Wrapped Cauchy distribution

In probability theory, the Mills ratio (or Mills's ratio) of a continuous random variable $X$ is the function

$m(x):=\frac{\overline{F}(x)}{f(x)},$

where $f(x)$ is the probability density function, and

$\overline{F}(x):=\mathrm{Pr}[X>x]=\underset{x}{\overset{+\mathrm{\infty }}{\int }}f(u)du$

is the complementary cumulative distribution function (also called survival function). The concept is named after John P. Mills. The Mills ratio is related to the hazard rate h(x) which is defined as

$h(x):=\underset{\delta \to 0}{\lim }\frac{1}{\delta }\mathrm{Pr}[x<X\le x+\delta |X>x]$

by

$m(x)=\frac{1}{h(x)}.$

Example

If $X$ has standard normal distribution thenTemplate:Cn

$m(x)\sim 1/x,$

where the sign $\sim$ means that the quotient of the two functions converges to 1 as $x\to +\mathrm{\infty }$. More precise asymptotics can be given.

See also

• Inverse Mills ratio

References

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