Adaptive density estimation: a curse of support?

TL;DR

This paper investigates the challenges of density estimation on the real line with infinite support, revealing a support-related curse similar to the curse of dimensionality, and proposes an adaptive wavelet-based method to address it.

Contribution

It introduces a new wavelet thresholding approach that adapts to support size and density regularity, mitigating practical issues caused by infinite support.

Findings

Support-dependent methods are significantly affected by the size of the density support.

The proposed wavelet thresholding method is robust and adaptively handles infinite support.

Calibrated thresholds improve practical performance close to theoretical optima.

Abstract

This paper deals with the classical problem of density estimation on the real line. Most of the existing papers devoted to minimax properties assume that the support of the underlying density is bounded and known. But this assumption may be very difficult to handle in practice. In this work, we show that, exactly as a curse of dimensionality exists when the data lie in $\R^d$, there exists a curse of support as well when the support of the density is infinite. As for the dimensionality problem where the rates of convergence deteriorate when the dimension grows, the minimax rates of convergence may deteriorate as well when the support becomes infinite. This problem is not purely theoretical since the simulations show that the support-dependent methods are really affected in practice by the size of the density support, or by the weight of the density tail. We propose a method based on a biorthogonal wavelet thresholding rule that is adaptive with respect to the nature of the support and the regularity of the signal, but that is also robust in practice to this curse of support. The threshold, that is proposed here, is very accurately calibrated so that the gap between optimal theoretical and practical tuning parameters is almost filled.