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Let $\theta_0<\theta_1$, and let the ratio of likelihood functions for the random variables $X_1,...,X_n$ be $$\frac{L(x ; \theta_1)}{L(x; \theta_0)}=\frac{I(x_{(n)}<\theta_1}{I(x_{(n)}<\theta_0}.$$ Supposing that in this case, the n-th order statistic is sufficient for $\theta$, then as $x_{(n)}$ increases, the ratio goes from 1 to infinity (1/0, by convention) to 1 again (0/0, by convention).

My question is, do I need to consider this last change of the ratio's value when $x_{(n)}$ is greater than both values of theta, or is this considered an impossibility? I believe this does constitute the MLR increasing property, but I can't really justify that belief.

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What is important to notice is what your likelihood functions indicate. If either of them is 0, then the $x_{(n)}$ is not possible under that hypothesis. This is why 1/0 is equaled to $\infty$: If the observation is impossible with $\theta_0$ but possible with $\theta_1$, then the observation is of course infinitely much more likely to occur under $\theta_1$ compared to $\theta_0$. However, if the observation is impossible under both $\theta$, then both are equally impossible, so the ratio is not defined, and not 1. This deviates from the usual convention. So, no, do not consider the last change, it is impossible

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