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  "body": "\n## Definition\n\nLet $\\Theta \\subseteq \\R$ and let $\\set{f_\\theta : \\theta \\in \\Theta}$ be a family of [[densities|04YG690N5V]] (or probability mass functions) on a sample space $\\mathcal{X}$.\nLet $T \\colon \\mathcal{X} \\to \\R$ be a [[statistic|036756NFD6]].\nThe family has a *monotone likelihood ratio* in $T$ if, for all $\\theta_1 \u003c \\theta_2$ in $\\Theta$:\n\n1. $f_{\\theta_1} \\ne f_{\\theta_2}$;\n2. there is a nondecreasing function $g \\colon \\R \\to [0, \\infty]$ with $f_{\\theta_2}(x)/f_{\\theta_1}(x) = g(T(x))$ for every $x$ with $f_{\\theta_1}(x) \u003e 0$ or $f_{\\theta_2}(x) \u003e 0$.\n\nHere $c/0 = \\infty$ for $c \u003e 0$.\n\n## Remarks\n\nThe ratio in clause 2 is the [[likelihood ratio|05FSTXM92W]] of $\\theta_2$ to $\\theta_1$ at the data $x$.\nExample: let $f_\\theta(x) = h(x) \\exp\\big(\\eta(\\theta) T(x) - A(\\theta)\\big)$ be a one-parameter exponential family with $\\eta$ strictly increasing.\nThen $f_{\\theta_2}(x)/f_{\\theta_1}(x) = \\exp\\big((\\eta(\\theta_2) - \\eta(\\theta_1)) T(x) - A(\\theta_2) + A(\\theta_1)\\big)$, which is increasing in $T(x)$.\nThis covers the binomial law in $p$ with $T(x) = x$, and $\\mathcal{N}(\\theta, 1)$ with $T(x) = x$.\nNon-example: the Cauchy location family $f_\\theta(x) = 1/(\\pi(1 + (x - \\theta)^2))$ fails clause 2 with $T(x) = x$.\nFor $\\theta_1 = 0$ and $\\theta_2 = 1$, the ratio $(1 + x^2)/(1 + (x - 1)^2)$ tends to $1$ as $x \\to \\pm\\infty$ and is larger than $1$ at $x = 1$, so it is not monotone.\n",
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