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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.

A smoothing bandwidth $h>0$ determines the scale over which a regression kernel averages data, or a kernel for density estimation spreads each observation. It has the same units as the observations. In kernel density estimation, changing $h$ changes the visible detail while preserving the total integral one. For a Gaussian density kernel, each bump has standard deviation $h$ and its individual contribution has peak height $1/(nh\sqrt{2\pi })$.

If the smoothing bandwidth is too small relative to the spacings in a statistical sample, the kernel density estimator shows narrow spikes tied to individual observations: apparent peaks may be sampling accidents. If it is too large relative to features of the unknown probability density function, it blurs distinct groups together and spreads fitted probability far from the observations. An intermediate smoothing bandwidth can reveal broad structure, but visual smoothness alone cannot establish accuracy.

The first three panels use exactly $(-1.8,-1.4,-1.1,-0.7,0.7,1.1,1.4,1.8)$ with smoothing bandwidths $0.08$, $0.35$ and $1.2$, respectively. Every curve has area one; the two groups in the observed statistical sample become less distinct as $h$ increases. No true probability density function is specified in this illustration, so the middle panel is illustrative, not a certified optimal fit. The fourth panel uses $(0.1,0.4,0.9)$ and $h=0.5$ to illustrate boundary bias of a kernel density estimator.

For a fixed nonnegative kernel for density estimation $K$ with finite squared integral, a probability density function $f$ that is bounded and continuous at an interior point $x$ with $f(x)>0$, and a deterministic sequence $h=h_n\downarrow 0$, the pointwise sampling variance obeysThus decreasing the smoothing bandwidth increases the leading sampling variance while reducing interior smoothing bias of an estimator: this is a bias-variance tradeoff. The conditions $h_n\to 0$ and $n{h}_n\to \infty$ express two goals, increasingly fine resolution and enough observations contributing locally. Practical methods such as plug-in bandwidth selection use the data to choose $h$; pointwise optimal kernel bandwidth depends on local features of the unknown probability density function. The smoothing bandwidth remains a tuning choice in nonparametric statistics, and knowing the support of a probability distribution can require more than a width adjustment near a boundary.