= Kernel for density estimation
{title2=$K,\quad K_h(u)=h^{-1}K(u/h)$}
= Kernels for density estimation
{synonym}
A kernel for <density estimation> is an <integrable function> $K:\mathbb R\to\mathbb R$ with unit <integral>. For an ordinary <kernel density estimator> that is itself a <probability density function>, take
$$
K(u)\geq0,\qquad \int_{\mathbb R}K(u)\,du=1,\qquad K_h(u)=\frac1hK\!\left(\frac uh\right),\quad h>0.
$$
Thus $K$ specifies the bump's shape and the <smoothing bandwidth> $h$ specifies its width. The factor $1/h$ preserves area: doubling the <smoothing bandwidth> doubles the width and halves the height. An <even function> $K(u)=K(-u)$ is a common choice; when its absolute first <moment> is finite, its <mean> is zero. Symmetry is useful but is not required for normalization.
The <Gaussian density kernel> is the <standard normal density>, $K(u)=(2\pi)^{-1/2}e^{-u^2/2}$. Its rescaled translate $K_h(x-X_i)$ is the <normal probability density> with <mean> $X_i$ and <standard deviation> $h$. It is positive everywhere, whereas an <Epanechnikov kernel> has <compact support>. With the <Gaussian density kernel>, the <smoothing bandwidth> is a <standard deviation>, not a cutoff beyond which observations have no influence.
The broader convention also allows signed <kernels of order ell>, such as the <flat-top kernel for density estimation>, to cancel <moments> and reduce the <bias of an estimator>. Their unit <integral> still normalizes total area, but a resulting <kernel density estimator> can take negative values and need not be a <probability density function>. Nonnegativity is therefore an additional condition, not a consequence of unit <integral>.
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