Minimax estimation of norms of a probability density: I. Lower bounds

TL;DR

This paper establishes fundamental lower bounds for the minimax risk in estimating the $L_p$-norm of a probability density, revealing the influence of whether $p$ is integer and introducing a versatile technique for nonlinear functional estimation.

Contribution

It provides new lower bounds for $L_p$-norm estimation in a nonparametric setting and introduces a general method applicable to various nonlinear functional estimation problems.

Findings

Lower bounds depend on whether $p$ is integer.

Developed a general technique for lower bounds on nonlinear functionals.

Demonstrated the technique's applicability to $L_p$-norm estimation.

Abstract

The paper deals with the problem of nonparametric estimating the $L_p$--norm, $p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from independent observations. The unknown density %to be estimated is assumed to belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax approach, and derive lower bounds on the minimax risk. In particular, we demonstrate that accuracy of estimation procedures essentially depends on whether $p$ is integer or not. Moreover, we develop a general technique for derivation of lower bounds on the minimax risk in the problems of estimating nonlinear functionals. The proposed technique is applicable for a broad class of nonlinear functionals, and it is used for derivation of the lower bounds in the~$L_p$--norm estimation.