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  "body": "\n## Definition\n\nLet $X$ be data from a model with parameter $\\theta$ in an open set $\\Theta \\subseteq \\R^k$, and let $\\theta_0 \\in \\Theta$.\nLet $\\hat\\theta$ be the [[maximum likelihood estimate|0FREPFKNS7]] of $\\theta$, and let $I(\\hat\\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$ evaluated at $\\hat\\theta$, assumed invertible.\nThe *Wald statistic* for $H_0 \\colon \\theta = \\theta_0$ is\n$$W = (\\hat\\theta - \\theta_0)^{T} I(\\hat\\theta) (\\hat\\theta - \\theta_0).$$\nThe *Wald test* at level $\\alpha$ rejects $H_0$ when $W$ 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, $W$ converges in distribution to $\\chi^2_k$, so the test has asymptotic level $\\alpha$.\nFor $k = 1$ the test rejects when $\\abs{\\hat\\theta - \\theta_0}/\\widehat{\\mathrm{se}} \u003e z_{1 - \\alpha/2}$, with $\\widehat{\\mathrm{se}} = I(\\hat\\theta)^{-1/2}$ and $z_{1 - \\alpha/2}$ a standard normal quantile.\nThe observed information may replace $I(\\hat\\theta)$ without changing the limit law.\nExample: for $X$ binomial with parameters $n$ and $p$, one has $\\hat p = X/n$ and $W = (\\hat p - p_0)^2 n/(\\hat p(1 - \\hat p))$.\nUnlike the [[likelihood-ratio test|YM1BNA80RC]], the Wald test is not invariant under a reparametrization of $\\theta$.\n",
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