Computing Truncation Corrections With the Inverse Mills Ratio

Why the Inverse Mills Ratio Appears

When an estimate is reported only after it crosses a threshold, its distribution among reported estimates differs from its unconditional sampling distribution. Even if an estimator is unbiased before selection, its conditional mean among threshold-crossing results will generally be shifted. The inverse Mills ratio expresses that shift for a normal estimate in a compact form. It is useful to know not only how to evaluate the ratio, but what its inputs mean: the standardized distance from the true mean to the selection threshold.

This tutorial considers a single normally distributed estimate subject to a fixed threshold. It does not account for dependence among several selected estimates or for a sequence of selection decisions. Those features require a selection event more complicated than one threshold crossing.

The denominator \(1-\Phi(a)\) is the probability of passing the threshold, \(P(X>c)\). The numerator is the standard normal density evaluated at the standardized threshold. The ratio itself is dimensionless; multiplying it by \(\sigma\), which has the same units as \(X\), gives the conditional-mean shift in the original units.

Conditional Mean and Correction

For upper-threshold selection, the selected estimate has conditional mean \(E[X\mid X>c]=\theta+\sigma\lambda_U(a)\). Thus the conditional bias relative to \(\theta\) is positive and equals \(\sigma\lambda_U(a)\). If \(\theta\), \(\sigma\), and \(c\) are specified, this is a direct calculation. In practice \(\theta\) is usually unknown, so a plug-in correction replaces it in the standardized threshold with an estimate. That substitution is an approximation, not a guarantee that the corrected estimate is unbiased.

The standardized threshold \(a\) describes how far \(c\) is from \(\theta\) in standard deviation units. For example, \(a=1.5\) means that the threshold is 1.5 standard deviations above the mean. It does not mean that the selected estimate is 1.5 units above the mean. The factor \(\sigma\) is needed to convert the dimensionless ratio back into the estimate's units.

Calculating the Upper-Tail Correction

The correction can be larger than the threshold's distance above the mean. In the example, \(a=1.5\), while \(\lambda_U(a)\approx1.9387\). This is not an inconsistency: conditioning on \(X>c\) retains the entire upper tail, including estimates well above \(c\).

Using the Ratio in a Plug-In Correction

The theoretical formula depends on \(\theta\), which is generally unknown. A plug-in calculation uses the observed estimate \(x\) in its place: \(\hat a=(c-x)/\sigma\), followed by a correction \(\sigma\lambda_U(\hat a)\). For upper-tail selection, the corresponding corrected estimate is \(x-\sigma\lambda_U(\hat a)\). This implements the plug-in approach to selection correction: it estimates the conditional shift using the observed estimate in the standardized threshold.

The subtraction sign follows from the direction of the theoretical conditional bias. Conditional on crossing an upper threshold, the estimate tends to be high relative to its mean, so the estimated shift is subtracted. However, \(\hat a\) uses the selected observation itself; it is not the unknown true standardized threshold \(a\). The resulting correction should therefore be described as a plug-in estimate, not as the exact amount of bias for that realized estimate.

Lower-Threshold Selection and Sign

If selection instead requires \(X<c\), the correction changes direction. With the same \(a=(c-\theta)/\sigma\), the lower-tail inverse Mills ratio is \(\lambda_L(a)=\phi(a)/\Phi(a)\), and the conditional mean is \(E[X\mid X<c]=\theta-\sigma\lambda_L(a)\). The conditional bias is negative: selected values lie, on average, below \(\theta\).

Justification, Checks, and Limitations

The formula follows directly by standardizing the estimate. Write \(Z=(X-\theta)/\sigma\), so \(Z\sim N(0,1)\), and \(X>c\) is equivalent to \(Z>a\). The conditional mean of \(X\) is \(\theta+\sigma E[Z\mid Z>a]\). Since the standard normal density satisfies \(\phi'(z)=-z\phi(z)\), integrating gives the numerator of that conditional expectation:

The denominator is essential: dividing by \(P(Z>a)=1-\Phi(a)\) changes an unconditional tail integral into a conditional mean. For lower-tail selection, the same density identity gives \(\int_{-\infty}^a z\phi(z)\,dz=-\phi(a)\); dividing by \(\Phi(a)\) yields the negative shift stated above.

A useful diagnostic is to check the direction and units. For finite \(a\), both \(\phi(a)\) and \(1-\Phi(a)\) are positive, so \(\lambda_U(a)>0\). Thus the upper-tail correction is positive before subtraction, and \(\sigma\lambda_U(a)\) has the units of the estimate. Also, the conditional mean given \(X>c\) must exceed \(c\); a computed value below the threshold signals a sign, tail-probability, or arithmetic error.

For thresholds far into the upper tail, \(1-\Phi(a)\) can be very small. Rounding that probability too aggressively can make the ratio inaccurate or even produce division by zero in software. Use a reliable normal survival-function calculation or a numerically stable log-tail calculation for extreme thresholds, and retain sufficient precision in intermediate values. A second check is that the ratio should be positive, and that the correction should be in the same units as \(\sigma\).

The inverse Mills ratio gives a closed-form result under a normal model with a single fixed threshold. Direct integration or simulation can be alternatives when the distribution or selection event is more complicated, though simulation adds Monte Carlo error and requires the model to be specified. The plug-in correction is distinct from the theoretical correction: the latter uses the true \(\theta\), whereas the former substitutes an estimate. Neither formula by itself accounts for a nonnormal sampling distribution, a threshold chosen adaptively from the same data, or selection among correlated arms.

Common Mistakes

• Using the wrong tail. For selection when \(X>c\), the denominator is \(1-\Phi(a)\), not \(\Phi(a)\). For selection when \(X<c\), it is \(\Phi(a)\).
• Leaving out \(\sigma\). The inverse Mills ratio is dimensionless. Multiplying by \(\sigma\) produces the correction in the estimate's units.
• Reversing the correction's sign. Upper-tail selection produces a positive conditional shift, which is subtracted in a plug-in correction. Lower-tail selection produces a negative shift.
• Interpreting a plug-in value as exact. Substituting \(x\) for \(\theta\) estimates the standardized threshold; it does not reveal the true conditional bias.
• Using a rounded denominator in an extreme tail. A near-zero rounded survival probability can distort the ratio. Use a stable tail-probability computation and check the result's scale and direction.