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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with an unknown [[probability density function|04YG690N5V]] $f$.\nLet $K \\colon \\R \\to \\R$ be an integrable function with $\\int_{\\R} K(u) \\, du = 1$, called the kernel, and let $h \u003e 0$, called the bandwidth.\nThe *kernel density estimator* of $f$ is\n$$\\hat f_h(x) = \\frac{1}{n h} \\sum_{i=1}^{n} K\\Big(\\frac{x - X_i}{h}\\Big), \\quad x \\in \\R.$$\n\n## Remarks\n\nThe substitution $u = (x - X_i)/h$ gives $\\int_{\\R} \\hat f_h(x) \\, dx = 1$; if moreover $K \\ge 0$, then $\\hat f_h$ is a probability density.\nCommon kernels are the Gaussian kernel $K(u) = e^{-u^2/2}/\\sqrt{2\\pi}$ and the Epanechnikov kernel $K(u) = \\frac{3}{4}(1 - u^2)$ for $\\abs{u} \\le 1$, $0$ otherwise.\nExample: with $K = \\frac{1}{2} \\mathbf{1}_{[-1, 1]}$, $\\hat f_h(x)$ is the fraction of data in $[x - h, x + h]$ divided by $2h$.\nNon-example: $K = \\mathbf{1}_{[-1, 1]}$ has integral $2$, so the resulting $\\hat f_h$ integrates to $2$ and is not a density estimate.\nIf $f$ is twice continuously differentiable and $K$ is a bounded symmetric density with finite variance, the mean integrated squared error is smallest for $h$ of order $n^{-1/5}$.\nIn $\\R^d$, one uses $\\frac{1}{n h^d} \\sum_i K((x - X_i)/h)$ with a kernel on $\\R^d$.\n",
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