Annals of the Institute of Statistical Mathematics

, Volume 38, Issue 2, pp 195–204

# Inequalities for a distribution with monotone hazard rate

• Ryoichi Shimizu
Article

## Summary

LetX be a positive random variable with the survival function$$\bar F$$ and the densityf. LetX have the moments μ=E(X) and μ2=E(X2) and put ε=|1-μ2/2μ2|. Put$$q(x) = f(x)/\bar F(x)$$ and$$q_1 (x) = \bar F(x)/\int_x^\infty {\bar F(u)du}$$. It is proved that the following inequalities hold:$$|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon e)$$, for allx>0, ifq(x) is monotone and that$$\int_0^\infty {|\bar F(x) - e^{ - x/\mu } |} dx \leqq 2\varepsilon \mu$$, ifq1(x) is monotone. It is also shown that Brown's inequality$$|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon )$$ which holds wheneverq1(x) is increasing is not valid in general whenq1 is decreasing.

## Key words

Characterization exponential distribution hazard rate mean residual life

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