Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Exponential approximations in the classes of life distributions

Exponential approximations in the classes of life distributions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Exponential approximations in the classes of life distributions

Acta Mathematicae Applicatae Sinica , Volume 4 (3) – Jul 13, 2005

Loading next page...
 
/lp/springer-journals/exponential-approximations-in-the-classes-of-life-distributions-Uq7EKkGMS0

References (2)

Publisher
Springer Journals
Copyright
Copyright © 1988 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02006220
Publisher site
See Article on Publisher Site

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 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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

There are no references for this article.