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  "body": "\n## Statement\n\nLet $\\theta \\in \\R$ and $\\sigma^2 \u003e 0$.\nLet $(T_n)$ be a sequence of real random variables such that\n$$\\sqrt{n}\\,(T_n - \\theta) \\xrightarrow{d} \\mathcal{N}(0, \\sigma^2),$$\nwhere $\\xrightarrow{d}$ is [[convergence in distribution|0JDAQHWA9N]] and $\\mathcal{N}(0, \\sigma^2)$ is the [[normal distribution|07RXBVTVZB]].\nLet $g \\colon \\R \\to \\R$ be [[differentiable|0A61B60B5R]] at $\\theta$.\nThen\n$$\\sqrt{n}\\,\\big(g(T_n) - g(\\theta)\\big) \\xrightarrow{d} \\mathcal{N}\\big(0, g'(\\theta)^2 \\sigma^2\\big).$$\n\n## Remarks\n\nIdea of proof: by differentiability, $g(T_n) - g(\\theta) = g'(\\theta)(T_n - \\theta) + r(T_n)(T_n - \\theta)$ with $r(t) \\to 0$ as $t \\to \\theta$.\nSince $T_n \\to \\theta$ in probability, $r(T_n) \\to 0$ in probability, and $\\sqrt{n}(T_n - \\theta)$ is bounded in probability.\nSo the remainder times $\\sqrt{n}$ tends to $0$ in probability, and Slutsky's lemma gives the limit $g'(\\theta)\\,\\mathcal{N}(0, \\sigma^2)$.\n\nMultivariate form: if $\\sqrt{n}(T_n - \\theta) \\xrightarrow{d} \\mathcal{N}_k(0, \\Sigma)$ in $\\R^k$, with a [[multivariate normal|0G558GJQMW]] limit, and $g \\colon \\R^k \\to \\R^m$ is [[differentiable|0135XZZQ8M]] at $\\theta$ with [[Jacobian matrix|047NDTEPCV]] $J$, then $\\sqrt{n}(g(T_n) - g(\\theta)) \\xrightarrow{d} \\mathcal{N}_m(0, J \\Sigma J^T)$.\n\nExample: for i.i.d. [[Poisson|0VEFSSKX18]] variables with mean $\\lambda \u003e 0$, the [[central limit theorem|0EPZC4Z1FC]] gives $\\sqrt{n}(\\bar{X}_n - \\lambda) \\xrightarrow{d} \\mathcal{N}(0, \\lambda)$.\nWith $g(t) = 2\\sqrt{\\abs{t}}$, $g'(\\lambda) = 1/\\sqrt{\\lambda}$, so $\\sqrt{n}\\big(2\\sqrt{\\bar{X}_n} - 2\\sqrt{\\lambda}\\big) \\xrightarrow{d} \\mathcal{N}(0, 1)$.\nThe limit variance no longer depends on $\\lambda$: the square root is a variance-stabilizing transformation for Poisson data.\n\nIf $g'(\\theta) = 0$, the limit is the point mass at $0$, and one needs a second-order expansion: if $g''(\\theta) \\ne 0$ exists, then $n(g(T_n) - g(\\theta)) \\xrightarrow{d} \\frac{1}{2} g''(\\theta) \\sigma^2 \\chi^2_1$.\n\n## Explanations\n\n### L1 human\n\nIf an estimate is approximately normal around the truth, then a smooth function of the estimate is also approximately normal around the function of the truth.\nNear the true value, a smooth function looks like a straight line with slope $g'(\\theta)$, and a straight line stretches the spread by that slope.\nSo the standard deviation is multiplied by $\\abs{g'(\\theta)}$.\n\n### L2 human\n\nThe delta method turns the central limit theorem for averages into limit laws for smooth functions of averages: ratios, logarithms, correlation coefficients, odds ratios.\nIt gives approximate standard errors and confidence intervals for such quantities.\nIt is also the step that passes from the asymptotic normality of an estimator $\\hat{\\theta}$ to that of $g(\\hat{\\theta})$, for example for maximum likelihood estimators.\nVariance-stabilizing transformations are chosen with it: one picks $g$ so that $g'(\\theta)^2 \\sigma^2(\\theta)$ does not depend on $\\theta$.\n",
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