José-María Ruiz

Stochastic Comparisons of Generalized Order Statistics

In this article, we give several results on (multivariate and univariate) stochastic comparisons of generalized order statistics. We give conditions on the underlying distributions and the parameters on which the generalized order statistics are based, to obtain stochastic comparisons in the stochastic, dispersive, hazard rate, and likelihood ratio orders. Our results generalize some recent results for order statistics, record values, and generalized order statistics and provide some new results for other models such as k-record values and order statistics under multivariate imperfect repair.

STOCHASTIC COMPARISONS OF NONHOMOGENEOUS PROCESSES

The purpose of this article is to describe various conditions on the parameters of pairs of nonhomogeneous Poisson or pure birth processes under which the corresponding epoch times or interepoch intervals are stochastically ordered in various senses. We derive results involving the usual stochastic order, the multivariate hazard rate order, the multivariate likelihood ratio order, as well as the dispersive and the mean residual life orders. A sample of applications involving generalized Yule processes, load-sharing models, and minimal repairs in reliability theory illustrate the usefulness of the new results.

ON PARTIAL ORDERINGS BETWEEN COHERENT SYSTEMS WITH DIFFERENT STRUCTURES

In this article, we give some results on the preservation of orderings between the components under the formation of coherent systems with different structures. We consider the stochastic, failure rate, reversed failure rate, and likelihood ratio orderings to compare two coherent structures formed from a set of components and from two sets of components, in both cases, when components are independent and either identically distributed or not necessarily identically distributed. As a consequence, we get new stochastic comparisons of systems with different k-out-of-n structures, and some results of partial orderings given by other authors may be also obtained from ours.

Stochastic comparisons of multivariate mixture models

In this paper we consider sufficient conditions in order to stochastically compare random vectors of multivariate mixture models. In particular we consider stochastic and convex orders, the likelihood ratio order, and the hazard rate and mean residual life dynamic orders. Applications to proportional hazard models and mixture models in risk theory are also given.

The dilation order, the dispersion order, and orderings of residual lives

One purpose of this paper is to study the relationship of the dilation order ([less-than-or-equals, slant]dil) to two other stochastic orders: the mean residual life order ([less-than-or-equals, slant]mrl) and the increasing convex order ([less-than-or-equals, slant]icx). Regarding these orders, it is already known that X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]icxY. In this paper we show that for non-negative random variables we actually have X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]dilY => X [less-than-or-equals, slant]icxY (the first implication holds under the assumption that at least one of the two underlying random variables satisfies some aging property). Thus, we refine the result of Theorem 3.A.13 in Shaked and Shanthikumar (1994). Another purpose of this paper is to identify conditions under which all the residual lives, that are associated with two random variables X and Y, are ordered according to the dilation or the dispersion orders. Some of these results extend parts (a) and (b) of Theorem 2.B.13 in Shaked and Shanthikumar (1994).

On allocation of redundant components for systems with dependent components

In this paper we consider the problem of optimal allocation of a redundant component for series, parallel and k-out-of-n systems of more than two components, when all the components are dependent. We show that for this problem is naturally to consider multivariate extensions of the joint bivariates stochastic orders. However, these extensions have not been defined or explicitly studied in the literature, except the joint likelihood ratio order, which was introduced by Shanthikumar and Yao (1991). Therefore we provide first multivariate extensions of the joint stochastic, hazard rate, reversed hazard rate order and next we provide sufficient conditions based on these multivariate extensions to select which component performs the redundancy.

Dispersive orderings and characterization of ageing classes

The dispersive orderings arise in a great variety of contexts. In this paper we bridge the gap between dispersive orderings and ageing classes. We give characterizations of the IFR(DFR) and DMRL(IMRL) classes in terms of the dispersive and dilation orderings.

A family of tests for right spread order

In this paper we address the problem of testing the equality in the right spread order of two random variables. We develop a family of tests statistics for right spread order. This technique is also used for testing NBUE [NWUE] aging classes. For these tests, we provide asymptotic results, and also, for the NBUE [NWUE] test, the exact distribution. We show the performance of these tests by comparing them with existing tests. Applications of the new proposed tests to some data sets are given.

Multivariate properties of random vectors of order statistics

The purpose of this paper is to extend some known results about stochastic comparisons of univariate order statistics to the case of random vectors of order statistics. These multivariate stochastic comparisons are given under conditions of stochastic orders, like the hazard rate order, likelihood ratio order and dispersive order, of the parent distributions. We obtain also some other results about multivariate classification of random vectors of order statistics, in particular, we give conditions on the parent distribution to classify the random vector of order statistics in the MIFR or MPF2 classes. Applications to stochastic comparisons of some transformations of order statistics are also given.

A characterization of continuous multivariate distributions by conditional expectations

For given real, continuous and monotone functions his, we obtain the necessary and sufficient conditions in order that any Rn-valuated function Ψ(x) be the conditional expextation E(h(X)/X>x) of a continuous random vector X, where h(X)=(h1(X1),…,hn(Xn)).