Minimax properties of gamma kernel density estimators under $L^p$ loss and $\beta$-H\"older smoothness of the target

TL;DR

This paper analyzes the asymptotic minimax properties of gamma kernel density estimators for nonnegative data under $L^p$ loss and $eta$-H"older smoothness, identifying conditions for optimality.

Contribution

It establishes the conditions under which gamma kernel estimators achieve the minimax rate, and when they do not, based on $p$ and $eta$ values.

Findings

Achieves minimax rate for certain $(p, eta)$ ranges.

Fails to be minimax for $p otin [1,4)$ or $eta > 2$.

Provides theoretical bounds for gamma kernel estimators.

Abstract

This paper considers the asymptotic behavior in $\beta$-H\"older spaces, and under $L^p$ loss, of the gamma kernel density estimator introduced by Chen [Ann. Inst. Statist. Math. 52 (2000), 471-480] for the analysis of nonnegative data, when the target's support is assumed to be upper bounded. It is shown that this estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever $(p,\beta)\in [1,3)\times(0,2]$ or $(p,\beta)\in [3,4)\times ((p-3)/(p-2),2]$. It is also shown that this estimator cannot be minimax when either $p\in [4,\infty)$ or $\beta\in (2,\infty)$.