两类寿命分布函数与指数分布的贴近性

THE PROXIMITY OF TWO CLASS OF LIFE DISTRIBUTIONS TO THE EXPONENTIAL DISTRIBUTIONS

摘要: 本文讨论了两类寿命分布与指数分布之间的贴近性.记F是均值为1,二阶矩为μ₂的寿命分布,F=1-F,\vecG(t)=\int_t^\infty \barF(u) d u.主要结果为:（1）若F∈DMRL,记 p=1-（μ²）/2,则a)\sup _t \geqslant 0\left|F(t)-e^-t\right| \leqslant 1-e^-2 p b)0 \leqslant e^-t-\barG(t) \leqslant \rho c)0 \leqslant \widetildeF(t)-\barG(t) \leqslant 2 \rho（2）若F∈HNWUE,,且μ²<<∞。,则\sup _l \geqslant 0\left|\vecF(t)-e^-1\right| \leqslant \sqrt836 \rho /(3+2 \sqrt\rho)^2, \rho=\frac\mu_22-1.进一步此上界还可改善为（13）。最后,还讨论了\vecG(t)-e^-t的界。

Abstract: In this paper we discuss the proximity of two classes of life distributions to the exponential distributions. Let F be a life distribution with mean 1, and denote \mu_2 is the seoond moment of F, \barG(t)=\int_t^\infty \barF(u) d u, \barF=1-F. The maim results are: (1) If F \in DMRL, then a) \sup _t>0\left|\barF(t)-e^-t\right| \leqslant 1-e^-2 \rho, b) 0 \leqslant e^-t-\barG(t) \leqslant \rho, e) 0 \leqslant \barF(t)-\barG(t) \leqslant 2 \rho, where \rho=1-\frac\mu_22. (2) If F \in \mathrmHNWUE, and \mu_2 is finite, then \sup _t>0\left|\barF(t)-e^-t\right| \leqslant as (6). Finally, we discuss the bound of \barG(t)-e^-t.