= Kernel density estimation
{title2=$\widehat f_{n,h,K}$}
{wiki}
= Kernel density estimator
{synonym}
Let $X_1,\ldots,X_n$ be real <independent and identically distributed random variables> with unknown <probability density function> $f$, let $n\geq1$, and choose a nonnegative, unit-<integral> <kernel for density estimation> $K$ and a <smoothing bandwidth> $h>0$. The <kernel density estimator> is
$$
\widehat f_{n,h,K}(x)=\frac1{nh}\sum_{i=1}^nK\!\left(\frac{x-X_i}{h}\right)=\frac1n\sum_{i=1}^nK_h(x-X_i),\qquad x\in\mathbb R.
$$
Place a bump at every observation and add their contributions. Each bump $K_h(x-X_i)/n$ has area $1/n$, so for every realized <statistical sample>
$$
\widehat f_{n,h,K}(x)\geq0,\qquad \int_{\mathbb R}\widehat f_{n,h,K}(x)\,dx=\frac1n\sum_{i=1}^n\int_{\mathbb R}K(u)\,du=1.
$$
The <change of variables> is $u=(x-X_i)/h$. Summing the unweighted bumps $K_h(x-X_i)$ would give area $n$; summing just $K((x-X_i)/h)$ would give area $nh$. Those raw sums do not have the normalization of a <probability density function>.
Unlike a <histogram>, the construction does not require choosing bin origins, and a <Gaussian density kernel> gives a smooth curve. Its appeal is that identical local contributions can combine into a <probability density function> with several peaks without first choosing their number. This makes <kernel density estimation> useful for exploring an unknown <probability distribution> and estimating interval <probabilities>, while the <smoothing bandwidth> controls how much detail is visible.
\Image[/kernel-density-components.png]
{title=Three Gaussian contributions, each of area one third, add to a normalized kernel density estimate}
Before the <statistical sample> is observed, $\widehat f_{n,h,K}$ is a random <function>; afterwards it is a fitted <probability density function>. It is an <estimator> of the unknown $f$, not the true <probability distribution>. For fixed $K,h$, its <expectation> is the <convolution> $K_h*f$, generally different from $f$, and repeating the sampling changes the fitted curve. The <exact mean integrated squared error of a kernel density estimator> separates this smoothing <bias of an estimator> from sampling <variance>. Allowing signed <kernels of order ell> retains the formula but can lose nonnegativity.
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