On Some Issues Of Stochastic Comparisons & Their Applications

Abstract

One of the important objectives of statistics is the comparison of random quantities. These comparisons are mainly based on the comparison of some measures associated with these random quantities. For example, it is very common to compare two random variables in terms of their means, medians, or variances. In some situations, comparisons based only on two single measures are not very informative. The necessity of providing more detailed comparisons of two random quantities has motivated the development of the theory of stochastic orders, which has grown significantly during the last 50 years. Stochastic order refers to comparing two random quantities in some stochastic sense. It is an important tool used in many diverse areas of statistics, reliability, economics, etc.

Reliability theory and actuarial science are two most important areas where stochastic orders are studied extensively. Usual stochastic ordering, hazard rate ordering, reversed hazard rate ordering for lifetimes of series and parallel systems with heterogeneous and dependent componentshave beenestablished. Dispersive and star order for one heterogeneous and one homogeneous dependent series or parallel systems havealso been established. For two finite mixture models’ comparison, results have beenestablished under the usual stochastic order, hazard rate order and reversed hazard rate order. Various up and down-shifted ordering results have beenestablished for two important continuous mixture models: Frailty and Resilience. Data analysis has been done for illustration purposes. To find the optimal set of redundant components or systems, the usual stochastic ordering, hazard rate ordering and reversed hazard rate ordering of systems lifetimes under active redundancy allocation have beenestablished. Data analysis has also been donein this contextfor illustration purposes.

Actuarial science is another area where stochastic ordershave extensivepotential for application. Usual stochastic ordering and star ordering results have beenestablished for the largest and aggregate claim amounts of two heterogeneous portfolios. Numerical exampleshave been provided in this context to illustrate the results thus obtained.