Exact Global Minimax Risk near Nonidentification

Abstract

We study squared-error estimation when a nuisance parameter is unidentified on a submodel but affects a target that vanishes there. For models with higher-order profile expansions, we give sufficient conditions under which the largest critical-path minimax risk is the exact asymptotic minimax risk over the full parameter space. An efficient observable feature, negligible calibration error and stable recovery yield one estimator attaining the leading constant uniformly, with the profile coordinates unknown. The estimator uses calibration estimates, a local feature mean and an empirical recovery feature. The constant is expressed through a bounded-normal decision problem. We verify the conditions in two models. In centered Gaussian logistic errors-in-variables regression with one surrogate and two unknown variances, we determine the fifth-order global coefficient risk and its exact constant. A sparse extension separates variance-decomposition uncertainty from direction estimation and specifies when the scalar constant is preserved. In a degree-balanced replicated-network model, we determine the sixth-order global edge-contrast risk when block proportions are unknown. Triangles identify a parameter combination; a bowtie supplies the additional information needed for the contrast.