Density targets and the kernel density estimator

Fix an integer $d\ge 1$. A density version on ${\mathbb{R}}^d$ is a Borel function $f:{\mathbb{R}}^d\to [0,\mathrm{\infty })$ with ${\int }_{{\mathbb{R}}^d}fd{\lambda }_d=1.$ Write $P_f=fd{\lambda }_d$ for its probability law (The measure with density $f$ relative to $\mu$). Let $\mathrm{\Theta }$ be any set of such density versions. For sample size $n\ge 1$, the iid experiment has data space $(({\mathbb{R}}^d{)}^n,{⨂}_{i=1}^n\mathcal{B}({\mathbb{R}}^d))$ and sampling law $P_f^{\otimes n}$ at parameter $f\in \mathrm{\Theta }$ (Statistical experiments, Parametric statistical models, Identical distribution and IID families, Independent random elements, The product sigma-algebra and its finite iterates, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique). The pointwise target at $x\in {\mathbb{R}}^d$ is the estimand ${\psi }_x(f)=f(x):\mathrm{\Theta }\to \mathbb{R}$ (Estimands on a statistical model).

A normalized kernel is a Borel function $K:{\mathbb{R}}^d\to \mathbb{R}$ with $K\in L^1({\lambda }_d)$ and $\int Kd{\lambda }_d=1$. It may be signed. For bandwidth $h>0$ and observations $X_1,\dots ,X_n$, the kernel density estimate is the indexed family ${\widehat{f}}_h(x)=\frac{1}{n{h}^d}{\sum }_{i=1}^{n}K\text{\hspace{0.17em}⁣}\left(\frac{x-X_i}{h}\right),x\in {\mathbb{R}}^d\mathrm{.}$ For each fixed $x$, this is intended as a real-valued estimator of ${\psi }_x$; the next local lemma supplies its measurability (Statistics and estimators).

Assuming ${\mathrm{A}\mathrm{C}}_{\omega }$, if $K$ is pointwise nonnegative the estimate is a probability density for every sample, as proved in A nonnegative normalized kernel estimate integrates to one ↗. For a signed kernel the estimate need not be nonnegative. Point evaluation depends on the selected density version: this definition does not assert that $f\mapsto f(x)$ is identifiable from the law $P_f$.