Mills quotient

In probability theory , the Mills quotient of a continuous random variable describes a function ${\displaystyle X}$

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

where the probability density function and ${\displaystyle f(x)}$

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

denote the complementary distribution function (also called the survival function) of . The concept is named after John P. Mills . The Mills quotient is related to the failure rate , which is defined as ${\displaystyle X}$ ${\displaystyle h(x)}$

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

by

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

example

If is normally distributed , then asymptotically applies ${\displaystyle X}$

${\displaystyle m(x)\sim 1/x,\,}$

ie . ${\displaystyle \lim _{x\to \infty }m(x)x=1}$

Web links

• Mills Ratio (MathWorld)

Individual evidence

1. ↑ Ulrich Müller-Funk : Mathematical Statistics II , Springer, 2013, p. 25.
2. ^ John P. Mills: Table of the Ratio: Area to Bounding Ordinate, for Any Portion of Normal Curve . In: Biometrika . 18, No. 3/4, 1926, pp. 395-400. doi : 10.1093 / biomet / 18.3-4.395 .
3. ↑ Klein, JP, Moeschberger, ML: Survival Analysis: Techniques for Censored and Truncated Data , Springer, 2003, p.27