Ratio of CDFs $F(x)/x$ property

Question

In my research project is useful to classify cumulative distributions functions of random variables with support in $[a,b]$ with $0\le a<b\le+\infty$ depending on whether the ratio, $$\dfrac{F(x)}{x}$$ is increasing (or decreasing or constant) everywhere in [a,b].

Of course several random variables do not fit in this classification. Also if $0<a$ then the ratio can never be everywhere constant nor decreasing. My question is:

1. Does the property of $F(x)/x$ being increasing has a known name?
2. As we can interpret this ratio as a ratio of the CDF of our original random variable and the CDF of a Uniform $[0,b]$ if $b<\infty$, is there any known property/name regarding the ratio of two CDFs being increasing?
3. If yes to 1 or 2, any references?

Answer 1

First, a few comments:

1. As shown in the comment by whuber, the assertion that if $a<0$, then $F(x)/x$ cannot decrease on $[a,b]$ is false. I have no problem with the assertion that $F(x)/x$ cannot be constant.
2. The CDF of a uniform random variable on [0,b] (with $b<\infty$) is given by $F(x)=\frac xb$. Thus, if $b$ is finite, then $b\cdot \frac{F(x)}x$ is the ratio of the CDFs you mention. Note, however, that $\frac{F(x)}x$ will be increasing/decreasing if and only if $b\cdot \frac{F(x)}x$ is increasing/decreasing. If $b=\infty$, then there is no such thing as a uniform random variable on $[0,b)$, hence your intuitive interpretation of the ratio can only make sense when $F$ is supported on a finite interval.

As for some sort of classification, if you assume the variables you're considering have a density (i.e., $F$ is differentiable), then coming up with a nice algebraic test to see if $F(x)/x$ increases/decreases shouldn't be too difficult:

Suppose that the density $f(x)=F'(x)$ exists. According to the quotient rule for differentiation, one has $$\left(\frac{F(x)}x\right)'=\frac{F'(x)x-F(x)}{x^2}=\frac{f(x)x-F(x)}{x^2}$$

Given that a differentiable function $g$ is increasing on $[a,b]$ if and only if $g'(x)\geq0$ for all $x\in[a,b]$ and decreasing if and only if $g'(x)\leq0$ for all $x\in [a,b]$, then we obtain

On $[a,b]$, $F(x)/x$ is

1. increasing if and only if $f(x)x\geq F(x)$ for every $x\in[a,b]$;

2. decreasing if and only if $f(x)x\leq F(x)$ for every $x\in[a,b]$; and

3. constant if and only if $f(x)x=F(x)$ for every $x\in[a,b]$.

Maybe this does not offer an entirely satisfying classification, but it seems like a good place to start.

Answer 2

Reading the book "Stochastic Orders" I found the answer. Two random variables $X$ and $Y$ are ranked by the reverse hazard rate, $X\stackrel{rhr}{\le} Y$, if and only if: $$ \dfrac{F_X(x)}{F_Y(x)}\text{ is non-decreasing in } [\min(a_X,a_Y),+\infty)$$ where $a_X$ and $a_Y$ are the lower bound of the supports of the random variables.

This is equivalent to $\dfrac{f_Y}{F_Y}\ge \dfrac{f_X}{F_X}$ (which are the respective reverse hazard rates).

The classification above is similar (but not entirely identical) to ranking the the uniform $[0,b]$ and the original random variable according to the reverse hazard rate order.