF of the copula model, respectively, byand
If the R.V.s and follow the Bivariate Weibull claim distribution, then its CDF is given bywhere The corresponding PDF be
Figures 1 and 2 show different PDF and CDF values of the bivariate Weibull claim model for . Figure 3, on the other hand, shows the SF and HF of the bivariate Weibull distribution model.
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4. Parameter Estimation
Many studies were examined in the articles to estimate the three unknown parameters. Let be a sample from the Weibull claim distribution.
4.1. Maximum Likelihood Method
The likelihood function corresponding to equation (7) is given by
The log-likelihood function corresponding to equation (13) is given by
The equation in equation (14) can be solved by using Newton–Raphson method. The corresponding partial derivatives are
And,
The maximum likelihood estimators for the parameters of the Weibull distribution and are the solutions of the above nonlinear equations.
Then, we can calculate the asymptotic confidence intervals of the parameters , and . The covariance matrix of the observed variance for the MLEs of the parameters was assumed as follows:
two-sided approximate confidence intervals for the parameters , , and are then given byandrespectively, where , , and are the estimated variances of , , and , which are given by the diagonal elements of , and is the upper percentile of the standard normal distribution.
Next, obtaining the bootstrap C.I. for boot-p for the unknown parameters , we apply the following algorithms (for more details, see [11, 12]).
Boot-p interval’s algorithm is as follows: Step 1. Generate from the Weibull claim distribution and derive an estimate of . Step 2. Generate another sample using . Then, derive the updated bootstrap estimate from . Step 3. Repeat step 2 with a given number B of repetitions. Step 4. Using , i.e., the CDF of , the C.I. of is given by where and is prefixed.
4.2. Bayesian Estimation
Bayesian inference is an appropriate method to work with the full samples of the Weibull claim distribution. Given that Weibull claim distributions are so rare, prior information is very useful. We assume that , and are R.V.s following prior DFs: Uniform , Uniform , and Uniform , respectively. The posterior DF of and the data under the uniform priors can take the form as follows:where is the normalizing constant. Next, we suppose that , and are R.V.s that follow the prior DF Gamma , Gamma , and Gamma , respectively, where and are positive constants and . The posterior DF of and the data under the Gamma priors can take the forms as follows:where is the marginal probability DF of .
4.3. MCMC Method
In the following algorithm, we use Metropolis Hastings (M-H) procedure with normal DF to simulate samples from the distributions:(1)Set the initial values , and . Then, simulate sample of size from Weibull claim distribution, next let .(2)Simulate , and . using the proposal distributions , , and .(3)Obtain the probability .(4)Simulate from Uniform (0, 1).(5)If , then . If , then .(6)Set .(7)Iterate Steps 2–6, M repetitions, and get , and for .
In order to conduct a Bayesian analysis, usually quadratic loss function is considered. A very popular quadratic loss is the squared error loss function given by , where is an estimate of the unknown parameter against SE loss function. By using the generated random samples from the above Gibbs sampling technique and for is the n burn, then the Bayes estimator of , say , can be obtained as
The second loss function is the LINEX loss function, given by
The approximate Bayes estimate of under LE loss function based on the Gibbs sampling technique becomes
Finally, the general entropy (GE) loss function is given by
The approximate Bayes estimate of the parameters is given by
MCMC HPD credible interval algorithm is as follows:(1)Sort , and in rising values.(2)The lower bounds of , and in the rank .(3)The lower bounds of , and in the rank .(4)Iterate the previous steps times. Get the average value of the lower and upper bounds of , and .
5. Simulation Study
The simulation study is conducted as follows:(1)Random samples of size from the Weibull claim distribution are simulated.(2)Parameters were estimated using maximum likelihood and Bayesian methods.(3)Two LINEX loss functions are used: when and when .(4)M iterations are performed to obtain the absolute biases and expected risk (ER) of these estimators.(5)The point estimates of the parameters using MLE and MCMC methods are obtained.(6)We obtain 95% and 90% HPD credibility intervals using MCMC methods.(7)The biases and ERs are, respectively, given by and
The MLE and Bayes simulation results of the Weibull claim distribution parameters , and are presented in Tables 1 and 2.
5.1. Application of the Weibull Claim Model
The data set represents the lifetimes of Kevlar 49/epoxy strands subjected to constant sustained pressure at 90% stress level until strand failure. For previous studies on data, see [13–16]. The data are as follows: 0.0251, 0.0886, 0.0891, 0.2501, 0.3113, 0.3451, 0.4763, 0.5650, 0.5671, 0.6566, 0.6748, 0.6751, 0.6753, 0.7696, 0.8375, 0.8391, 0.8425, 0.8645, 0.8851, 0.9113, 0.9120, 0.9836, 1.0483, 1.0596, 1.0773, 1.1733, 1.2570, 1.2766, 1.2985, 1.3211, 1.3503, 1.3551, 1.4595, 1.4880, 1.5728, 1.5733, 1.7083, 1.7263, 1.7460, 1.7630, 1.7746, 1.8275, 1.8375, 1.8503, 1.8808, 1.8878, 1.8881, 1.9316, 1.9558, 2.0048, 2.0408, 2.0903, 2.1093, 2.1330, 2.2100, 2.2460, 2.2878, 2.3203, 2.3470, 2.3513, 2.4951, 2.5260, 2.9911, 3.0256, 3.2678, 3.4045, 3.4846, 3.7433, 3.7455, 3.9143, 4.8073, 5.4005, 5.4435, 5.5295, 6.5541, and 9.0960.
Table 3 presents the point estimation of the Kevlar 49/epoxy strands parameters while Table 4 presents the corresponding interval estimation. Table 5 compares the Weibull claim distribution based on some detection criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and consistent Akaike information criterion (CAIC). The goodness-of-fit results of the Weibull claim model (W-claim) are compared with those of some other models, including the Exponential claim distribution (Exp-claim), the Weibull distribution (W-D), and the exponential distribution (EXP-D). Table 6 compares the W-claim with the Kolmogorov–Smirnov test for one sample.
6. Discussion and Future Framework
The experimental side showed, depending on the two statistical criteria, the mean square errors and the bias in estimating the parameters, that the Bayes method ranks first in computing the parameter estimates and that the MLE method ranks second, especially for small samples. As the sample size increases, the MSE values for each experiment decrease. The values of the cumulative distribution are between zero and one, and they increase and are directly proportional to time. Based on the test results, it was shown that the proposed distribution best represents the data compared with competing distributions based on some AIC, BIC, HQIC, and CAIC. The goodness-of-fit results of the Weibull claim model are compared with Exp-claim, W-D, and EXP-D. The results in Tables 5 and 6 suggest that the W-claim distribution provides a better fit than other competing models and could be chosen as a suitable model for analyzing heavy-tail engineering data.
We recommend the use of the Bayes method for parameter estimation. In the future, it is possible to expand the scientific aspects associated with the application, such as the medical, engineering, and industrial aspects. Other estimation methods can also be used to estimate the parameters of the proposed new Weibull claim model and compare them with the methods used in this study.
7. Concluding Remarks
In this paper, we introduced a new extension of the Weibull distribution, the Weibull claim model. We determined the maximum likelihood estimators for the parameters of the Weibull claim distribution and performed a Bayesian Monte Carlo simulation study. The performances of the Bayesian estimators are better than those of the corresponding ML estimators. The new Weibull claim model outperformed many other old models. The performance of the Bayesian estimates is more excellent than that of the corresponding ML estimators. We found that the proposed model provides a better characterization of industry events by analyzing the Kevlar 49/epoxy strands data. Therefore, it can be a perfect model for predicting future cases.
Data Availability
All the datasets used in this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The study was funded by Researchers Supporting Project number (RSP2022R488), King Saud University, Riyadh, Saudi Arabia.