For a couple of lifetimes (
X
1,
X
2) with an exchangeable joint survival function
F̄, attention is focused on notions of bivariate aging that can be described in terms of properties of the level ...curves of
F̄. We analyze the relations existing among those notions of bivariate aging, univariate aging, and dependence. A goal and, at the same time, a method to this purpose is to define axiomatically a correspondence among those objects; in fact, we characterize notions of univariate and bivariate aging in terms of properties of dependence. Dependence between two lifetimes will be described in terms of their survival copula. The language of copulæ turns out to be generally useful for our purposes; in particular, we shall introduce the more general notion of
semicopula. It will be seen that this is a natural object for our analysis. Our definitions and subsequent results will be illustrated by considering a few remarkable cases; in particular, we find some necessary or sufficient conditions for Schur-concavity of
F̄, or for IFR properties of the one-dimensional marginals. The case characterized by the condition that the survival copula of (
X
1,
X
2) is Archimedean will be considered in some detail. For most of our arguments, the extension to the case of
n>2 is straightforward.
Tropical planar networks Gaubert, Stéphane; Niv, Adi
Linear algebra and its applications,
06/2020, Volume:
595
Journal Article
Peer reviewed
Open access
We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network.
We deduce a uniqueness theorem for the ...factorization of a tropical totally positive in terms of elementary Jacobi matrices.
In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding ...light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory.
Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first ...recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems.
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