We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Multivariate convex orderings, dependence, and stochastic equality

Moments inequalities of aging families of distributions with hypotheses testing applications

In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.

A probabilistic analogue of the mean value theorem and its applications to reliability theory

The equilibrium distribution arises as the limiting distribution of the forward recurrence time in a renewal process. The purpose of this article is to study the relationships between the equilibrium distributions (including its higher derivates) and the original distributions. Some stochastic order relations and the relations between their aging properties are investigated and some applications in the field of insurance and financial investments are given. In addition, the relation between the equilibrium distribution of a series system and the series system of equilibrium distribution is investigated. Bivariate equilibrium distribution whose reliability properties are consistent with those of the univariate equilibrium distribution is defined.

Role Of Equilibrium Distribution In Reliability Studies

Moment inequalities derived from comparing life with its equilibrium form

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions. EXPONENTIAL DISTRIBUTIONS

On weak convergence within the hnbue family of life distributions

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

Families of life distributions characterized by two moments

Interesting characterizations of the exponential distribution have been obtained in classes of life distributions important in reliability theory. The results strengthen some of the analogous conclusions already existing in the literature. AMS (1991) Subject Classification No. Primary 62NOS: Secondaey 90825. 60F99.

Characterizing the Exponential Law Under Laplace Order Domination

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

Manish C. Bhattacharjee

Papers

On weak convergence within the hnbue family of life distributions

Families of life distributions characterized by two moments

Characterizing the Exponential Law Under Laplace Order Domination

On a characterization of gamma distributions via exponential mixtures

Aging with Laplace Order Conserving Survival Under Perfect Repairs