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

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Exponential approximations in the classes of life distributions

Authors:	Kan Cheng Zongfu He

Institution:	(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 $(\sqrt3]{{4\rho }}.\sqrt3]{{4\rho }})$ . Furthermore, the improved upper bound is $\sqrt3]{{36\rho /(3 + 2\sqrt \rho )^2 }}$ . 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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