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  "body": "\n## Definition\n\nLet $(X_1, Y_1), \\dots, (X_n, Y_n)$ be [[i.i.d.|0AJN7WHBJZ]] pairs of real random variables with $E\\abs{Y_1} \u003c \\infty$, and with regression function $m(x) = E[Y_1 \\mid X_1 = x]$, a [[conditional expectation|0TTDGTEFMN]].\nLet $K \\colon \\R \\to [0, \\infty)$ be a kernel as in [[kernel density estimation|ZV65XMJ9A4]], and let $h \u003e 0$ be a bandwidth.\nThe *Nadaraya–Watson kernel regression estimator* of $m$ is\n$$\\hat m_h(x) = \\frac{\\sum_{i=1}^{n} K\\big((x - X_i)/h\\big)\\, Y_i}{\\sum_{i=1}^{n} K\\big((x - X_i)/h\\big)},$$\ndefined at each $x$ where the denominator is not $0$.\n\n## Remarks\n\nThe weights $K((x - X_i)/h) / \\sum_j K((x - X_j)/h)$ are nonnegative and sum to $1$, so $\\hat m_h(x)$ is a weighted average of the $Y_i$.\nDividing the numerator and the denominator by $nh$ shows the denominator is the kernel density estimate of the law of $X$ at $x$.\nExample: with the box kernel $K = \\frac{1}{2} \\mathbf{1}_{[-1, 1]}$, $\\hat m_h(x)$ is the plain average of the $Y_i$ with $\\abs{X_i - x} \\le h$.\nNon-example: the numerator alone, divided by $nh$, estimates $m(x) f_X(x)$ rather than $m(x)$, where $f_X$ is the density of $X$.\nLocal polynomial regression generalizes this estimator; the Nadaraya–Watson estimator is the local constant case.\n",
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