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)} ,

\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]

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

Example

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]

↑ Grimmett, G.; Stirzaker, S. (2001). Probability Theory and Random Processes (3rd ed.). Cambridge. p. 98. ISBN 0-19-857223-9.
↑ Mills, John P. (1926). "Table of the Ratio: Area to Bounding Ordinate, for Any Portion of Normal Curve". Biometrika. 18 (3/4): 395–400. doi:10.1093/biomet/18.3-4.395. JSTOR 2331957.
↑ 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.
↑ 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. .

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