Extended Marshall–Olkin Model and Its Dual Version

Abstract

We propose an extension of the generalized bivariate Marshall-Olkin model assuming dependence between the random variables involved. Probabilistic, aging properties, and survival copula representation of the extended model are obtained and illustrated by examples. Bayesian analysis is performed and possible applications are discussed. A dual version of extended Marshall-Olkin model is introduced and related stochastic order comparisons are presented. IntroductionA variety of bivariate (multivariate) extensions of the univariate exponential distribution have been considered in the literature. These include the distributions of [6,10,11,24], see a full review in [2].The vector (X 1 , X 2 ) meets the Marshall-Olkin model (MO hereafter) whenever it admits the stochastic representationwhere the random variables T i are independent and exponentially distributed with parameters λ i > 0, respectively, i = 1, 2, 3. The random variables T 1 and T 2 in (6.1) can be interpreted as the arrival time of individual shocks for two different components, while T 3 represents the time of arrival of a shock common to both components. Several applications of (6.1) can be mentioned. In reliability theory it can describe the lifetime of a system of two components operating in a random environment and subject to three independent

A weak version of bivariate lack of memory property

“…With this additional source of dependence extended Marshall-Olkin's distributions allow modeling both positive and negative quadrant dependence between its components. Moreover, the model may be non-exchangeable even if the marginals have the same distribution, consult Pinto and Kolev (2015) for details.…”

“…The corresponding joint distributions do not possess BLMP, i.e., are "aging". As a further step, Pinto and Kolev (2015) introduced the Extended MO model assuming dependence between variables T 1 and T 2 , but keeping T 3 independent of them. The motivation is that the individual shocks might be dependent if the items share a common environment.…”

“…Recall that Pinto and Kolev (2015) introduced an Extended MO model with singularity along the line {x 1 = x 2 } in R 2 + . The important consequence of Lemma 4.3 is that it gives us an idea how to define a modified Extended MO model, but with a possible singularity along the line L ω : {x 1 = ωx 2 }.…”

Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

“…In 2014 the model was extended to the multidimensional case by Lin and Li [4]. As an added step, in 2015 Pinto and Kolev [5] introduced the extended BLMP model assuming the dependence between individual shocks, but keeping the third one independent of the previous two. Their motivation is that the individual shocks might be dependent if the items share a common environment.…”

Joint Life Insurance Pricing Using Extended Marshall–olkin Models

“…More specifically, let the distribution of the pair (X m , X f ) be defined via the joint survival functionF X m ,X f (x, y) = P(X m > x, X f > y) for x, y ≥ 0, and consider a continuous non-negative random variable Z with survival functionF Z (x) = P(Z > x), independent of X m and X f . Thus, we arrive to the Extended Marshall-Olkin (EMO) model introduced in Pinto and Kolev (2015) and defined as follows:…”