Exponential approximations in the classes of life distributions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Exponential approximations in the classes of life distributions

Authors:	Kan Cheng Zongfu He

Affiliation:	(1) Institute of Applied Mathematics, Academia Sinica, China;(2) Air Force Engineering College, China

Abstract:	In this paper we discuss the approximation of life distributions by exponential ones. The main results are: (1) F NBUE, where its mean is 1, we have $\|bar F(t) - e^{ - t} \| leqslant 1 - e^{ - sqrt {20} } ,forall t geqslant 0$ , 0, where = 1 - ₂/2, ₂ being the second moment ofF. The inequality is sharp. (2) In the case ofFIFR, the upper bound is $1 - e^{ - tfrac{rho }{{1 - rho }}}$ . (3) For the HNBUE class, the upper bound is min. Furthermore, the improved upper bound is $sqrt[3]{{36rho /(3 + 2sqrt rho )^2 }}$ . In addition, we show $mathop {sup }limits_{t > 0} \|bar G(t) - e^{ - t} \| leqslant sqrt {frac{rho }{2}}$ , where $bar G(t) = int_t^infty {bar F} (u)du$ (4) For the IMRL class, the upper bound is /(1+) ([1]). Here we give a simple proof.Project supported by the National Natural Science Fund of China.

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