Monotone Likelihood ratio and Karlin-Rubin test I'm studying the Karlin-Rubin test on the book Statistical Inference by Casella and Berger. There the MLR property for a family of pdf is defined as:
$\forall \theta_2>\theta_1\:\: \frac{g(t,\theta_2)}{g(t,\theta_1)}$ is a monotone function of $t$ on the union of the supports. 
Then Karlin-Rubin test says that if $T$ is a sufficient statistic for $\theta$ and his family of densities has the MLR, then a UMP level $\alpha$ test for testing $H_0: \theta\leq\theta_0$ vs $H_0: \theta>\theta_0$ has a rejection region with shape $\{T>t_0\}$ (for an appropriate $t_0$). 
My question: 
If I choose $-T$ then I will still have a sufficient statistic for $\theta$ with the MLR. However, using the theorem I will get a region $\{-T>t_1\}$ which is equivalent to $\{T<-t_1\}$ that has got an opposite shape to the one above. 

Is this correct? I believe that the definition of MLR given there is wrong ...

... and that one should require that the ratio is increasing, but I'm not sure about that. Also I'm not sure about the contradiction in what I got.
 A: This is a late answer to the question, but given the ubiquity of the Casella and Berger text, it seems worthy of an answer.
My copy of Casella and Berger reads the same as yours: monotone likelihood ratio is explicitly defined as either a non-increasing or non-decreasing ratio in Definition 8.3.16, but the statement of Karlin–Rubin in Theorem 8.3.17 is not consistent with this definition. This allows for precisely the sort of contradictory results you describe.
You'll note that the proof of the theorem given in the text is actually dependent on an increasing ratio, as seen in this line:
$$
T > t_0 \iff \frac{g(t|\theta')}{g(t|\theta_0)} > k'
$$
This line is corrected to the following in the errata, but this correction is unrelated to your question and still assumes a non-decreasing ratio:
$$
\left\{ t : \frac{g(t|\theta')}{g(t|\theta_0)} > k' \right \} \subset \{t : t > t_0\} \subset \left\{ t : \frac{g(t|\theta')}{g(t|\theta_0)} \geq k' \right \}
$$

I believe that the definition of MLR given there is wrong ...

I don't think the MLR definition is necessarily wrong. Looking around, I see definitions of MLR that require various forms of monotonicity—some require monotonicity in a certain direction and some don't.
An example of a textbook that defines MLR similarly to Casella and Berger is Introduction to Mathematical Statistics by Hogg, McKean, and Craig, where the ratio is allowed to be non-decreasing or non-increasing. (The ratio they consider is the reciprocal of the one given in Casella and Berger, but both allow for monotonicity in either direction.) The construction of a UMP test is then described for the case when the ratio is decreasing, with a note stating that the inequalities flip if the ratio is increasing:

Assume that our likelihood function $L(\theta, \mathbf{x})$ has a monotone decreasing likelihood ratio in the statistic $y = u(\mathbf{x})$. Then the ratio in (2.2) is equal to $g(y)$, where $g$ is a decreasing function. The case where the likelihood function has a monotone increasing likelihood ratio (i.e., $g$ is an increasing function) follows similarly by changing the sense of the inequalities below.

In short, the direction of the inequality you need for your rejection region is dependent on the direction of the MLR. Unfortunately, the theorem in Casella and Berger doesn't directly state this.
