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. The Mills ratio is related[2] 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.[3]

See also

References

  1. ↑ G. Grimmett, S. Stirzaker. Probability Theory and Random Processes. 3rd ed. Cambridge. Page 98.
  2. ↑ Klein, J.P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data, Springer, 2003, p.27
  3. ↑ Small, Christopher G. (2010), Expansions and Asymptotics for Statistics, Monographs on Statistics & Applied Probability 115, CRC Press, pp. 48, 50–51, 88–90, ISBN 9781420011029.

External links