oral-presentation-nps

\(\vec{x}=(x^{(1)},x^{(2)},\dots,x^{(d)})^{T}\)

\begin{align} \hat{f}_{H}(\vec{x})=\frac{1}{n|H|}\sum_{i=1}^{n}K(H^{-1}(\vec{x}-\vec{X_{i}})\\ \end{align}
\begin{align} h_{opt}=\Bigg{(}\frac{R(K)}{\kappa_{2}^{2}(K)R(f^{{}^{\prime\prime}})}\Bigg{)}^{1/5}n^{-1/5}\\ \end{align}
\begin{align} R(f^{{}^{\prime\prime}})^{1/5}\\ \end{align}
\begin{align} CV(h)=\frac{1}{n^{2}h}\sum_{i=1}^{n}\sum_{j=1}^{n}K^{*}\Bigg{(}\frac{X_{i}-X_{j}}{h}\Bigg{)}-\frac{2}{n(n-1)h}\sum_{i=1}^{n}\sum_{j\neq i}K\Bigg{(}\frac{X_{i}-X_{j}}{h}\Bigg{)}\\ \end{align}
\begin{align} \hat{h}_{CV}=arg\inf_{h>0}CV(h)\\ \end{align}