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

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

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

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

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

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

by

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

Upper and Lower Bounds

When ${\displaystyle X}$ has a standard normal distribution then the following bounds hold for ${\displaystyle x>0}$:

${\displaystyle {\frac {x}{x^{2}+1}}<m(x)<{\frac {1}{x}}}$^[4][5]

Example

If ${\displaystyle X}$ has standard normal distribution then

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

where the sign ${\displaystyle \sim }$ means that the quotient of the two functions converges to 1 as ${\displaystyle x\to +\infty }$, see Q-function for details. More precise asymptotics can be given.^[6]

Inverse Mills ratio

The inverse Mills ratio is the ratio of the probability density function to the complementary cumulative distribution function of a distribution. Its use is often motivated by the following property of the truncated normal distribution. If X is a random variable having a normal distribution with mean μ and variance σ², then

${\displaystyle {\begin{aligned}&\operatorname {E} [\,X\,|\ X>\alpha \,]=\mu +\sigma {\frac {\phi {\big (}{\tfrac {\alpha -\mu }{\sigma }}{\big )}}{1-\Phi {\big (}{\tfrac {\alpha -\mu }{\sigma }}{\big )}}},\\&\operatorname {E} [\,X\,|\ X<\alpha \,]=\mu -\sigma {\frac {\phi {\big (}{\tfrac {\alpha -\mu }{\sigma }}{\big )}}{\Phi {\big (}{\tfrac {\alpha -\mu }{\sigma }}{\big )}}},\end{aligned}}}$

where ${\displaystyle \alpha }$ is a constant, ${\displaystyle \phi }$ denotes the standard normal density function, and ${\displaystyle \Phi }$ is the standard normal cumulative distribution function. The two fractions are the inverse Mills ratios.^[7]

See also

• Heckman correction

References

External links

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