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Assume a symmetric second-order probability density function kernel for density estimation and enough smoothness for the displayed derivative functionals. The benchmark smoothing bandwidth is $h_0=[R(K)/(n{\mu }_2(K{)}^{2}R(f^{\prime \prime })){]}^{1/5}$, balancing $R(K)/(nh)$ against $h^4{\mu }_2(K{)}^{2}R(f^{\prime \prime })/4$. Three widely used selection methods are as follows.

For least-squares density cross-validation, expand integrated squared error as $R({\hat{f}}_h)-2\int {\hat{f}}_{h}f+R(f)$. Estimate the middle integral by leaving out the observation at which the estimate is evaluated, and minimizeThe left-out observation is independent of its estimator, so the second term has expectation $-2\int (K_h\ast f)f$. Thus $\mathbb{E}\mathrm{CV}(h)=\mathrm{MISE}(h)-R(f)$ exactly. The subtraction of the self-contribution is essential. Computation uses $R({\hat{f}}_h)=n^{-2}{\sum }_{i,j}(K_h\ast {\tilde{K}}_h)(X_i-X_j)$, where ${\tilde{K}}_h(u)=K_h(-u)$. This method directly targets integrated error without estimating a curvature functional first. Under standard smoothness and a suitable search interval its selected smoothing bandwidth is ratio-consistent with the MISE optimum, and hence with $h_0$. Relative fluctuations are ordinarily $O_p(n^{-1/10})$, so the selected values can be quite variable in finite samples.

For biased density cross-validation, estimate curvature from the second-derivative probability density function estimator and subtract its diagonal variance term:Differentiating the kernel for density estimation estimator and applying the integrated-variance identity gives the exact expectation $\mathbb{E}R({\hat{f}}_h^{\prime \prime })=R(K^{\prime \prime })/(n{h}^5)+(1-n^{-1})R(K_h\ast f^{\prime \prime })$. Under the derivative regularity assumptions, $R(K_h\ast f^{\prime \prime })=R(f^{\prime \prime })+O(h^2)$, so $\mathbb{E}{\hat{A}}_h=R(f^{\prime \prime })+O(h^2)+O(n^{-1})$. Minimize BCV over a reasonable smoothing bandwidth range. It is a smoothed criterion directed at AMISE; the name reflects its finite-sample bias rather than an exact unbiased-risk identity. Standard conditions again give ratio consistency and relative fluctuation order $n^{-1/10}$. It often has a smoother objective than ordinary least-squares cross-validation, though numerical behavior and relative variance depend on the probability density function and kernel for density estimation.

For plug-in bandwidth selection, obtain an independent pilot scale g, estimate $A=R(f^{\prime \prime })$, then setThe pilot should be positive and stable; consistent positive estimates make this expression ratio-consistent by continuity. A simple normal-reference initialization uses $R(f^{\prime \prime })=3/(8\sqrt{\pi }{\sigma }^5)$ under a normal reference model, substituting a sample scale for sigma. For a Gaussian density kernel this gives $h\approx 1.06\hat{\sigma }{n}^{-1/5}$. A pure normal-reference rule is quick but is asymptotically correctly calibrated only when its assumed curvature matches the actual probability density function; it is useful as the first stage of a data-driven plug-in procedure.

A more adaptive fourth-derivative pilot estimate of density curvature uses $R(f^{\prime \prime })=\int f{f}^{(4)}$, obtained by two integrations by parts, and estimates it byThe diagonal expectation contributes $K^{(4)}(0)/(n{g}^5)$; the leading smoothing bias is $-g^2{\mu }_2(K)R(f^{\prime \prime \prime })/2$. Balancing these terms produces the stated pilot scale $g\asymp R(f^{\prime \prime \prime }{)}^{-1/7}{n}^{-1/7}$, with another pilot or reference model supplying $R(f^{\prime \prime \prime })$. Under sufficient derivatives and tail control, its error is bounded byat that pilot scale. Taylor expansion of the fifth-root formula gives ${\hat{h}}_{\mathrm{PI}}/h_0-1=-(\hat{A}-A)/(5A)+o_p(|\hat{A}-A|)$, so it inherits that bound. A calibrated cancellation of leading pilot bias can give a faster rate. This illustrates why an appropriately constructed plug-in selector can converge more quickly than cross-validation, while requiring more smoothness and pilot choices. For each ratio-consistent method, its attained AMISE divided by the minimal AMISE tends to one.