Local-polynomial estimator

Assume AC as in Regression target and chosen version. Fix $n\ge 1$, a point $x\in \mathbb{R}$, bandwidth $h>0$, degree $p\in {\mathbb{N}}_0$, a finite-valued nonnegative Borel function $K$, and fallback $a_0\in \mathbb{R}$. Set $u_i=(x-X_i)\mathrm{/}h$, $r_p(u)=(1,u,\dots ,u^p{)}^{\mathsf{T}}$, let $Z$ be the matrix whose $i$th row is $r_p(u_i{)}^{\mathsf{T}}$, let $W=\mathrm{diag}(K(u_1),\dots ,K(u_n))$, and put $M=Z^{\mathsf{T}}WZ$. If $M$ is nonsingular, define $\widehat{\beta }=M^{-1}{Z}^{\mathsf{T}}W(Y_1,\dots ,Y_n{)}^{\mathsf{T}},{\widehat{m}}_{p,h}(x)={\widehat{\beta }}_0\mathrm{.}$ If $M$ is singular, define ${\widehat{m}}_{p,h}(x)=a_0$. For fixed $x,h,n,p$ and Borel $K$, this is a real statistic on $({\mathbb{R}}^2{)}^n$ with its product Borel sigma-algebra: the matrix entries are Borel, the nonsingular-matrix set is Borel, matrix inversion is continuous there, and the fixed fallback gives the other branch. 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 estimates ${\psi }_x$ in the sense of Statistics and estimators. The case $p=1$ is the local-linear estimator. No claim that $M$ is nonsingular for a given sample is implicit in this definition.

Newey, “Locally Linear Regression,” PDF pp. 1–4; Tibshirani, §3.1, PDF p. 6.