Theory and practice of a bivariate trigonometric Burr XII distribution

The precise modeling of bivariate continuous characteristics remains an actual challenge in probability and statistics. In this paper, we explore a new strategy based on the combination of a simple polynomial-sine copula and the Burr XII distribution. The idea is to use the oscillating functionalities of the polynomial-sine copula and the flexibility of the Burr XII distribution to propose a serious bivariate solution for the modelling of bivariate lifetime phenomena. Both theory and practice are developed. In particular, we determine the main functions related to the distribution, like the cumulative distribution function, probability density function, conditional density function, and hazard rate function, and perform a moment analysis, including various useful measures for bivariate modeling. On the practical plan, we derive the maximum likelihood and Bayes estimates for the unknown parameters. Also, the bootstrap confidence interval and the highest posterior density interval are obtained. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, one data set is considered to illustrate its flexibility for real-life applications.