On the Generalization of the Bivariate Extended Standard U-quadratic Distribution

Abstract

This paper derives the generalized version of the Bivariate extended standard U-quadratic  distribution using the compounding method. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape distributions with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ and $Y$ have inverted bathtub shapes; $(iii)$ $X$ and $Y$ have constant shapes; $(iv)$ $X$ has a constant distribution and $Y$ has a bathtub shape; $(v)$ $X$ has a constant distribution and $Y$ has inverted shape; and $(vi)$ $X$ has an inverted bathtub and $Y$ has a bathtub shape. Moreover, two special cases of the generalized BeSU distribution are constructed, and these are called the Special Bivariate Extended Standard U Quadratic Type I (SBeSU-Type I) and Special Bivariate Extended Standard U Quadratic Type II (SBeSU-Type II) distributions. Further, some properties of this proposed distribution are derived such as the marginal distribution, conditional distribution, conditional moments, conditional mean, conditional variance, product and ratio moments, Pearson correlation coefficient, joint moment generating function, Kendall's tau coefficient, Spearman's rho, and the stress - strength parameter. In addition, maximum likelihood estimation is performed to estimate the parameters of the derived distribution. A simulation study is carried out to evaluate the behavior of the parameter estimates. Finally, the proposed generalization of the  Bivariate eSU distribution is applied to simulated data and compared with the Bivarite extended Standard U-quadratic distribution. The results show that the proposed generalization of the bivariate eSU distribution provides a better fit on the simulated data set than the bivariate eSU distribution.