Carolina Martínez-Riquelme

Comparison of hazard rates for dependent random variables

In this paper, we define and study a new notion for the comparison of the hazard rates of two random variables taking into account their mutual dependence. Properties, applications and the comparison for a data set are given.

On sufficient conditions for the comparison in the excess wealth order and spacings

The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

On sufficient conditions for mean residual life and related orders

A well known sufficient condition for the mean residual life order of two random variables is the hazard rate order of the two random variables. The hazard rate order is characterized by the monotonicity of the ratio of the two survival functions. However in many cases this ratio is non monotone and the hazard rate order does not hold. The purpose is to show that, in some situations, this non monotonicity is still a sufficient condition for the mean residual life order, under some additional mild conditions. Applications to compare some parametric models of distributions and generalized order statistics are provided. Similar results are given for the mean inactivity time order. Highlights? We give conditions for the mean residual life order based on the ratio of the survival functions. ? The result can be easily applied to provide results for parametric models and ordered data. ? Similar results are given for the mean inactivity time.

On sufficient conditions for the comparison of some quantile-based measures

ABSTRACT In this article, we focus on characterizations and sufficient conditions for the comparison of some quantile-based measures. Mainly we focus on crossing conditions and changes of monotonicity of the quantile functions. The results are given for the comparison of the tail value at risk, total time on test, and the expected proportional shortfall measures.

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.

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.