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 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 $β\in (2,\infty)$.

💡 Research Summary

The paper investigates the asymptotic performance of the gamma kernel density estimator introduced by Chen (2000) when the target density is supported on a compact interval, assumed to be (