摘要:
本文讨论了在某些随机序下寿命分布函数之间差的界. 若 $F$ 为寿命分布, 其均值、二阶矩分别记作 $\mu\left(F^{\prime}\right), \mu_2(F)$ 。主要结果为1) 若 $F< G, \mu(F)=\mu(G)=\mu$ ，则$$\frac{1}{\mu} \int_0^{\infty}|\vec{F}(t)-\bar{G}(t)|d t \leqslant 2 \sqrt{\rho}$$这里$$\rho=p / \mu^2, p=\frac{1}{2}\left[\mu_2(G)-\mu_2(F)\right].$$2) 若 $F< G, \mu(F)=\mu(G)=\mu, G$ 满足 Lipschitz 条件, 即$$\forall x_1, x_2 \geqslant 0, \quad\left|G\left(x_1\right)-G\left(x_2\right)\right|\leqslant M\left|x_1-x_2\right|, \quad M>0$$常数，则$$\sup _{t>0}|\bar{F}(t)-\bar{G}(t)|\leqslant \sqrt[8]{(2 \bar{M})^2 p}$$最后，还在特殊的一类寿命分布族中讨论了用 Weibull 分布作近似的界。
Abstract:
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 $$ \frac{1}{\mu} \int_0^{\infty}|\bar{F}(t)-\bar{G}(t)|d t \leqslant 2 \sqrt{\bar{\rho}} $$ where $$ \rho=p / \mu^2, p=\frac{1}{2}\left[\mu_2(G)-\mu_2(F)\right].$$ 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 $$ \sup _{t>0}|\vec{F}(t)-\bar{G}(t)|\leqslant \sqrt[3]{(2 M)^2 p}.$$ Finally, we discuss the using of Weibull distribution as an approximate bound of a special class of life distributions.
下载:
