Reversing conditional orderings

We analyze some specific aspects concerning conditional orderings and relations among them. To this purpose we define a suitable concept of reversed conditional ordering and prove some related results. In particular we aim to compare the univariate stochastic orderings ≤ st, ≤ hr, and ≤ lr in terms of differences among different notions of conditional orderings. Some applications of our result to the analysis of positive dependence will be detailed. We concentrate attention to the case of a pair of scalar random variables X, Y ​. Suitable extensions to multivariate cases are possible

Some Positive Dependence Orderings involving Tail Dependence

In this paper we discuss the properties of the orderings of positive dependence introduced by Hollander et al. (1990) as generalizing the bivariate positive dependence concepts of left-tail decreasing (LTD) and right-tail increasing (RTI) studied by Esary and Proschan (1972). We show which of the postulates proposed by Kimeldorf and Sampson (1987) for a reasonable positive dependence ordering are satisfied and how the orders can be studied by restricting them to copulas, and we give some examples. We also investigate the relationship of these orders with some other orderings which have appeared in the literature and generalize the same notions of positive dependenceCopula; Fréchet class; positive dependence stochastic ordering; right-tail decreasing (RTI); left-tail decreasing (LTD)

Multivariate hazard orderings of discrete random vectors

The task of comparing two random vectors with respect to some multivariate stochastic ordering usually involves an infinite number of comparisons. Dyckerhoff and Mosler (1997) proved that, when the random vectors have finite supports, this task, for some orderings, can be simplified by considering only a small finite number of comparisons. In this paper we extend their results to two multivariate hazard rate stochastic orderings.Multivariate stochastic orderings, dependence orderings, weak hazard rate (whr) ordering, lower orthant decreasing ratio (lodr) ordering, upper orthant increasing ratio (uoir) ordering, multivariate hazard function, hazard gradient.

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are describe

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails