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

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.