SOME INEQUALITIES UNDER CERTAIN STOCHASTIC ORDER

In this paper we discuss the bounds of difference between two life distribations under certain stochastic orders.Let F be a life distribution, \mu(F) and \mu_2(F) the first and the second moments of F, respectively.The main results are as follows.1) If F< G, \mu\left(F^\prime\right)=\mu(G)=\mu, then \frac1\mu \int_0^\infty|\barF(t)-\barG(t)|d t \leqslant 2 \sqrt\bar\rho where \rho=p / \mu^2, p=\frac12\left\mu_2(G)-\mu_2(F)\right. 2) If F _v G, \mu(F)=\mu(G)=\mu, and G satisfies Lipschitz condition, i.e., \forall x_1, x_2 \geqslant 0, \quad\left|G\left(x_1\right)-G\left(x_2\right)\right|\leqslant M\left|x_1-x_2\right|, M>0 is a constant, then \sup _t>0|\vecF(t)-\barG(t)|\leqslant \sqrt3(2 M)^2 p. Finally, we discuss the using of Weibull distribution as an approximate bound of a special class of life distributions.