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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be real random variables, and let $X_{(1)} \\le X_{(2)} \\le \\dots \\le X_{(n)}$ be their [[order statistics|58NHGZDSRE]].\nAn *L-estimator* is an [[estimator|0TTNRP5PFQ]] of the form\n$$T_n = \\sum_{i=1}^{n} c_{n,i} \\, X_{(i)},$$\nwhere the coefficients $c_{n,1}, \\dots, c_{n,n} \\in \\R$ are fixed numbers that do not depend on the data.\n\n## Remarks\n\nCommon examples:\n\n1. the sample mean, with $c_{n,i} = 1/n$;\n2. the sample median, with $c_{n,(n+1)/2} = 1$ for odd $n$, and $c_{n,n/2} = c_{n,n/2+1} = 1/2$ for even $n$;\n3. the $k$-trimmed mean, with $c_{n,i} = 1/(n - 2k)$ for $k \u003c i \\le n - k$ and $c_{n,i} = 0$ otherwise;\n4. the midrange $(X_{(1)} + X_{(n)})/2$ and the range $X_{(n)} - X_{(1)}$.\n\nExample: data $(4, 1, 100, 2, 3)$, sorted $(1, 2, 3, 4, 100)$.\nThe $1$-trimmed mean drops $1$ and $100$ and gives $(2 + 3 + 4)/3 = 3$, while the mean is $22$.\n\nNon-example: the sample variance $\\frac{1}{n-1}\\sum_i (X_i - \\bar X)^2$ is a quadratic function of the order statistics, not a linear one, so it is not an L-estimator.\n\nOften the weights come from a function $J$ on $[0, 1]$ through $c_{n,i} = \\frac1n J\\big(\\frac{i}{n+1}\\big)$; then $T_n$ estimates $\\int_0^1 J(u) F^{-1}(u) \\, du$, where $F^{-1}$ is the quantile function.\n",
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