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  "body": "\n## Definition\n\nLet $X$ be data with log-likelihood $\\ell(\\theta) = \\log L(\\theta \\mid X)$, differentiable in $\\theta$ in an open set $\\Theta \\subseteq \\R^k$, and let $\\theta_0 \\in \\Theta$.\nLet $U(\\theta) = \\nabla \\ell(\\theta)$ be the [[score|CP97GRVTQ0]] and let $I(\\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$, assumed invertible at $\\theta_0$.\nThe *score statistic* for $H_0 \\colon \\theta = \\theta_0$ is\n$$S = U(\\theta_0)^{T} I(\\theta_0)^{-1} U(\\theta_0).$$\nThe *score test* at level $\\alpha$ rejects $H_0$ when $S$ exceeds the $1 - \\alpha$ quantile of the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.\n\n## Remarks\n\nUnder $H_0$ and the usual regularity conditions for i.i.d. samples, $S$ converges in distribution to $\\chi^2_k$, so the test has asymptotic level $\\alpha$.\nFor a composite $H_0$ with nuisance parameters, one evaluates the score and the information at the maximum likelihood estimate under $H_0$.\nExample: for $X$ binomial with parameters $n$ and $p$, one has $U(p) = (X - np)/(p(1 - p))$ and $I(p) = n/(p(1 - p))$.\nSo $S = (X - np_0)^2/(np_0(1 - p_0))$, the square of the usual z-statistic for a proportion.\nNon-example: the statistic $U(\\theta_0)^T U(\\theta_0)$, without the factor $I(\\theta_0)^{-1}$, does not have a $\\chi^2_k$ limit in general, so it is not the score statistic.\n",
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