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

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 (Annals of the Institute of Statistical Mathematics, 52, 471-480, 2000) 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 an analytical device in the definition of the function class and the risk, not 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 additional upper-endpoint leakage bias. The estimator attains the minimax rate for $(p,β)\in [1,3)\times(0,2]$ and part of $[3,4)\times(0,2]$, but fails for $p\geq 4$ or $β>2$, leaving a small region unresolved.