The augmented van Trees inequality

TL;DR

This paper introduces an augmented van Trees inequality that provides tighter, more flexible lower bounds on the minimax risk in estimation problems, improving upon classical bounds and enabling sharper constants in various statistical models.

Contribution

The paper presents a novel augmented van Trees inequality that yields uniformly tighter bounds and handles broader conditions than the classical version, including boundary densities and non-Gaussian models.

Findings

Achieves sharper minimax lower bounds in nonparametric estimation.

Provides asymptotic minimax pointwise mean squared error with improved constants.

Extends applicability to models beyond Gaussian and loss functions beyond squared error.

Abstract

We introduce an augmented form of the van Trees inequality, that yields uniformly tighter lower bounds on the minimax squared Bayes risk of estimators compared with the classical van Trees inequality. Our augmented inequality also accommodates prior distributions whose densities need not vanish at the boundaries of their supports. We demonstrate how this refinement can be utilised for elementary proofs of a number of minimax lower bounds for nonparametric estimands, that also often attain sharper constants than those obtained by the alternative Le Cam convergence of experiments theory and the classical van Trees inequality, and in some cases obtain exact constants. As an example, our augmented van Trees inequality can be used to obtain the asymptotic minimax pointwise mean squared error when estimating the regression function in the model with normal errors: when the regression function is univariate and differentiable with Lipschitz derivative we obtain this quantity up to a constant factor of $1.37$; and in the high dimensional regime with a H\"older smooth regression function of smoothness $\beta\in(0,2]$ we obtain exact constants. Both these results do not follow from an application of the classical van Trees inequality. The flexibility of our augmented van Trees inequality accommodates lower bounds for models beyond Gaussianity, loss functions beyond the squared error loss, and we are also able to incorporate this augmentation into generalised versions of the van Trees inequality for irregular models.