Using the fact that the asymptotic mean integrated squared error of the kernel estimator with bandwidth \(h\) is given by,

\[\operatorname{AMISE}\left(\tilde{f}_{n, h}, fight)=(n h)^{-1} R(k)+\frac{1}{4} h^{4} \sigma_{k}^{4} R\left(f^{\prime \prime}ight),\]

show that the asymptotically optimal bandwidth is given by \[h_{\mathrm{opt}}=n^{-1 / 5}\left[\frac{R(k)}{\sigma_{k}^{4} R\left(f^{\prime \prime}ight)}ight]^{1 / 5} .\]