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Mark Brown, Guangping Ge (1984)
Exponential Approximations for Two Classes of Aging Distributions.Annals of Probability, 12
Mark Brown (1983)
Approximating IMRL Distributions by Exponential Distributions, with Applications to First Passage TimesAnnals of Probability, 11
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 ρ = 1 - μ2/2, μ2 being the second moment ofF. The inequality is sharp. (2) In the case ofF∈IFR, the upper bound is $$1 - e^{ - \tfrac{\rho }{{1 - \rho }}} $$ . (3) For the HNBUE class, the upper bound is min $$(\sqrt[3]{{4\rho }}.\sqrt[3]{{4\rho }})$$ . Furthermore, the improved upper bound is $$\sqrt[3]{{36\rho /(3 + 2\sqrt \rho )^2 }}$$ . 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 ρ/(1+ρ) ([1]). Here we give a simple proof.
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Jul 13, 2005
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