= Solution
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 $h_0=[R(K)/(n\mu_2(K)^2R(f''))]^{1/5}$, balancing $R(K)/(nh)$ against $h^4\mu_2(K)^2R(f'')/4$. Three widely used selection methods are as follows.
For <least-squares density cross-validation>, expand integrated squared error as $R(\widehat f_h)-2\int\widehat f_h f+R(f)$. Estimate the middle integral by leaving out the observation at which the estimate is evaluated, and minimize
$$
\operatorname{CV}(h)=R(\widehat f_h)-\frac2n\sum_i\widehat f_{h,-i}(X_i),\qquad\widehat f_{h,-i}(x)=\frac1{n-1}\sum_{j\ne i}K_h(x-X_j).
$$
The left-out observation is independent of its estimator, so the second term has expectation $-2\int(K_h*f)f$. Thus $\mathbb E\operatorname{CV}(h)=\operatorname{MISE}(h)-R(f)$ exactly. The subtraction of the self-contribution is essential. Computation uses $R(\widehat f_h)=n^{-2}\sum_{i,j}(K_h*\widetilde K_h)(X_i-X_j)$, where $\widetilde K_h(u)=K_h(-u)$. 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:
$$
\widehat A_h=R(\widehat f_h'')-\frac{R(K'')}{nh^5},\qquad\operatorname{BCV}(h)=\frac{R(K)}{nh}+\frac{\mu_2(K)^2h^4}{4}\widehat A_h.
$$
Differentiating the <kernel for density estimation> estimator and applying the integrated-variance identity gives the exact expectation $\mathbb ER(\widehat f_h'')=R(K'')/(nh^5)+(1-n^{-1})R(K_h*f'')$. Under the derivative regularity assumptions, $R(K_h*f'')=R(f'')+O(h^2)$, so $\mathbb E\widehat A_h=R(f'')+O(h^2)+O(n^{-1})$. 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 $A=R(f'')$, then set
$$
\boxed{\widehat h_{\rm PI}=\left[\frac{R(K)}{n\mu_2(K)^2\widehat A}\right]^{1/5}.}
$$
The pilot should be positive and stable; consistent positive estimates make this expression ratio-consistent by continuity. A simple normal-reference initialization uses $R(f'')=3/(8\sqrt\pi\,\sigma^5)$ under a normal reference model, substituting a sample scale for sigma. For a <Gaussian density kernel> this gives $h\approx1.06\widehat\sigma n^{-1/5}$. 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 $R(f'')=\int ff^{(4)}$, obtained by two integrations by parts, and estimates it by
$$
\widehat R_g^{(2)}=\frac1n\sum_i\widehat f_g^{(4)}(X_i)=\frac1{n^2}\sum_{i,j}K_g^{(4)}(X_i-X_j).
$$
The diagonal expectation contributes $K^{(4)}(0)/(ng^5)$; the leading smoothing <bias> is $-g^2\mu_2(K)R(f''')/2$. Balancing these terms produces the stated pilot scale $g\asymp R(f''')^{-1/7}n^{-1/7}$, with another pilot or reference model supplying $R(f''')$. Under sufficient derivatives and tail control, its error is bounded by
$$
O_p\left(g^2+\frac1{ng^5}+n^{-1/2}+\frac1{ng^{9/2}}\right)=O_p(n^{-2/7})
$$
at that pilot scale. <Taylor expansion> of the fifth-root formula gives $\widehat h_{\rm PI}/h_0-1=-(\widehat A-A)/(5A)+o_p(|\widehat A-A|)$, 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.
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