Monotone likelihood ratio

Remarks

The ratio in clause 2 is the likelihood ratio of $\theta_2$ to $\theta_1$ at the data $x$. Example: 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. Then $f_{\theta_2}(x)/f_{\theta_1}(x) = \exp\big((\eta(\theta_2) - \eta(\theta_1)) T(x) - A(\theta_2) + A(\theta_1)\big)$, which is increasing in $T(x)$. This covers the binomial law in $p$ with $T(x) = x$, and $\mathcal{N}(\theta, 1)$ with $T(x) = x$. Non-example: the Cauchy location family $f_\theta(x) = 1/(\pi(1 + (x - \theta)^2))$ fails clause 2 with $T(x) = x$. For $\theta_1 = 0$ 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.