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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] random variables with values in a measurable space $E$.\nLet $r \\le n$ be a positive integer, and let $h \\colon E^r \\to \\R$ be a measurable function that is symmetric in its $r$ arguments, the *kernel*.\nThe *U-statistic* with kernel $h$ is\n$$U_n = \\binom{n}{r}^{-1} \\sum_{1 \\le i_1 \u003c i_2 \u003c \\dots \u003c i_r \\le n} h(X_{i_1}, \\dots, X_{i_r}).$$\nThe integer $r$ is the degree of the kernel.\n\n## Remarks\n\nIf $E\\abs{h(X_1, \\dots, X_r)} \u003c \\infty$, then $U_n$ is an unbiased estimator of $\\theta = E[h(X_1, \\dots, X_r)]$, by linearity of [[expectation|0DCT0QZVQ1]].\nLet $h_1(x) = E[h(x, X_2, \\dots, X_r)]$ and $\\zeta_1 = \\operatorname{Var}(h_1(X_1))$.\nIf $E[h^2] \u003c \\infty$, Hoeffding's theorem gives $\\sqrt{n}(U_n - \\theta) \\to N(0, r^2 \\zeta_1)$ in distribution.\n\nExample: $r = 2$ and $h(x, y) = (x - y)^2 / 2$.\nThe identity $\\sum_{i \u003c j} (x_i - x_j)^2 = n \\sum_i (x_i - \\bar x)^2$ gives $U_n = \\frac{2}{n(n-1)} \\cdot \\frac{n}{2} \\sum_i (X_i - \\bar X)^2 = \\frac{1}{n-1} \\sum_i (X_i - \\bar X)^2$, the sample variance.\nHere $\\theta = \\operatorname{Var}(X_1)$.\nWith $r = 1$ and $h(x) = x$, $U_n$ is the sample mean.\n\nNon-example: the average $\\frac{1}{n^2} \\sum_{i, j} h(X_i, X_j)$ over all ordered pairs, including $i = j$, is a V-statistic, not a U-statistic.\nFor $h(x, y) = (x - y)^2/2$ it equals $\\frac{1}{n} \\sum_i (X_i - \\bar X)^2$, which is biased.\n\nTwo-sample U-statistics average a kernel over one subset from each of two samples; the [[Mann–Whitney statistic|CKF9KDXBP7]] is one.\n",
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