*J. Japan Statist. Soc.,* Vol. 32 (No. 1), pp. 95-105, 2002

## Preservation of Some New Partial Orderings Under Poisson and Cumulative Damage Shock Models

S. E. Khider, A.-H. N. Ahmed and M. K. Mohamed

**Abstract. **
Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity *l*. Let _{k} denote the probability that the first device will survive the *k* shocks and _{k} denote such a probability for second device. Let (*t*) and (*t*) denote the survival functions of the first and second device respectively. In this paper we show that some new partial ordering, namely dual (*D*), dual stochastic (*DST*), dual weak likelihood ratio (*DWLR*), increasing failure rate (*IFR*), dual mean residual lives (*DMRL*) and dual convex (*DCX*) orderings between the shock survival probabilities _{k} and _{k} are preserved by the corresponding survival function (*t*) and (*t*). We also obtain sufficient condition under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

*Key words and phrases*:
Stochastic order; Dual stochastic order; Dual weak likelihood order; Increasing failure rate; Dual mean residual lives; Dual convex; Shock models.

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