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Mills Ratio

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The Mills ratio is defined as

m(x)=1/(h(x))
(1)
=(S(x))/(P(x))
(2)
=(1-D(x))/(P(x)),
(3)

where h(x) is the hazard function, S(x) is the survival function, P(x) is the probability density function, and D(x) is the distribution function.

For example, for the normal distribution,

m_(normal)(x)=e^((x-mu)^2/(2sigma^2))sqrt(pi/2)[1erf((x-mu)/(sqrt(2)sigma))],
(4)

where erf is the error function, which simplifies to

m_(standard normal)(x)=e^(x^2/2)sqrt(pi/2)erfc(x/(sqrt(2))),
(5)

where erfc is the complementary error function, for the standard normal distribution. The latter function has the particularly simple continued fraction representation

m_(standard normal)(x)=1/(x+K_(k=1)^inftyk/x)
(6)
=1/(x+1/(x+2/(x+3/(x+4/(x+...)))))
(7)

(Cuyt et al. 2010, p. 376).

See also

Distribution Function, Hazard Function, Probability Density Function, Survival Function

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References

Baricz, Á. "Mills' Ratio: Reciprocal Convexity and Functional Inequalities." Acta Univ. Sapientiae, Mathematica 4, 26-35, 2012.Boyd, A. V. "Inequalities for Mills' Ratio." Rep. Stat. Appl. Res. (Union Japan. Sci. Eng.) 6, 44-46, 1959.Cuyt, A.; Brevik Petersen, V.; Verdonk, B.; Waadeland, H.; and Jones, W. B. Handbook of Continued Fractions for Special Functions. New York: Springer, 2010.Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, p. 13, 2000.Grimmett, G. and Stirzaker, D. Probability and Random Processes, 3rd ed. Oxford, England: Oxford University Press, p. 98, 2001.Mills, J. P. "Table of the Ratio: Area to Bounding Ordinate, for Any Portion of Normal Curve." Biometrika 18, 395-400, 1926.Savage, I. R. "Mill's [sic] Ratio for Multivariate Normal Distributions." it J. Res. Nat. Bureau Standards--B. Mathematics and Mathematical Physics 66B, 93-96, 1962.

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Mills Ratio

Cite this as:

Weisstein, Eric W. "Mills Ratio." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MillsRatio.html

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