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  "body": "\n## Definition\n\nLet $\\set{f(\\cdot; \\theta) : \\theta \\in \\Theta}$ be a parametric family of densities with $\\Theta \\subseteq \\R$ an open interval, regular enough for the [[Cramér–Rao bound|1JFKTEQVRX]] to hold.\nLet $X = (X_1, \\dots, X_n)$ be the data, and let $I_n(\\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$, assumed finite and positive.\nLet $T$ be an unbiased [[estimator|0TTNRP5PFQ]] of $\\theta$ (its [[bias|01BD2TFFP4]] is $0$) with finite positive [[variance|007QED3E0Y]].\n\n1. The *efficiency* of $T$ at $\\theta$ is $e(T, \\theta) = \\dfrac{1/I_n(\\theta)}{\\operatorname{Var}_\\theta(T)}$.\n2. $T$ is *efficient* if $e(T, \\theta) = 1$ for every $\\theta \\in \\Theta$.\n3. For two unbiased estimators $T_1, T_2$ of $\\theta$, the *relative efficiency* of $T_1$ with respect to $T_2$ is $\\operatorname{Var}_\\theta(T_2)/\\operatorname{Var}_\\theta(T_1)$.\n\n## Remarks\n\nBy the Cramér–Rao bound, $0 \u003c e(T, \\theta) \\le 1$.\n\nExample: for i.i.d. $X_i \\sim \\mathcal{N}(\\mu, \\sigma^2)$ with $\\sigma^2$ known, $I_n(\\mu) = n/\\sigma^2$ and the [[sample mean|0QNRBHWV35]] has variance $\\sigma^2/n$, so $e(\\bar{X}, \\mu) = 1$ and $\\bar{X}$ is efficient.\n\nNon-example: in the same model, $T = X_1$ is unbiased with variance $\\sigma^2$, so $e(T, \\mu) = 1/n$, and $T$ is not efficient when $n \\ge 2$.\n\nThe *asymptotic relative efficiency* compares two estimators $T_{1,n}, T_{2,n}$ with $\\sqrt{n}(T_{j,n} - \\theta) \\xrightarrow{d} \\mathcal{N}(0, v_j(\\theta))$ by the ratio $v_2(\\theta)/v_1(\\theta)$.\nFor normal data, the asymptotic relative efficiency of the sample median with respect to the sample mean is $2/\\pi \\approx 0.64$.\n",
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