カーネル密度推定 📖 Wikipedia

${\displaystyle {\hat {f}}_{h}(x)={\frac {1}{nh}}\sum _{i=1}^{n}K\left({\frac {x-x_{i}}{h}}\right)}$

${\displaystyle K(x)={1 \over {\sqrt {2\pi }}}\,e^{-x^{2}/2}}$

${\displaystyle R(f,{\hat {f}}(x))\approx {\frac {1}{4}}\sigma _{k}^{4}h^{4}\int (f''(x))^{2}\,dx+{\frac {\int K^{2}(x)\,dx}{nh}}\quad {\text{ where }}\quad \sigma _{K}^{2}=\int x^{2}K(x)\,dx}$

${\displaystyle h^{*}={\frac {c_{1}^{-2/5}c_{2}^{1/5}c_{3}^{-1/5}}{n^{1/5}}}}$

${\displaystyle c_{1}=\int x^{2}K(x)\,dx}$
${\displaystyle c_{2}=\int K(x)^{2}\,dx}$
${\displaystyle c_{3}=\int (f''(x))^{2}\,dx}$