Minimax Linear Estimation at a Boundary Point

TL;DR

This paper derives the minimax linear estimator for boundary point function values in a Gaussian white noise model with Lipschitz continuous derivatives, and applies it to optimize estimators in regression discontinuity designs.

Contribution

It provides a theoretical characterization of the minimax linear estimator at boundary points and applies it to improve regression discontinuity analysis.

Findings

Derived the minimax linear estimator at boundary points.

Applied the estimator to regression discontinuity models.

Established optimality under Lipschitz continuity constraints.

Abstract

This paper characterizes the minimax linear estimator of the value of an unknown function at a boundary point of its domain in a Gaussian white noise model under the restriction that the first-order derivative of the unknown function is Lipschitz continuous (the second-order H\"{o}lder class). The result is then applied to construct the minimax optimal estimator for the regression discontinuity design model, where the parameter of interest involves function values at boundary points.