Salem Al-Dokar

Salem Al-Doukar

DISSERTATION TITLE:

"Characterization of Some Stochastic Orders in Reliability Theory"

Co-supervisor: Professor Shaban Abo Yousef

INTRODUCTION:

Stochastc comparisons between probability distributions play a fundamental rol in probability, statistics and some related areas, such as reliability theory, survival analysis, economics and actuarial science. As a result, several more refined orders have been comprehensively discussed in the literature. Of the most commonly used comparisons we find the total time on test transform (TTT) comparison and the excess wealth comparison. The TTT transform is a theoretical version of the empirical TTT transform that is often used in statistical reliability theory. Based on the TTT-transform a new stochastic order has been introduced and studied in reliability and life testing. Recently, another ordering called excess wealth order has also come to use in reliability and life testing. For any random variable X, let X_t=[X-t | X>t] denote a random variable whose distribution is the same as the conditional distribution of given that . When X is the lifetime of a device, X_t can be regarded as the residual lifetime of the device at time t, given that the device has survived up to time t. The comparison of the residual life at different times has been used to give characterizations of stochastic orders in statistical reliability theory.

For instance, suppose F is the distribution function of the lifetime of a unit, which could be a living organism or a mechanical component or a system. To determine whether the component is aging with time for the TTT-transform associated with this distribution, Ahmad et al. (2005) proposed a new aging notion related to the TTT-transform order called new better than used in the total time on test transform order. By the same idea(for the new better than used in the total time on test transform order class) we can compare the residual lifetime at different times via "total time on test transform order" study the proposed ageing classes.

OBJECTIVE:

The aim of this dissertation is to to study in this Theises some new stochastic orders appeared recently in literature. In general, by willing of Allah, we will study the following two stochastic orders:

(i) The total time on test order, and

(ii) The excess wealth order.

RRSEARCH PLAN:

[1] Study “the total time on test transform” order and the “excess wealth” order.

[2] Extend some published results, available in the literature.

[3] Study the two stochastic orders of their residual life.

[4] Provide the relationship between ordering of the residual life of the two orders and other stochastic orders.

[5] Provide some applications in the relability theory .

APPLIED FIELDS:

The expected results may be will find practical applications in reliability theory and Life Testing.

EXPECTED TIME OF FINALIZATION:

Finished, April, 2008