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  "body": "\n## Definition\n\nLet $X$ be data with values in a set $\\mathcal{X}$, whose law $P_\\theta$ depends on a parameter $\\theta \\in \\Theta$, and let $\\alpha \\in (0, 1)$.\nFor each $\\theta \\in \\Theta$, choose an acceptance region $A(\\theta) \\subseteq \\mathcal{X}$ with $P_\\theta(X \\in A(\\theta)) \\ge 1 - \\alpha$.\nThe *Neyman construction* is the set-valued statistic\n$$C(X) = \\set{\\theta \\in \\Theta : X \\in A(\\theta)}.$$\n\n## Remarks\n\nFor each $\\theta$, the events $\\set{\\theta \\in C(X)}$ and $\\set{X \\in A(\\theta)}$ are equal, so $P_\\theta(\\theta \\in C(X)) \\ge 1 - \\alpha$.\nSo $C(X)$ is a [[confidence set|0K4NMMJBG2]] of level $1 - \\alpha$; its coverage is exactly $1 - \\alpha$ when each $A(\\theta)$ has probability exactly $1 - \\alpha$.\nEquivalently, $A(\\theta)$ is the acceptance region of a [[test|087P03NEND]] at level $\\alpha$ of the hypothesis that $\\theta$ is the true value.\nThen $C(X)$ is the set of values that these tests do not reject.\nExample: for $X \\sim \\mathcal{N}(\\theta, 1)$ and $A(\\theta) = [\\theta - 1.96, \\theta + 1.96]$, the construction gives $C(X) = [X - 1.96, X + 1.96]$, of level about $0.95$.\nNon-example: for $\\alpha = 0.05$, the choice $A(\\theta) = [\\theta - 1, \\theta + 1]$ fails the condition, since $P_\\theta(X \\in A(\\theta)) \\approx 0.68$; the resulting set has level only about $0.68$.\n",
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