Asymptotic mean integrated squared error ($\mathrm{AMISE}(h)=R(K)/(nh)+{\mu }_2(K{)}^{2}R(f^{\prime \prime })h^4/4$)

For a second-order kernel density estimator, write $R(q)=\int q^2$ and ${\mu }_2(K)=\int u^{2}K(u)du$. Under the usual smoothness and shrinking-bandwidth assumptions, the leading mean integrated squared error is the displayed sum of integrated variance and squared bias of an estimator. Its minimizer is $h=[R(K)/(n{\mu }_2(K{)}^{2}R(f^{\prime \prime })){]}^{1/5}$, giving order $n^{-4/5}$. An added $O(n^{-1})$ integrated covariance term is lower order at this bandwidth and does not change its leading constant.