Inequalities for a distribution with monotone hazard rate

  • Ryoichi Shimizu


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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Copyright information

© The Institute of Statistical Mathematics, Tokyo 1986

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  • Ryoichi Shimizu

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