On Bivariate Exponentiated Extended Weibull Family of Distributions

In this paper, we introduce a new class of bivariate distributions called the bivariate exponentiated extended Weibull distributions. The model introduced here is of Marshall-Olkin type. This new class of bivariate distributions contains several bivariate lifetime models. Some mathematical properties of the new class of distributions are studied. We provide the joint and conditional density functions, the joint cumulative distribution function and the joint survival function. Special bivariate distributions are investigated in some detail. The maximum likelihood estimators are obtained using the EM algorithm. We illustrate the usefulness of the new class by means of application to two real data sets.Comment: arXiv admin note: text overlap with arXiv:1501.03528 by other author

Generalized Marshall-Olkin Distributions, and Related Bivariate Aging Properties

National Natural Science Foundation of China [10771090]A class of generalized bivariate Marshall-Olkin distributions, which includes as special cases the Marshall-Olkin bivariate exponential distribution and the Marshall-Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well. (C) 2011 Elsevier Inc. All rights reserved

Revisiting Relations between Stochastic Ageing and Dependence for Exchangeable Lifetimes with an Extension for the IFRA/DFRA Property

We first review an approach that had been developed in the past years to introduce concepts of "bivariate ageing" for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing. A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. "Archimedean" models have an important role in the present framework. In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of "Pseudo-Archimedean" models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD

A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows $${\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.$$ Under mild conditions on $\xi$, we show that the following factorization of exponential functionals $${\rm{I}}_{\xi}\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}$$ holds, where, $\times$ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $\xi$ and $Y$ is a spectrally positive L\'evy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of ${\rm{I}}_{\xi}$ for a large class of L\'evy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes which is of independent interest on its own. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process