Unconditional Bias-Corrected Estimators

Why Calibrate Across the Complete Stopping Rule?

A sequential trial does not always produce an estimate at the same sample size. Some trials stop early; others continue to later analyses. If the ordinary estimate is systematically pulled away from the treatment effect by this data-dependent stopping, one possible target is to calibrate an estimator so that its repeated-sampling expectation under the entire design equals the effect. This is an unconditional target: it averages over every possible stopping time and every data path that can lead to it.

The calibration must use the full sampling procedure, not just the distribution conditional on the look actually observed. That distinction matters because the stopping decision carries information about the data. A correction that works on average across all paths need not work within any one stopping group.

This definition is a precise target, not a guarantee that a suitable estimator exists or is useful. It also depends on the assumed sampling model and the actual rule: changing the boundaries, analysis times, or treatment of a stopping decision can change the expectation that must be calibrated.

Writing the Calibration Equation

Suppose the possible observed paths are indexed by \(\omega\). A path records the observed data and the look at which the trial stops. If the design and model specify its probability \(q_\theta(\omega)\), then the unconditional calibration requirement is

For continuous data, the sum is replaced by an integral over the possible paths, using their sampling density under \(\theta\). Crucially, paths at different stopping looks are all included. The probabilities of those paths must reflect the stopping rule; it is not enough to calculate an expectation as if every trial had reached the maximum sample size.

A useful way to formulate the task is to start with a familiar estimator \(\hat\theta\) and find a fixed correction \(c(D)\) such that

The quantity on the right is the bias of the ordinary estimator under the stopping rule. The correction must be a function of observable data and known design features, not of the unknown true \(\theta\). Simply subtracting a bias expression evaluated at the true parameter would not define an estimator. A plug-in correction may be useful in some settings, but does not by itself establish exact unconditional unbiasedness.

Worked Example: Calculating Bias Under a Two-Stage Rule

Worked Example: Solving for an Unconditionally Calibrated Estimator

Worked Example: Checking the Unconditional Target at a Fixed Parameter

Justification, Assumptions, and Practical Computation

The central justification is direct. If a fixed correction function satisfies the calibration equation, linearity of expectation gives

This establishes unbiasedness for every parameter value at which the correction equation holds. It does not establish small variance, good performance in a particular stopping group, or robustness to a misspecified model.

In practice, calibration requires a specified sampling model and the complete stopping rule. One can enumerate all paths when the data are discrete and the design is small. For more complex models, calculate the expectation by numerical integration or simulate complete trials under a grid of parameter values. A numerical solution should be checked over the parameter space, not just at one fitted value. Simulation also introduces Monte Carlo error: close agreement in a finite simulation is evidence of approximate calibration, not proof of exact unbiasedness.

For a parametric calibration, one may express \(c(D)\) using a chosen set of observable path features and solve the resulting expectation equations for its coefficients. The equations must hold for the parameter values claimed. Fitting them only at a few grid points may match those points while leaving bias between them. If nuisance parameters affect the path probabilities, the calibration target must specify whether unbiasedness is required at every nuisance-parameter value; that requirement can make the equations harder or impossible to satisfy.

What Unconditional Calibration Does—and Does Not—Guarantee

Unconditional calibration differs from several other common targets. An estimator calibrated conditional on the observed stopping look targets an expectation within that selected group. A median-unbiased estimator targets a probability statement about whether its estimate lies above or below the parameter, rather than matching the mean. Conditional likelihood methods use a likelihood conditioned on the stopping event. These methods answer different questions; none is automatically a substitute for unconditional mean calibration.

Unbiasedness is a statement about repeated sampling under a model, not a claim that any one estimate is close to the unknown effect. An unbiased estimator may have substantial variance or take values outside a scientifically plausible range. In the worked example, assigning 0 to a continuation path is an algebraic consequence of the selected target and design, not evidence that a continuing trial’s effect is zero.

The estimator may also fail to exist in a requested class. For example, restrictions such as nonnegative estimates, a bounded range, or dependence only on a particular summary statistic can conflict with the expectation equations. Even when solutions exist, they may not be unique. The choice among them may require considering variance, interpretability, and performance across parameter values, in addition to unbiasedness.

Diagnostics and Common Pitfalls

• Omitting stopping paths. The expectation must include every possible stopping time and the probability of reaching it. Calculate path probabilities from the complete rule.
• Conditioning without noticing. An expectation given that the trial stopped at a particular look is not the unconditional expectation. Keep the target explicit.
• Using the unknown parameter in the correction. A correction that depends on the true \(\theta\) is not an operational estimator. Solve for fixed functions of observed data and known design quantities.
• Checking only one parameter value. Unbiasedness requires the expectation equation across the parameter space, unless a narrower pointwise target is explicitly intended.
• Assuming unbiased means precise. Unbiasedness controls the mean over repetitions; it does not control variance, interval coverage, or the behavior of estimates after a particular stopping outcome.
• Ignoring design changes or misspecification. Recalculate or reassess calibration if boundaries, analysis timing, or the sampling model differ from those used to derive it.

The key practical question is therefore not just whether a correction removes a calculated bias, but under which model, for which complete stopping rule, and over which parameter values the expectation identity has been verified.