Least-squares cross-validation for density bandwidth ($\mathrm{CV}(h)=R({\hat{f}}_h)-\frac{2}{n}{\sum }_i{\hat{f}}_{h,-i}(X_i)$)

Minimize the displayed criterion using a Leave-one-out cross-validation estimate in the second term. Its expectation is the exact mean integrated squared error minus $R(f)$, which does not depend on h. With standard smoothness and appropriate search ranges, the chosen bandwidth is asymptotically optimal. Its relative fluctuations are ordinarily of order $n^{-1/10}$, considerably larger than the estimation error achievable by well-chosen plug-in pilots.