Abstract
Copulas are simply equivalent structures to joint distribution functions. Then, we propose modified structures that depend on classical probability space and concepts with respect to copulas. Copulas have been presented in equivalent probability measure forms to the classical forms in order to examine any possible modern probabilistic relations. A probability of events was demonstrated as elements of copulas instead of random variables with a knowledge that each probability of an event belongs to [0,1]. Also, some probabilistic constructions have been shown within independent, and conditional probability concepts. A Bay's probability relation and its properties were discussed with respect to copulas. Moreover, an extension of multivariate constructions of each probabilistic copula has been presented. Finally, we have shown some examples that explain each relation of copula in terms of probability space instead of distribution functions.
Highlights
Recently, copulas have played an essential role in many applications that their structures need a perfect statistical inference
Copulas are used as an efficient statistical instrument to describe the dependence structures of random variables
Copula is a modern phenomenon in the world of statistics and especially in statistical inference of non-linear data
Summary
Copulas have played an essential role in many applications that their structures need a perfect statistical inference. This work is an attempt to demonstrate copulas in association with the classical probability space This means that our concern focuses on the language of events as elements of assigned copulas rather than the language of random variables and distribution functions. Basic Concepts we present the most common definitions, and properties related to probability space structure, copulas and their constructions. We propose constructions of copulas in terms of probability measure space instead of distributing functions (whether univariate or bivariate distribution functions). It is an approach that allows us to investigate the properties of those functions via probability measure space and test some characteristics that might differ from classical properties. Definition 5: A function PC: [0,1]2 → [0,1] is said to be a bivariate probabilistic intersection copula, denoted by b.p.i.c, if the following conditions hold:
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