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

Abstract

This paper considers the asymptotic behavior in $β$-Hölder spaces, and under $L^p$ loss, of the non-modified gamma kernel density estimator introduced by Chen [Ann. Inst. Statist. Math. 52 (2000), 471-480] for the analysis of nonnegative data, in the situation where the target may have a finite effective or true upper endpoint but the estimator itself is left untruncated and treats the support as $[0,\infty)$. The finite endpoint is used as an analytical device in the definition of the function class and the risk, not as information supplied to the estimator. The functional classes are chosen so that the target density matches smoothly to zero at the upper endpoint, which isolates the behavior at the origin and avoids an additional upper-endpoint leakage bias. 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 whenever $3 \leq p < 4$ and $(p-3)/(p-2) < β\leq 2$. It is also shown that this estimator cannot be minimax when either $p\in [4,\infty)$ or $β\in (2,\infty)$. The remaining region $\left\{(p,β): 3 < p < 4,\ 0 < β\leq (p-3)/(p-2)\right\}$ is an open case.