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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a family of distributions on a set $\\mathcal{X}$, with $\\Theta \\subseteq \\R^k$.\nAn *estimating function* is a map $g \\colon \\mathcal{X} \\times \\Theta \\to \\R^k$.\nIt is *unbiased* if $E_\\theta[g(X, \\theta)] = 0$ for every $\\theta \\in \\Theta$, where $X \\sim P_\\theta$ and $E_\\theta$ is the [[expectation|0DCT0QZVQ1]].\nFor [[i.i.d.|0AJN7WHBJZ]] observations $X_1, \\dots, X_n$, the *estimating equation* is\n$$\\sum_{i=1}^n g(X_i, \\theta) = 0,$$\nand an [[estimator|0TTNRP5PFQ]] $\\hat{\\theta}_n$ that solves it is an estimating-equation estimator.\n\n## Remarks\n\nExample: the [[score|CP97GRVTQ0]] $g(x, \\theta) = \\nabla_\\theta \\log f(x; \\theta)$ is an unbiased estimating function under the usual regularity conditions, and its estimating equation gives the likelihood equations.\n\nExample: in the [[method of moments|09J1H07VAH]] for a Poisson mean, $g(x, \\lambda) = x - \\lambda$ has $E_\\lambda[X - \\lambda] = 0$, and the equation $\\sum_i (X_i - \\lambda) = 0$ gives $\\hat{\\lambda} = \\bar{X}$.\n\nNon-example: for the Poisson mean, $g(x, \\lambda) = x - 2\\lambda$ has $E_\\lambda[g(X, \\lambda)] = -\\lambda \\ne 0$, so it is not unbiased, and its root $\\bar{X}/2$ converges to $\\lambda/2$, not to $\\lambda$.\n\nEstimators of this kind are the $\\psi$-type [[M-estimators|4503R8BVF8]].\nUnder regularity conditions, $\\sqrt{n}(\\hat{\\theta}_n - \\theta) \\xrightarrow{d} \\mathcal{N}(0, A^{-1} B A^{-T})$ with $A = E_\\theta[\\nabla_\\theta g(X, \\theta)]$ and $B = E_\\theta[g(X, \\theta) g(X, \\theta)^T]$.\n",
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