Copulas for statistical signal processing (Part I): Extensions and generalization

• Extensions and generalization of copulas families are proposed for exponential, Rayleigh, Weibull, log-normal, Nakagami-m and Rician distributions.
• To explore the dependence between these distributions, their associated bivariate copula functions are derived from their bivariate probability distributions.
• To prove the copula functions of bivariate exponential, Rayleigh and Weibull distributions are equivalent and defined using one parameter.
• To prove log-normal copula is equivalent to the Gaussian one; also simplify the Rician copula function with two parameters.
• Associated copula density functions are derived; the work fills many blank areas in this field (see in Table 1).

Existing works on multivariate distributions mainly focus on limited distribution functions and require that the associated marginal distributions belong to the same family. Although this simplifies problems, it may fail to deal with practical cases when the marginal distributions are arbitrary. To this end, copula function is employed since it provides a flexible way in decoupling the marginal distributions and dependence structure for random variables. Among different copula functions, most researches focus on Gaussian, Student's t and Archimedean copulas for simplicity. In this paper, to extend bivariate copula families, we have constructed new bivariate copulas for exponential, Weibull and Rician distributions. We have proved that the three copula functions of exponential, Rayleigh and Weibull distributions are equivalent, constrained by only one parameter, thus greatly facilitating practical applications of them. We have also proved that the copula function of log-normal distribution is equivalent to the Gaussian copula. Moreover, we have derived the Rician copula with two parameters. In addition, the modified Bessel function or incomplete Gamma function with double integrals in the copula functions are simplified by single integral or infinite series for computational efficiency. Associated copula density functions for exponential, Rayleigh, Weibull, log-normal, Nakagami-m and Rician distributions are also derived.