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I have come across a needed condition on a continuous probability distribution defined over $[0, \infty]$ and wonder whether it has a name. For a distribution with CDF $F$ and pdf $f$ I need that the quantity: $$\phi(x) \equiv \frac{f(x)}{F(x)+xf(x)}$$ be monotonically non-increasing. Putting the condition in another form (by taking derivatives), the requirement is that for all $x \in [0,\infty]$ such that $f'(x) > 0$: $$f(x) \geq \sqrt{\frac{F(x) f'(x)}{2}}$$

This seems to be a commonly satisfied property, so does it have a name? It is related to, but distinct from a monotone hazard rate condition.

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    $\begingroup$ Have you looked at monotonicity of the mean residual life? (Maybe that's how you derived the conditions in the first place?) $\endgroup$ – guest Dec 19 '11 at 17:52
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    $\begingroup$ Can you tell us a bit about how the quantity ϕ comes about? $\endgroup$ – r.e.s. Dec 19 '11 at 18:38
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This is almost the condition for the cumulative distribution function to be log-concave , which is a very useful property with many applications. But almost.

A function $F(x)$ is log-concave if

$$\frac {\partial^2 \ln F(x)}{\partial x^2} \le 0 \Rightarrow F''(x)F(x) - \left[F'(x)\right]^2 \le 0$$

Write $\phi(x)$ in terms of $F(x)$

$$\phi(x) \equiv \frac{F'(x)}{F(x)+xF'(x)}$$

and we want

$$\frac {\partial \phi(x)}{\partial x} \le 0 \Rightarrow F''(x)\Big(F(x)+xF'(x)\Big)-F'(x)\Big(F'(x)+F'(x) +xF''(x)\Big) \le 0$$

$$\Rightarrow F''(x)F(x)-2\left[F'(x)\right]^2 \le 0 $$

...which is not enough for log-concavity, due to the existence of the factor $2$.

Assume that the condition is satisfied. If we divide by $[F(x)]^2$ and rearrange we obtain

$$\frac {\partial \phi(x)}{\partial x} \le 0 \Rightarrow \frac {\partial^2 \ln F(x)}{\partial x^2} \le \left( \frac{F'(x)}{F(x)}\right)^2 = \left(\frac {\partial \ln F(x)}{\partial x}\right)^2$$

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