Kernel Density Estimation under $C^{1,1}$ Regularity: AMISE, Weak Curvature, and Plug-in Bandwidths

Classical kernel density estimation usually derives the AMISE and optimal bandwidth from a pointwise Taylor expansion, which requires twice continuous differentiability. This assumption is stronger than necessary and excludes natural densities arising from threshold models, regime changes, and robust mixture models, where the first derivative may be Lipschitz while the curvature is kinked, discontinuous, or only weakly defined. We show that the classical AMISE theory remains valid under the weaker condition $f\in C^{1,1}(\mathbb{R})$. The pointwise $C^2$ Taylor expansion is replaced by an integral Taylor representation based on the weak second derivative, so that $R(f'')$ is interpreted as a weak-curvature functional. Under $f\in C^{1,1}(\mathbb{R})$ and $f''\in L^2(\mathbb{R})$, we recover the classical AMISE formula, the $n^{-1/5}$ optimal bandwidth, and Epanechnikov kernel optimality without assuming a continuous classical second derivative. We also propose a generalized-curvature plug-in bandwidth selector, prove its first-order AMISE equivalence under ratio-consistent curvature estimation, and establish consistency of a leave-one-out U-statistic curvature estimator. A multivariate extension using weak Hessians recovers the scalar-bandwidth rate $n^{-4/(d+4)}$.