Families with a monotone likelihood ratio

Assume AC. Let $\mathcal{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment with $\mathrm{\Theta }\subseteq \mathbb{R}$, dominated by a fixed sigma-finite positive measure $\mu$. Let $T:(\mathcal{X},\mathcal{A})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a statistic. Choose the simultaneous finite-valued nonnegative density representatives $p_{\theta }=d{P}_{\theta }\mathrm{/}d\mu$ from Likelihood densities.

For ${\theta }_1<{\theta }_2$, define the extended likelihood ratio on $U_{{\theta }_1,{\theta }_2}=\{x:p_{{\theta }_1}(x)+p_{{\theta }_2}(x)>0\}$ by $R_{{\theta }_2,{\theta }_1}(x)=\{\begin{array}{ll}{p}_{{\theta }_2}(x)\mathrm{/}{p}_{{\theta }_1}(x),& p_{{\theta }_1}(x)>0,\\ +\mathrm{\infty },& p_{{\theta }_1}(x)=0<p_{{\theta }_2}(x)\mathrm{.}\end{array}$ The family has a monotone likelihood ratio (MLR) in $T$ when, for each ordered pair ${\theta }_1<{\theta }_2$, there is a nondecreasing Borel function $r_{{\theta }_2,{\theta }_1}:\mathbb{R}\to [0,+\mathrm{\infty }]$ such that $R_{{\theta }_2,{\theta }_1}(x)=r_{{\theta }_2,{\theta }_1}(T(x))$ for $\mu$-almost every $x\in U_{{\theta }_1,{\theta }_2}$. The set where both densities vanish is irrelevant. No common exceptional null set for all parameter pairs is asserted. This definition concerns the likelihood ratio of the full data; $T$ need not be sufficient.

Because the ratio is nondecreasing in $T$, increasing the parameter favors larger values of $T$. The convention $R=+\mathrm{\infty }$ where the lower-parameter density vanishes records any alternative-only mass and keeps the direction of the upper-tail rule explicit.