The Inverse Mills Ratio

Inverse Mills ratio: some properties

The Inverse Mills Ratio (IMR) is defined as the ratio of the standard normal density, \(\phi\), divided by the standard normal cumulative distribution function, \(\Phi\):

\[\begin{eqnarray} \operatorname{IMR}(x) = \dfrac{\phi(x)}{\Phi(x)},\,\,\, x \in {\mathbb R}. \end{eqnarray}\]

This quantity appears in many statistical models such as Sample Selection models, survival analysis, principal component analysis, among others. The IMR has some interesting properties [1,2]:

• \(\operatorname{IMR}(x)\) is decreasing.
• Its derivative is given by \(\operatorname{IMR}^{'}(x) = - x \operatorname{IMR}(x) - \operatorname{IMR}(x)^2\).
• \(\operatorname{D}(x) = -\operatorname{IMR}^{'}(x)\) is positive, decreasing, and bounded on \((0,1)\).

References

1. Principal component analysis of binary data by iterated singular value decomposition

2. Exact bounds on the inverse Mills ratio and its derivatives

3. Sample Selection as a Specification Error

4. Some Inequalities on Mill’s Ratio and Related Functions

Illustration in R

# The Inverse Mills Ratio (IMR) function
IMR <- Vectorize( function(x) exp( dnorm(x,log=T) - pnorm(x,log.p = T) ) )
curve(IMR, -15, 15, n = 10000, lwd = 2, cex.axis=1.5, cex.lab=1.5, xlab="x", ylab="IMR", main = "Inverse Mills Ratio")

# -First derivative of IMR
D <- Vectorize( function(x) x*IMR(x) + IMR(x)^2 )
curve(D, -15, 15, n = 10000, lwd = 2, cex.axis=1.5, cex.lab=1.5, xlab="x", ylab="D", main = "-First Derivative of IMR")