The Mean Squared Error Trade-Off for Truncation

Bias and variance answer different questions

Clipping limits the largest weights, but changing a weight also changes the weighted contribution it multiplies. The resulting estimator may be less variable across samples and, at the same time, systematically shifted away from its target. Whether clipping improves estimation therefore depends on the size of the bias squared relative to the variance saved, not on boundedness alone.

The earlier material on clipping bias establishes why clipping can change a weighted target. Here the goal is to combine that bias with sampling variance in a finite-sample mean squared error (MSE) comparison. We begin with a sample-mean estimator, so the calculation is exact rather than an approximation for a nonlinear estimator.

Exact MSE decomposition for a clipped weighted mean

Suppose \((W_i,R_i)\), \(i=1,\ldots,n\), are independent and identically distributed pairs. Let \(W_i\geq 0\) be a weight and \(R_i\) the contribution multiplied by that weight. For a fixed cutoff \(c>0\), define \(C_i=\min(W_i,c)\). Consider the untruncated estimator \(\hat{\theta}_U=n^{-1}\sum_i W_iR_i\) and the clipped estimator \(\hat{\theta}_C=n^{-1}\sum_i C_iR_i\), both evaluated against the target \(\theta=E[WR]\).

Assume \(E[WR]=\theta\), \(E[(WR)^2]<\infty\), and \(E[(CR)^2]<\infty\). These conditions give finite means and variances for both contributions. The cutoff is fixed, and the pairs are identically distributed; this calculation does not account for uncertainty from estimating the weights.

Proof. By identical distribution and linearity of expectation, \(E[\hat{\theta}_U]=E[WR]=\theta\), while \(E[\hat{\theta}_C]=E[CR]=\theta+b_c\). Independence gives \(\operatorname{Var}(\hat{\theta}_U)=\sigma_U^2/n\) and \(\operatorname{Var}(\hat{\theta}_C)=\sigma_C^2/n\). For any square-integrable estimator, write \(\hat{\theta}-\theta=(\hat{\theta}-E[\hat{\theta}])+(E[\hat{\theta}]-\theta)\). Squaring and taking expectations eliminates the cross term because \(E[\hat{\theta}-E[\hat{\theta}]]=0\). Thus MSE equals variance plus squared bias. Applying this identity gives both displayed MSEs. Comparing them and multiplying by \(n>0\) gives the strict-improvement condition. \(\square\)

The condition has an important implication: if \(\sigma_U^2\leq\sigma_C^2\), clipping cannot strictly improve MSE in this comparison, because its squared bias is nonnegative. If \(\sigma_U^2>\sigma_C^2\), the variance reduction must still be large enough to exceed the squared bias. The theorem does not assert that clipping always reduces variance.

Worked calculations

What the comparison assumes—and what to check

The exact inequality is useful only when its quantities match the estimator and target under study. It compares estimators based on the same independent, identically distributed observations, a fixed cutoff, and a finite-variance sample mean. In a weighted marginal structural model, a coefficient may be obtained by solving a weighted estimating equation rather than averaging one fixed contribution. A ratio estimator also has a random denominator. For either case, the general identity \(\operatorname{MSE}=\operatorname{Var}+\operatorname{Bias}^2\) still holds when the estimator has a finite second moment, but the sample-mean variance formulas above do not automatically apply.

For a fitted coefficient or a ratio, the relevant comparison must use the bias and variance of that actual estimator, not just the variance of the weights. Likewise, estimated weights can introduce additional sampling variation, and dependence across repeated observations from the same person violates the independent-observation calculation unless the sampling structure is handled appropriately.

In practice, diagnostics should address both sides of the trade-off. Examine how much weight mass is clipped and how the weighted contributions change; then assess whether the resulting variance reduction is plausible relative to the induced shift in the target. A sample variance of observed contributions can help describe variability, but it cannot by itself measure clipping bias: the unobserved contribution beyond the cutoff is precisely what has been altered. Comparisons across plausible cutoffs can be informative, but choosing a cutoff from estimated MSE is a separate procedure.

Clipping, deleting observations with large weights, and leaving weights untruncated are not interchangeable. Clipping retains observations but changes their contributions; deletion removes observations and changes the empirical target; no truncation preserves the original contributions but may leave high variability. Weight normalization is another distinct operation: rescaling weights does not have the same effect as limiting their largest values. Each alternative requires evaluating the estimator’s bias and variance for the target of interest.