Association of probability measures on partially ordered spaces

Lindqvist, Bo Henry

doi:10.1016/0047-259x(88)90076-0

Cited by 60 publications

(72 citation statements)

References 22 publications

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“…Consider an arbitrary renewal process [N (t), t 0]. Since for any t, N (t) is a decreasing functional of the sequence of independent interarrival times, (N (t 1 ), ..., N (t n )) is associated for any t 1 , ..., t n (see Lindqvist, 1988). Thus any renewal process is associated in time and therefore positively upper and lower orthant dependent in time.…”

Section: If [N(t) Tmentioning

confidence: 96%

“…Esary and Proschan (1970) introduced the notion of time association for stochastic processes (also see Lindqvist, 1988) in order to obtain bounds for the reliability of certain systems with dependent components. A realvalued stochastic process [X(t), t 0] is said to be associated in time, if for any set [t 1 , t 2 , ..., t n ], (X(t 1 ), X(t 2 ), ..., X(t n )) is associated.…”

Section: $ Is Puod (Plod Nuod Nlod) If and Only Ifmentioning

confidence: 99%

“…By systematically varying the parameters of a tree, we can obtain the shock loading vector that is positively upper (respectively, negatively upper, positively lower, and negatively lower) orthant dependent (PUOD) (respectively, NUOD, PLOD, and NLOD). We also introduce several notions of orthant dependence in time for a realvalued stochastic process, which are weaker and broader than the notion of association in time (Esary and Proschan, 1970;Lindqvist, 1988). These notions of orthant dependence in time for the shock arrival process allows us to compare the structural difference of two shock arrival processes and to characterize their autocorrelation properties.…”

mentioning

confidence: 98%

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Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

Li¹,

Xu²

2001

Journal of Multivariate Analysis

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Consider a system that consists of several components. Shocks arrive according to a counting process (which may be non-homogeneous and with correlated interarrival times) and each shock may simultaneously destroy a subset of the components. Shock models of this type arise naturally in reliability modeling in dynamic environments. Due to correlated shock arrivals, individual component lifetimes are statistically dependent, which makes the explicit evaluation of the joint distribution intractable. To facilitate the development of easily computable tight bounds and good approximations, an analytic analysis of the dependence structure of the system is needed. The purpose of this paper is to provide a general framework for studying the correlation structure of shock models in the setup of a multivariate, correlated counting process and to systematically develop upper and lower bounds for its joint component lifetime distribution and survival functions. The thrust of the approach is the interplay between a newly developed notion, majorization with respect to weighted trees, and various stochastic dependence orders, especially orthant dependence orders of random vectors and orthant dependence orders of stochastic processes. It is shown that the dependence nature of the joint lifetime is inherited from spatial dependence and temporal dependence; that is, dependence among various components due to simultaneous arrivals and dependence over different time instants introduced by the shock arrival process. The two types of dependency are investigated separately and their joint impact on the performance of the system is analyzed. The results are used to develop computable bounds for the statistics of the joint component lifetimes, which are tighter than the productform bounds under certain conditions. The shock model with a non-homogeneous Poisson arrival process is studied as an illustrative example. The result is also applicable to the cumulative damage model with multivariate shock arrival processes.2000 Academic Press AMS 1991 subject classifications: 60E15; 60K10. Key words and phrases: shock models, correlated shock arrivals, Marshall Olkin distributions, majorization with respect to weighted trees, positive and negative orthant dependence order, orthant dependence in time, non-homogeneous Poisson arrival processes, bounds.

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Section: If [N(t) Tmentioning

confidence: 96%

Section: $ Is Puod (Plod Nuod Nlod) If and Only Ifmentioning

confidence: 99%

mentioning

confidence: 98%

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Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

Li¹,

Xu²

2001

Journal of Multivariate Analysis

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“…Holley's paper [14] initiated much of the work in this area and Thomas' paper [32] is also related. For other results and extensions in the statistics and reliability literature, see Barlow and Proschan [2], Kamae, Krengel, and O'Brien [17], Lindquist [21], and Stoyan [29].…”

mentioning

confidence: 99%

Diffusion equation techniques in stochastic monotonicity and positive correlations

Herbst

Pitt

1991

Probab. Th. Rel. Fields

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Summary. A diffusion equation approach is investigated for the study of stochastic monotonicity, positive correlations and the preservation of Lipschitz functions. Necessary and sufficient conditions are given for diffusion semigroups to be stochastically monotonic and to preserve the class of positively correlated measures. Applications are given which discuss the shape of the ground state for Schr6dinger operators -A + V with FKG potentials V. O. IntroductionIn this paper we investigate the use of diffusion equations as a tool for establishing stochastic monotonicity and correlation inequalities. Our approach to these problems is through diffusion semigroups on IR n which preserve the cone ~/of bounded monotonic increasing functions on IR" and (or) which preserve the class of probability measures on 1R n which have positive correlations in the sense that [,f9 d# > 0 holds for all f g ~ ~/~ with ~fd# = ~9 dl~ = O.The problem of characterizing those diffusion generators G = 89 + ~bJ(x)c~j on IR" for which the corresponding semigroup preserves monotonic functions leads to conditions on the {alJ(x)} and {bJ(x)} of a specific intuitive nature. These are: each aiJ(x) depends only on the coordinates xl and x j, so aiJ(x) = a~J(xl, xj) and aU(x) = ai(xi) while each drift coefficient bJ(x) must be an increasing function of the other coordinates xl for i ~ j. Here, 1 < i, j < n.Since aii(x) = ai(xl), 89 ~2 generates a one-dimensional diffusion Xi(t) without drift. The independent diffusions {X~(t)} may be coupled with pair interactions so that the resulting multidimensional diffusion X(t)= (Xl(t),..., X,(t)) has generator 89 with the infinitesimal correlations a~J(x) satisfying a~J(x) = aiJ (x i, x J). Finally, if an order-preserving drift term ~ bJ(x)c~j is introduced, *

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“…, u s ) in multivariate risk models with positively associated claims. First, we review some notions of associations of stochastic processes and random vectors, which can be found, for example, in Tong (1980), Lindqvist (1988). Let E be a partially ordered Polish space (i.e., a complete, separable metric space) with a closed partial ordering ≤.…”

Section: Bounds For the Ruin Probabilitiesmentioning

confidence: 99%

Multivariate risk model of phase type

Cai

2005

Insurance: Mathematics and Economics

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Association of probability measures on partially ordered spaces

Cited by 60 publications

References 22 publications

Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

Diffusion equation techniques in stochastic monotonicity and positive correlations

Multivariate risk model of phase type

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