# UMP test for 1-sided hypothesis: Why the focus on the monotonicity in the sufficient statistic, and not in the parameters?

Let's assume that we have $X_1,...,X_n\sim^{iid} Exp(\theta)$.

The LRT is of the form $\Lambda(X,\theta_0,\theta_1)=e^{\left(\frac{1}{\theta_0}-\frac{1}{\theta_0}\right)\sum X_i}$. When $\theta_0<\theta_1$, the LRT is increasing in the sufficient statistic $T(X)=\sum X_i$.

Why the focus on the monotonicity of the LRT w.r.t. $\sum X_i$? Is it because then we can write the test the Neyman-Pearson test as $T(X)\in \text{Rej.Region}_{\alpha}$, and we usually know the distribution of $T(X)$?

I ask this because it would also seem important to focus on $\Lambda(X,\theta,\theta_1)$ being monotone w.r.t $\theta$.

Then testing hypothesis $H_0: \theta=\theta_0 \ vs \ H_1:\theta>\theta_0$, we could also test $H_0: \theta\leq\theta_0 \ vs \ H_1:\theta>\theta_0$, maintaining the size of the test, i.e. for this example the Neyman-Pearson test for the simple hypothesis is reject $H_0$ if $\sum X_i > k_{\alpha}$. Let's call it $\phi_{simple}(X)$. To extend to the composite hypothesis, we notice

$$Pr_{\theta}(\sum X_i > k_{\alpha})=Pr_{\theta}(\Lambda(X,\theta,\theta_1) > K_{\alpha})\leq Pr_{\theta_0}(\Lambda(X,\theta_0,\theta_1) > K_{\alpha})\leq\alpha \quad \forall \theta\leq \theta_0$$ due to the monotonicity of $\Lambda$, we still have $\phi_{simple}(X)$ with same size. Now, by the Neyman-Pearson theorem, under the simple hypothesis, we have that $E_{\theta_1}(\phi(X))\leq E_{\theta_1}(\phi_{simple}(X)) \quad \forall \theta_1> \theta_0$, for tests $\phi(X)$ of size $\alpha$. For the composite hypothesis, $\Theta_1$ set is exactly the same, so our $\phi_{simple}$ should be UMP for the composite hypothesis.

Couldn't we extend this reasoning into other examples, and make it a theorem?

The only practical problem would be to know the distribution of the LRT, I think.

• When it's monotone wrt the sufficient stat, the rejection region is a half of the real line. It's not all split up and weird – Taylor Jun 19 '16 at 1:28
• Your last paragraph is a bit confusing. Are you asking what kind of monotonicity you need in addition to NPL to get UMP? – Taylor Jun 19 '16 at 1:31
• @Taylor I've edited my question. I hope it's simpler. Also, regarding your first comment: You're right, but we can calculate probabilities for very strange types of intervals, so it wouldn't be that problematic to lose that nice half line shape. Do you agree? – An old man in the sea. Jun 19 '16 at 9:39
• @Taylor more problematic may be to find the distribution of certain statistics, right? – An old man in the sea. Jun 19 '16 at 9:49
• But if it's not monotone in the stat, you're not just shifting single rejection region cutoffs, so maybe you could gerrymander something that's more powerful than another thing without using some of these theorems in the composite versus composite case. However it probably wouldn't generalize well and be very problem specific – Taylor Jun 19 '16 at 13:15