Focused median bias reduction

2026-06-26

AI Summary

Existing median bias correction methods are often implicit, computationally intensive, and rely on fully specified perturbation parameters, limiting their applicability under parameter transformations. This work proposes an explicit, general third-order median-unbiased estimator for smooth scalar transformations of a reference parameter. Building on the Cornish–Fisher expansion, the method directly constructs a higher-order approximation to the centered maximum likelihood estimator without solving nonlinear systems. It requires only the maximum likelihood estimate, its gradient and Hessian, and expectations of products of log-likelihood derivatives—obtainable analytically or via Monte Carlo simulation. The approach accommodates arbitrary smooth transformations, integrates naturally with hull confidence intervals, and achieves near-nominal finite-sample coverage in regression, circular, and hierarchical models. It has been successfully applied to post-selection inference after FIC, Mahalanobis distances, and quantile estimation.

Abstract

Median bias reduction of maximum likelihood estimators can substantially improve estimation and inference. Existing generally applicable methods are, however, typically implicit, requiring the solution of nonlinear systems of estimating equations, which is computationally demanding. They also require a fully specified nuisance parameterization, and their application to transformations of parameters involves tedious algebra and bespoke implementations. We develop an explicit median bias-corrected estimator for focus parameters that are smooth scalar transformations of a chosen reference parameterization. The estimator is obtained by solving, to the required order, an equation based on the Cornish-Fisher expansion of the centred and scaled maximum likelihood estimator of the focus parameter. It requires only the maximum likelihood or an asymptotically equivalent estimator at the reference parameterization, the gradient and Hessian of the transformation, and expectations of products of log-likelihood derivatives. These expectations are available for many models from the existing bias reduction literature and can also be estimated by Monte Carlo simulation. The resulting estimators are third-order median unbiased and provide one-step approximations to estimators from implicit median bias reduction when the focus parameter is included in the reference parameterization. The method improves standard asymptotic inference and integrates naturally with hull-based confidence procedures, yielding intervals with near nominal finite-sample coverage under median bias control. We demonstrate the framework through post-selection inference using the Focused Information Criterion, Mahalanobis distances, quantiles, and other scalar focus parameters in regression, circular, and stratified models.

Problem

• median bias reduction
• maximum likelihood estimation
• computational complexity
• nuisance parameterization
• parameter transformation

Innovation

• median bias reduction
• explicit estimator
• Cornish-Fisher expansion
• focus parameter
• finite-sample inference