STOCHASTICS ORDERS & THEIR APPLICATIONS

BACKGROUND:

      Stochastic orders and inequalities are being used at an accelerated rate in many diverse areas of mathematics, probability and statistics. In the literature several concepts of stochastic ordering between random variables have been considered. The simplest way of comparing two-distribution functions is by the comparison of the associated means. However, such a comparison is based on only two single numbers (the means), and therefore it is often not very informative. In addition to this, the means sometimes do not exist. In many instances in applications one has more detailed information, for the purpose of comparison of two distribution functions, than just the two means.

      When one wishes to compare two distribution functions that have the same mean (or that are centered about the same value), one is usually interested in the comparison of the dispersive of these distributions. The simplest way of doing it is by the comparison of the associated standard deviations. However, such a comparison, again, is based on only two single numbers, and therefore its not very informative. In addition to this, again, the standard deviations sometimes do not exist. 

     As a result, the researcher need to introduce and study new stochastic orders give more detailed information than just the two means and the two standard deviations.

RECOMMENDED TEXT BOOKS

- Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. Spronger, New-Yourk

- Muller, A. and Stoyan, D. (2002). Comparison Methods for Queues and Other Stochastic Models. Wiley & Sons, New-Yourk.

 LECTURE NOTES:  

 ◘  Basic Notions of the Stochastic Order 

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