Kernel density estimation and bandwidth

For an iid sample from a Lebesgue density f on the real line, choose an integrable kernel K with integral one and bandwidth h>0. The estimate averages $h^{-1}K((x-X_i)/h)$. If K is nonnegative, the estimate is nonnegative and integrates to one. The factor h inverse is essential: widening a bump lowers its height while preserving its probability mass. A symmetric kernel has zero first moment when that moment is finite, but symmetry alone does not supply finite second moments or square integrability.

The expectation of the estimate is the convolution of f with the rescaled kernel, so smoothing creates bias before sampling variability enters. A signed higher-order kernel can cancel additional moments and reduce certain bias terms, but its estimate can be negative. Thus unit integral and nonnegativity are distinct requirements. Calling every finite curve a density because it has a plausible shape overlooks both normalization and support. A density value also has inverse measurement units; changing the units of the observations must change its numerical height.

Take a bounded, nonnegative, symmetric, compactly supported kernel with integral one, positive second moment $\mu_2(K)$, and $R(K)=\int K^2$. At a fixed point where f is twice continuously differentiable in a neighborhood, Taylor expansion gives bias $h^2\mu_2(K)f''(x)/2+o(h^2)$. With f continuous there, h tending to zero and nh tending to infinity, the variance is $f(x)R(K)/(nh)+o((nh)^{-1})$. These are asymptotic expansions, not exact finite-sample errors for every density.

For a density f in W²,²(R), the Sobolev space with two square-integrable weak derivatives, and the same compactly supported kernel, the leading mean integrated squared error balances a term proportional to h to the fourth power against a term proportional to one over nh. The optimizing order is n to the minus one-fifth in one dimension. Its constant depends on the unknown curvature of f, so the expression is an oracle guide rather than an implementable instruction by itself. Cross-validation and plug-in rules estimate different ingredients and can behave differently on finite samples.

If a density is supported on a half-line and has a positive boundary value, a symmetric kernel spills mass outside the support. At the boundary its expectation may approach half the correct height rather than the height itself. The interior Taylor calculation cannot be used across a discontinuity in the density's zero extension. Reflection, boundary kernels, transformations, or local-polynomial density methods address particular boundary settings; none is justified merely by shrinking the bandwidth while retaining the same incorrect boundary argument.

Bandwidth controls a spatial scale, whereas increasing the sample size supplies more information at that scale. Comparing estimates at several bandwidths can reveal instability but does not distinguish noise from a genuine narrow feature without assumptions. Multivariate product kernels also pay a variance cost involving the product of bandwidths, which becomes severe as dimension grows. The normalization, effective local sample size, and applicable smoothness class should be stated before declaring a visually detailed estimate more accurate than a smooth one.