Nadaraya–Watson estimator

Assume AC as in Regression target and chosen version. Fix a sample size $n\ge 1$, a prediction point $x\in \mathbb{R}$, a bandwidth $h>0$, a finite-valued nonnegative Borel function $K:\mathbb{R}\to [0,\mathrm{\infty })$, and a fixed fallback $a_0\in \mathbb{R}$. For data $(X_i,Y_i{)}_{i=1}^n$, put $D_{n,h}(x)={\sum }_{i=1}^{n}K((x-X_i)\mathrm{/}h)\mathrm{.}$ The Nadaraya–Watson estimator of the pointwise target $m(x)$ is ${\widehat{m}}_{\mathrm{N}\mathrm{W},h}(x)=\{\begin{array}{ll}{\displaystyle \frac{\sum _{i=1}^{n}K((x-X_i)\mathrm{/}h)Y_i}{D_{n,h}(x)}},& D_{n,h}(x)>0,\\ a_0,& D_{n,h}(x)=0.\end{array}$ For fixed $x,h,n$, this is a Borel statistic on $({\mathbb{R}}^2{)}^n$ with its product Borel sigma-algebra: each weight, product, and finite sum is Borel; the positive-denominator set is Borel, and division is continuous there. In any statistical experiment on this sample space whose pointwise regression estimand is ${\psi }_x(\theta )=m_{\theta }(x)$, with $m_{\theta }$ the selected version from Regression target and chosen version, it is an estimator of ${\psi }_x$ in the sense of Statistics and estimators. The fallback is part of the definition. Kernel regularity and bandwidth assumptions used for bias or variance results are stated in those results.