Mills ratio

In probability theory, the Mills ratio (or Mills's ratio) of a continuous random variable

X

{\displaystyle X}

is the function

m

(

x

)

:=

F

¯

(

x

)

f

(

x

)

,

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

where

f

(

x

)

{\displaystyle f(x)}

is the probability density function, and

F

¯

(

x

)

:=

Pr

[

X

>

x

]

=

∫

x

+

∞

f

(

u

)

d

u

{\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. The Mills ratio is related to the hazard rate h(x) which is defined as

h

(

x

)

:=

lim

δ

→

0

1

δ

Pr

[

x

<

X

≤

x

+

δ

|

X

>

x

]

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

by

m

(

x

)

=

1

h

(

x

)

.

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

Upper and lower bounds

When

X

{\displaystyle X}

has a standard normal distribution then the following bounds hold for

x

>

0

{\displaystyle x>0}

:

x

x

2

+

1

<

m

(

x

)

<

1

x

{\displaystyle {\frac {x}{x^{2}+1}}<m(x)<{\frac {1}{x}}}

Example

If

X

{\displaystyle X}

has standard normal distribution then

m

(

x

)

∼

1

/

x

,

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

where the sign

∼

{\displaystyle \sim }

means that the quotient of the two functions converges to 1 as

x

→

+

∞

{\displaystyle x\to +\infty }

, see Q-function for details. More precise asymptotics can be given.

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

E

⁡

[

X

|

X

>

α

]

=

μ

+

σ

ϕ

(

α

−

μ

σ

)

1

−

Φ

(

α

−

μ

σ

)

,

E

⁡

[

X

|

X

<

α

]

=

μ

−

σ

ϕ

(

α

−

μ

σ

)

Φ

(

α

−

μ

σ

)

,

{\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

α

{\displaystyle \alpha }

is a constant,

ϕ

{\displaystyle \phi }

denotes the standard normal density function, and

Φ

{\displaystyle \Phi }

is the standard normal cumulative distribution function. The two fractions are the inverse Mills ratios.

Use in regression

A common application of the inverse Mills ratio (sometimes also called “non-selection hazard”) arises in regression analysis to take account of a possible selection bias.