Research Article The Weibull Claim Model: Bivariate Extension, Bayesian, and Maximum Likelihood Estimations Walid Emam and Yusra Tashkandy Department of Statistics and Operations Research, Faculty of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Correspondence should be addressed to Walid Emam; walid_emam42@yahoo.com Received 6 January 2022; Revised 20 March 2022; Accepted 5 April 2022; Published 4 May 2022 Academic Editor: Emilio G´ omez-D´ eniz Copyright © 2022 Walid Emam and Yusra Tashkandy. is 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. Using a class of claim distributions, we introduce the Weibull claim distribution, which is a new extension of the Weibull distribution with three parameters. e maximum likelihood estimation method is used to estimate the three unknown pa- rameters, 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. e 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 e use of statistical distributions to model life phenomena has attracted considerable research interest. Recent articles have demonstrated the potential of statistical distributions in mod- elling life data. e Weibull distribution with two parameters is a well-known model that can be effectively used for data mod- elling in lifetime analysis. e Weibull distribution was intro- duced by Frechet [1] and first applied by Rosin and Rammler [2] to describe the distribution of a particle size. e Weibull distribution has many applications in most fields. Let X be a random variable (R.V.) that follows the two-parameter Weibull distribution (λ, c), then its cumulative distribution function (CDF), denoted by F(x; λ, c), is given by F(x; λ, c) 1 − exp − x λ  c  ,x ≥ 0; λ, c > 0. (1) e corresponding probability density function (PDF), survival function (SF), and hazard rate function (HRF) of the Weibull R.V. are given, respectively, by f(x; λ, c) c λ x λ  c− 1 exp − x λ  c  ,x > 0, S(x; λ, c) exp − x λ  c  ,x > 0, (2) and h(x; λ, c) c λ x λ  c− 1 ,x > 0. (3) 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. e Weibull claim distribution is introduced using the class of claim dis- tributions introduced by Ahmad et al. [4] based on the Marshall and Olkin mechanism. e CDF and the PDF of a class of claim distributions are, respectively, given by Hindawi Mathematical Problems in Engineering Volume 2022, Article ID 8729529, 10 pages https://doi.org/10.1155/2022/8729529