  In reliability and life data analysis, the Weibull distribution is widely
used to accommodate more data characteristics by changing the values of the
parameters. We frequently observe many zeros or close to zero data points in
reliability and life testing experiments. We call this phenomenon a nearly
instantaneous failure. Many researchers modified the commonly used univariate
parametric models such as exponential, gamma, Weibull, and log-normal
distributions to appropriately fit such data having instantaneous failure
observations. Researchers also find bivariate correlated life testing data
having many observations near a particular point while the remaining
observations follow some continuous distribution. This situation defines as
responses having early failures for such bivariate responses. If the point is
the origin, then we call the situation a nearly instantaneous failure for the
responses. Here, we propose a modified bivariate Weibull distribution that
allows early failure by combining bivariate uniform distribution and bivariate
Weibull distribution. The bivariate Weibull distribution is constructed 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 approaches to model such data
in the literature.
