Two Kinds of Stochastic Dependences, Bi-Variate Distributions General Case (original) (raw)

2016

Abstract

Abstract: Two classes of stochastic models in the form of bivariate probability distributions are constructed and their fusion into one class is obtained. The “departure point” of the first class of the models is the Aalen additive version of the famous Cox (1972) proportional hazard rates model. Unlike with the original Cox approach, use of the Aalen (1989) model allows to consider the two underlying marginal random variables in their mutual stochastic interaction i.e., each variable is explanatory for the other. The models, obtained in this way, turned out to have nice and very general form so that the construction yields to the general (and probably universal) characterization of any bivariate survival function. The latter fact resembles the copula representation methodology but our representation is different. The second class of bivariate survival functions we consider, is obtained by the ‘method of parameter dependence’ that we provided in our previous publications. The alternative for this method is the method of triangular (here pseudolinear) transformations where two independent (baseline) random variables are transformed into the random vector whose joint distribution is the same as that obtained by the parameter dependence method. If for this transformation, instead of the independent input random variables, we apply the input random vector, whose distribution belongs to the previously considered class related to the Cox-Aalen paradigms, one obtains fusion of the two classes of bivariate joint survival functions. The fusion, on the level of analytical form of the so obtained survival functions, has a nice property of factorization. One then can talk about two different types of stochastic dependences (mechanisms) comprised in one (composed) analytic model.

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