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Minimax properties of gamma kernel density estimators under $L^p$ loss and $β$-Hölder smoothness of the target

Frédéric Ouimet

TL;DR

This work analyzes Chen's gamma kernel density estimator for nonnegative data with densities supported on a compact interval, investigating minimax performance under $L^p$ loss within β-Hölder classes. By decomposing risk into bias and stochastic components and exploiting inhomogeneous smoothing, the authors identify precise regions in $(p,β)$ where the gamma estimator achieves the minimax rate $n^{-β/(2β+1)}$ with a bandwidth $b_n\asymp n^{-2/(2β+1)}$, and regions where minimaxity fails for any bandwidth. The results hinge on endpoint regularity and the behavior of mirrored gamma densities near $x=1$, with a detailed lower-bound analysis for non-minimaxity and tailored bias-control arguments. These findings parallel known results for beta and Dirichlet kernels and inform bandwidth choices and the feasibility of adaptation in this nonnegative-data setting.

Abstract

This paper considers the asymptotic behavior in $β$-Hölder 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 $(p,β)\in [1,3)\times(0,2]$ or $(p,β)\in [3,4)\times ((p-3)/(p-2),2]$. It is also shown that this estimator cannot be minimax when either $p\in [4,\infty)$ or $β\in (2,\infty)$.

Minimax properties of gamma kernel density estimators under $L^p$ loss and $β$-Hölder smoothness of the target

TL;DR

This work analyzes Chen's gamma kernel density estimator for nonnegative data with densities supported on a compact interval, investigating minimax performance under loss within β-Hölder classes. By decomposing risk into bias and stochastic components and exploiting inhomogeneous smoothing, the authors identify precise regions in where the gamma estimator achieves the minimax rate with a bandwidth , and regions where minimaxity fails for any bandwidth. The results hinge on endpoint regularity and the behavior of mirrored gamma densities near , with a detailed lower-bound analysis for non-minimaxity and tailored bias-control arguments. These findings parallel known results for beta and Dirichlet kernels and inform bandwidth choices and the feasibility of adaptation in this nonnegative-data setting.

Abstract

This paper considers the asymptotic behavior in -Hölder spaces, and under 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 or . It is also shown that this estimator cannot be minimax when either or .
Paper Structure (10 sections, 8 theorems, 171 equations, 3 figures)

This paper contains 10 sections, 8 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Main results
  4. Regularity of mirrored gamma densities truncated to be supported on [0,1]
  5. Future research directions
  6. Proofs
  7. Proof of Theorem \ref{['thm:minimax']}
  8. Proof of Proposition \ref{['prop:2']}
  9. Proof of Proposition \ref{['prop:3']}
  10. Proofs of technical lemmas

Key Result

Theorem 1

Let $L > 0$ be given. Define Assume that $(p,\beta)\in [1,3)\times(0,2]$ or that $(p,\beta)\in \mathcal{S}$. Let $b_n = c \, n^{-2/(2\beta + 1)}$ for all $n\in \mathbb{N}$ and some constant $c\in (0,\infty)$. Then i.e., the sequence $\{\hat{f}_{n,b_n}: n\in \mathbb{N}\}$ achieves the minimax rate over $\Sigma(\beta,L)$ under $L^p$ loss.

Figures (3)

  • Figure 1: Visualization of the mirrored gamma densities truncated to $[0,1]$ defined in \ref{['eq:mirrored-gamma']} for $\theta=0.2$ and various values of the shape parameter $\alpha$ (from $2.0$ to $6.0$ with $0.2$ increments). The left panel shows the full range $x\in[0,1]$, while the right panel zooms in on $x\in[0.9,1.1]$.
  • Figure 2: Visualization of the non-smooth test densities $f_0$ and $f_3$ defined in \ref{['eq:test-densities']} with $L=2$ (hence $\varepsilon=1/2$).
  • Figure A.3: Visualization of the preliminary (non-smooth at $x=1$) test density $f_{\beta,b}$ defined in \ref{['eq:fbeta-def']} for $L=2$, $b=0.0005$, and various values of $\beta \in (0,2]$. The oscillations represent the localized perturbations used to lower bound the maximal risk.

Theorems & Definitions (14)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • proof : Proof of Lemma \ref{['lem:prop2-key']}
  • Lemma 6
  • Lemma 7
  • ...and 4 more