Inequalities for a distribution with monotone hazard rate

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(X ²) 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$\int_0^\infty {|\bar F(x) - e^{ - x/\mu } |} dx \leqq 2\varepsilon \mu$, ifq ₁ (x) is monotone. It is also shown that Brown's inequality$|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon )$ which holds wheneverq ₁ (x) is increasing is not valid in general whenq ₁ is decreasing.