Julio Mulero

Bivariate Aging Properties under Archimedean Dependence Structures

Let X = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X t = [(X − t, Y − t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X t , with t ≥ 0, were studied in Pellerey (2008), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X t and X t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.

Univariate stochastic orders

In this chapter, we introduce and study some of the main multivariate stochastic orders that we can find in the literature. Again, the list is not exhaustive and the idea of this chapter is to give the reader a manageable contact with the topic of stochastic orders in the multivariate case. We provide formal definitions and some results on sufficient conditions for the multivariate stochastic orders to hold, with special emphasis in the case of random vectors with the same copula. We also focus our attention to the derivation of comparisons of convolutions when the addends are possibly dependent. Finally, we provide a section on applications to the comparison of conditionally independent random vectors and ordered data.

On conditional skewness with applications to environmental data

The statistical literature contains many univariate and multivariate skewness measures that allow two datasets to be compared, some of which are defined in terms of quantile values. In most situations, the comparison between two random vectors focuses on univariate comparisons of conditional random variables truncated in quantiles; this kind of comparison is of particular interest in the environmental sciences. In this work, we describe a new approach to comparing skewness in terms of the univariate convex transform ordering proposed by van Zwet (Convex transformations of random variables. Mathematical Centre Tracts, Amsterdam, 1964), associated with skewness as well as concentration. The key to these comparisons is the underlying dependence structure of the random vectors. Below we describe graphical tools and use several examples to illustrate these comparisons.

Chapter 1 – Preliminaries

In this chapter, we introduce several concepts for univariate and multivariate distributions, which will be used in the book. We provide the definition and interpretation of some functions in several contexts, like reliability, survival analysis, risks, and economics. These functions will be used to define several stochastic orders in Chapters 2 and 3. In order to provide examples based on some data sets, we also provide non-parametric estimators for some of these functions. Besides, we give the definition of some parametric models of univariate and multivariate distributions, as well as some results on total positivity theory and dependence.

Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions

Distorted distributions were introduced in the context of actuarial science for several variety of insurance problems. In this paper we consider the quantile-based probabilistic mean value theorem given in Di Crescenzo et al. [4] and provide some applications based on distorted random variables. Specifically, we consider the cases when the underlying random variables satisfy the proportional hazard rate model and the proportional reversed hazard rate model. A setting based on random variables having the ‘new better than used’ property is also analyzed.