Integrated mean squared error ($\mathrm{MISE}$)

The integrated mean squared error is $\mathbb{E}{\int }_D(\hat{m}(x)-m(x){)}^{2}dx$ over a specified domain $D$. The Tonelli theorem permits integration of the pointwise bias-variance decomposition of mean squared error.

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.