Stochastic ordering of medians in exchangeable trivariate normal vectors

In this paper, we establish the stochastic ordering of median from an exchangeable trivaxiate normal vector based on the strength of the correlation coefficient. Specifically, by considering two exchangeable trivariate normal vectors with different correlation coefficients, we show that the absolute value of the median in the vector with smaller correlation coefficient is stochastically smaller than the absolute value of the median in the vector with larger correlation coefficient. We prove this result by utilizing skew-normal distributions.

Analysis of a POMDP Model for an Optimal Maintenance Problem with Multiple Imperfect Repairs

I consider a system whose deterioration follows a discrete-time and discrete-state Markov chain with an absorbing state. When the system is put into practice, I may select operation (wait), imperfect repair, or replacement at each discrete-time point. The true state of the system is not known when it is operated. Instead, the system is monitored after operation and some incomplete information concerned with the deterioration is obtained for decision making. Since there are multiple imperfect repairs, I can select one option from them when the imperfect repair is preferable to operation and replacement. To express this situation, I propose a POMDP model and theoretically investigate the structure of an optimal maintenance policy minimizing a total expected discounted cost for an unbounded horizon. Then two stochastic orders are used for the analysis of our problem.

Representations for reliability functions of conditional coherent systems with INID components and ordered properties

This paper presents several useful mixture representations for the reliability function of the residual live of a coherent system with independent but non-identically distributed components. These presentations are based on order statistics, signatures and mean reliability functions. We then discuss some stochastic comparisons of residual lives between two systems based on the stochastic ordering of coefficient vectors （or components） of the two systems. These results form nice extensions of some known results for the case of independent and identically distributed components.

Risk Measure and Premium Distribution on Catastrophe Reinsurance

In this paper, we propose a new risk measure which is based on the Or- licz premium principle to characterize catastrophe risk premium. The intention is to develop a formulation strategy for Catastrophe Fund. The logarithm equivalent form of reinsurance premium is regarded as the retention of reinsurer, and the differential earnings between the reinsurance premium and the reinsurer＇s retention is accumu- lated as a part of Catastrophe Fund. We demonstrate that the aforementioned risk measure has some good properties, which are further confirmed by numerical simu- lations in R environment.

Power Hamza Distribution with Application to Lifetime Data

In this paper, a three-parameter lifetime distribution named power Hamza distribution (PH) is proposed. The PH distribution is a useful generalization of the Hamza distribution which accommodates heavy-tailed, upside-down bathtub and J-shaped hazard rates making it more flexible than the Hamza distribution for modelling various kinds of lifetime data. A comprehensive account of the properties of this distribution is presented. The maximum likelihood estimators of the unknown model parameters are discussed. Finally, a real-life data is analyzed for illustrative purpose proving that the PH outperforms the Hamza distribution and several other lifetime distributions.

THE SINGLE MACHINE STOCHASTIC SCHEDULING WITH THE WEIGHTED JOB TARDINESS MINIMIZATION

This paper considers scheduling n jobs on a single machine where the job processing times anddue dates are independent random variables with arbitrary distribution functions.We consider the casethat the weighted job tardiness in expectation is minimized.It is assumed that job's due dates arecompatible with processing times and weights.We show that the jobs should be sequenced indecreasing stochastic order of their due dates.

A LIMITED ORDER CAPACITY STOCHASTIC INVENTORY MODEL WITH A FIXED COST FOR ORDER: THE DISCOUNTED CASE

This paper considers a single-item, periodic-review inventory model with linear ordercosts, a convex function representing expected one-period costs, nonegative i.i.d. demandsand a fixed cost for order. Stockouts are backordered. All data are stationary Both finiteand infinite horizon problems are treated.

Stochastic comparisons on total capacity of weighted k-out-of-n systems with heterogeneous components

This paper carries out stochastic comparisons on the total capacity of weighted k-out-of-n sys-tems with heterogeneous components.The expectation order,the increasing convex/concave order and the usual stochastic order are employed to investigate stochastic behaviours of sys-tem capacity.Sufficient conditions are established in terms of majorisation-type orders between the vectors of component lifetime distribution parameters and the vectors of weights.Some examples are also provided as illustrations.

Stochastic ordering by g-expectations

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations.Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity,monotonicity,and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations.Applications to contingent claim price comparison under different hedging portfolio constraints are provided.

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

Two classes of multivariate DMRL distributions and a class of multivariate NBUE distributions are introduced in this paper by using conditional stochastic order.That is, a random vector belongs to a multivariate DMRL class of life distributions if its residual life(defined as a conditional random vector)is decreasing in time under convex or linear order.Some conservation properties of these classes are studied.

On controllability for a class of second order semilinear stochastic systems

Many dynamical systems are modeled as second order differential systems.In this paper we address the problem of controllability for second order semilinear stochastic systems in finite dimensional spaces.Sufficient criteria for the complete controllability are formulated under some appropriate assumptions.The results are obtained by using the Banach fixed point theorem.Finally an illustrative example is provided.

STATIONARY OSCILLATIONS OF SECOND ORDER PERIODIC SYSTEMS UNDER STOCHASTIC NOISE INTERFERENCE

In this paper,the existence of stationary oscillations of second order periodic systemsunder stochastlc noise interference is investigated,and the necessary and sufficient condition for the existence of stationary oscillations in the sense of mean-square stability is obtained via establishing equivalent vector stochastic systems and establishing deterministicequivalent systems in the sense of stability, and applying the existence theorem of stationary oscillations for the deterministic systems.

Ordering Results on Largest Order Statistics from Multiple-Outlier Gamma Variables

In this article,we carry out stochastic comparisons on the maximum order statistics arising from two batches of multiple-outlier gamma random variables with different shape and scale parameters.It is proved that,under certain conditions,the majorization order between the vectors of shape parameters together with the weak majorization order[p-larger order]between the vectors of scale parameters implies the likelihood ratio order[hazard rate order]between the largest order statistics.The results established here strengthen and generalize some known ones in the literature.

Optimal Grouping of Heterogeneous Components in Series and Parallel Systems Under Archimedean Copula Dependence

This paper considers series and parallel systems comprising n components drawn from a heterogeneous population consisting of m different subpopulations.The components within each subpopulation are assumed to be dependent,while the subpopulations are independent of each other.The authors also assume that the subpopulations have different Archimedean copulas for their dependence.Under this setup,the authors discuss the series and parallel systems reliability for three different cases,respectively.The authors use the theory of stochastic orders and majorization to establish the main results,and finally present some numerical examples to illustrate all the results established here.

Some Ordering Results for the Marshall and Olkin’s Family of Distributions

In this paper,we consider Marshall and Olkin’s family of distributions.The parent(baseline)distribution is taken to be a scaled family of distributions.Two models:(i)modified proportional hazard rate scale and(ii)modified proportional reversed hazard rate scale,are considered.Some stochastic comparison results in terms of the usual stochastic,hazard rate and reversed hazard rate orders are studied to compare order statistics formed from two sets of independent observations following these models.Most of the sufficient conditions are obtained depending on various majorizationtype partial orderings.Further,the setup with multiple-outlier model is taken.Various stochastic orders between the smallest and largest order statistics are developed.Several numerical examples are provided to illustrate the effectiveness of the established theoretical results.

Some Limit Theorems in Geometric Processes

Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {X n , n = 1, 2, · · ·} for which there exists a real number a 】 0, such that {a n−1 X n , n = 1, 2, · · ·} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S n with a 】 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.

Risk excess measures induced by hemi-metrics

The main aim of this paper is to introduce the notion of risk excess measure,to analyze its properties,and to describe some basic construction methods.To compare the risk excess of one distribution Q w.r.t.a given risk distribution P,we apply the concept of hemi-metrics on the space of probability measures.This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures.Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings.Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the valuesρ(Q),ρ(P)of a risk measureρ.We argue that the differenceρ(Q)−ρ(P)neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure.We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

SOME RESULTS ON THE RESIDUAL LIFE AT RANDOM TIME

In this paper, we consider the residual life at random time, i.e. XY=X-Y|X>Y, where X and Y are non-negative random variables. We establish a number of stochastic comparison properties for XY under various assumptions of X and Y. Under the assumption that Y has decreasing reverse hazard rate (DAHR), we show that if X is in any one of the classes IFR, DFR, DMRL or IMRL then XY is in the same class as X. We also obtain some useful bounds for the distribution and the moment of XY. Because the idle time in classical GI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system.

DEVELOPMENTS ON MTP_2 PROPERTIES OF ABSOLUTE VALUE MULTINORMAL VARIABLES WITH NONZERO MEANS

et X=(X1,...,Xn )' have a multivariate normal distribution with mean μ and covariance matrix Σ. In the case μ=0, Karlin and Rinott[6] obtained a necessary and sufficient condition on Σ for |X|=(|X1|,...,|Xn|)' to be MTP2. In this paper we consider the case μ≠0, and give some conditions under which |X| is MTP2. A necessary and sufficient condition is given for |X| to be TP2 when n=2 and μ≠0. Some results about the TP2 stochastic ordering are also given. The results are applied to obtain positive dependence and associated inequalities for multinormal and related distributions.