(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

(2007). Stochastic Ageing and Dependence for Reliability. Technometrics: Vol. 49, No. 2, pp. 222-222.

Stochastic Ageing and Dependence for Reliability

A class of bivariate distributions that generalize Marshall-Olkin's one is characterized through a functional equation which involves two associative operations. The obtained distributions concentrate positive mass on the linex=y when the two associative operations coincide; otherwise a positive mass is concentrated on a continuous monotone function.

Characterization of a Marshall-Olkin type class of distributions

Dependence structures of multiple risks play an important role in optimal allocation problems for insurance, quantitative risk management, and finance. However, in many existing studies on these problems, risks or losses are often assumed to be independent or comonotonic or exchangeable. In this paper, we propose several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks. These characterizations are related to the properties of arrangement increasing functions and the proposed notions of dependence incorporate many typical dependence structures studied in the literature for optimal allocation problems. We also develop the properties of these dependence structures. We illustrate the applications of these notions in the optimal allocation problems of deductibles and policy limits and in capital reserves problems. These applications extend many existing researches to more general dependent risks.

Some new notions of dependence with applications in optimal allocation problems

Likelihood ratio order of sample minimum from heterogeneous Weibull random variables

Consider an n–component coherent system having random lifetime \(T_{\boldsymbol{X}}\), where \(\boldsymbol{X} = (X_{1},\ldots,X_{n})\) is the vector of the non-independent components’ lifetimes. Stochastic comparisons of the residual life of \(T_{\boldsymbol{X}}\) at a fixed time t ≥ 0, conditioned on \(\{T_{\boldsymbol{X}} > t\}\) or on \(\{X_{i} > t,\forall i = 1,\ldots,n\}\), are investigated. Sufficient conditions on the vector \(\boldsymbol{X}\) that imply this comparison in the usual stochastic order are provided, together with sufficient conditions under which the lifetime \(T_{\boldsymbol{X}}\) satisfies the NBU aging property.

https://link.springer.com/content/pdf/10.1007%2F978-1-4614-6892-9_11.pdf

On Used Systems and Systems with Used Components

This paper studies the optimal allocation of one active redundancy to a k-out-of-n system in the context of statistically dependent component lifetimes. It is proved that assigning the redundancy t...

Ordering k-out-of-n systems with interdependent components and one active redundancy

We study here extremes of residuals of the bivariate lifetime and the residual of extremes of the two lifetimes. In the case of generalized Marshall–Olkin model and the total time transformed exponential model, we first present some sufficient conditions for the extremes of residuals to be stochastically larger than the residual of the corresponding extremes, and then investigate the stochastic order of the residual of extremes of the two lifetimes based on the majorization of the age vector of the residuals.

On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

Comments on "ordering properties of order statistics from heterogeneous populations"

Consider an n{components coherent system having random lifetime T(X), where X = (X1;:::;Xn) is the vector of the non-independent components’ lifetimes. Stochastic comparisons of the residual life of T(X) at a flxed time t ‚ 0, but conditioned on the difierent events fT(X) > tg and fXi > t;8i = 1;:::;ng, are discussed. Su‐cient conditions on the vector X that imply this comparison in the usual stochastic order are provided, together with su‐cient conditions under which the lifetime T(X) satisfles positive aging properties. The case of negative aging is discussed as well.

Yinping You

Papers

On Used Systems and Systems with Used Components

Ordering k-out-of-n systems with interdependent components and one active redundancy

On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models

Comments on "ordering properties of order statistics from heterogeneous populations"

Stochastic Orders Between Used Systems and Systems with Used Components