Plug-in bandwidth selection ($\hat{h}=[R(K)/({\mu }_2(K{)}^{2}n\hat{R}(f^{\prime \prime })){]}^{1/5}$)

Estimate the unknown curvature in the AMISE minimizer and substitute it into that formula. A consistent positive curvature estimate makes the selected bandwidth ratio-consistent. Pilot smoothing is needed because differentiating an undersmoothed density estimator magnifies variance. A normal-reference pilot can initialize an estimate of a higher derivative functional; the final density-curvature estimate is computed from the data rather than assuming the density is normal.

Twice integrating by parts gives $R(f^{\prime \prime })=\int f{f}^{(4)}$. Estimate this by averaging a fourth-derivative kernel density estimator at the observations. Its expectation contains diagonal term $K^{(4)}(0)/(n{g}^5)$ and leading smoothing bias $-g^2{\mu }_2(K)R(f^{\prime \prime \prime })/2$. Balancing them yields pilot scale $g\asymp R(f^{\prime \prime \prime }{)}^{-1/7}{n}^{-1/7}$. With suitable derivatives and tails, the estimation error is bounded by $O_p(g^2+(n{g}^5{)}^{-1}+n^{-1/2}+n^{-1}{g}^{-9/2})$, giving an $O_p(n^{-2/7})$ relative-bandwidth bound after substitution. More carefully calibrated bias cancellation can improve that bound.

// Target: statistical-inference.bigb