On Modeling Bivariate Lifetime Data in the Presence of Inliers

Abstract

Many items fail instantaneously or early in life-testing experiments, mainly in electronic parts and clinical trials, due to faulty construction, inferior quality, or non-response to treatments. We record the observed lifetime as zero or near zero, defined as instantaneous or early failure observations. In general, some observations may be concentrated around a point, and others follow some continuous distribution. In data, these kinds of observations are regarded as inliers. Some unimodal parametric distributions, such as Weibull, gamma, log-normal, and Pareto, are usually used to fit the data for analyzing and predicting future events concerning lifetime observations. The usual modelling approach based on uni-modal parametric distributions may not provide the expected results for data with inliers because of the multi-modal nature of the data. The correlated bivariate observations with inliers also frequently occur in life-testing experiments. Here, we propose a method of modelling bivariate lifetime data with instantaneous and early failure observations. A new bivariate distribution is constructed by combining the bivariate uniform and bivariate Weibull distributions. The bivariate Weibull distribution has been obtained by using a 2-dimensional copula, assuming that the marginal distribution is a two-parametric Weibull distribution. An attempt has also been made to derive some properties (viz. joint probability density function, survival (reliability) function, and hazard (failure rate) function) of the modified bivariate Weibull distribution so obtained. The model’s unknown parameters have been estimated using a combination of the Maximum Likelihood Estimation technique and machine learning clustering algorithm, viz. Density-Based Spatial Clustering of Applications with Noise (DBSCAN). Numerical examples are provided using simulated data to illustrate and test the performance of the proposed methodologies. Relevant codes and necessary computations have been developed using R and Python languages. The proposed method has been applied to real data with possible inflation. It has been observed that the data contain inliers with a probability of 0.57. The study also does a comparison test with the proposed method and the existing method in the literature, wherein it was found that the proposed method provides a significantly better fit than the base model (in literature) with a P value less than 0.0001.