Abstract

In reliability, the lifetime data is usually modeled using one or two parametric distributions, such as Weibull, gamma, log-normal, Pareto, etc., which are unimodal by nature. Sometimes, the data may contain many zeros or close to zero data points, defined as inliers (instantaneous or early failure observations) in the literature. The usual modeling approach using the uni-modal parametric distributions may not provide expected results for such data in the presence of inliers. Furthermore, correlated bivariate observations with inliers frequently occur in reliability; here, we propose a method of modeling bivariate lifetime data with instantaneous and early failure observations. We construct a new bivariate distribution function by combining bivariate uniform and Weibull distributions. The bivariate Weibull distribution is obtained using a 2-dimensional copula, assuming the marginal distributions as two parametric Weibull distributions. We derive some properties of that modified bivariate Weibull distribution, mainly the joint probability density function, the survival (reliability) function, and the hazard (failure rate) function. The model’s unknown parameters are estimated using the Maximum Likelihood Estimation (MLE) technique combined with a machine learning clustering algorithm. Numerical examples are provided using simulated data to illustrate and test the performance of the proposed methodologies. The method is also applied to real data and compared with existing methods in the literature.