Mills ratio

In probability theory, the Mills ratio (or Mills's ratio^[1]) of a continuous random variable $X$ is the function

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

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

$\bar{F}(x) := \Pr[X>x] = \int_x^{+\infty} f(u)\, du$

is the complementary cumulative distribution function (also called survival function). The concept is named after John P. Mills.^[2] The Mills ratio is related^[3] to the hazard rate h(x) which is defined as

$h(x):=\lim_{\delta\to 0} \frac{1}{\delta}\Pr[x < X \leq x + \delta | X > x]$

by

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

Example

If $X$ has standard normal distribution then

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

where the sign $\sim$ means that the quotient of the two functions converges to 1 as $x\to+\infty$. More precise asymptotics can be given.^[4]

See also

• Inverse Mills ratio

References

1. Grimmett, G.; Stirzaker, S. (2001). Probability Theory and Random Processes (3rd ed.). Cambridge. p. 98. ISBN 0-19-857223-9.
2. Mills, John P. (1926). "Table of the Ratio: Area to Bounding Ordinate, for Any Portion of Normal Curve". Biometrika 18 (3/4): 395–400. JSTOR 2331957.
3. Klein, J. P.; Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data. New York: Springer. p. 27. ISBN 0-387-95399-X.
4. Small, Christopher G. (2010). Expansions and Asymptotics for Statistics. Monographs on Statistics & Applied Probability 115. CRC Press. pp. 48, 50–51, 88–90. ISBN 978-1-4200-1102-9..

External links

• Weisstein, Eric W., "Mills Ratio", MathWorld.