= Uniformly most powerful test for a gamma rate
{title2=$\pi(\theta)=G_{n\lambda}(\theta c_\alpha)$}
For an independent <statistical sample> from a <gamma distribution> with known shape $\lambda>0$ and unknown rate $\theta>0$, test $\theta=\theta_0$ against $\theta>\theta_0$. The alternative-to-null <likelihood ratio> for any larger rate $\theta_1$ is $(\theta_1/\theta_0)^{n\lambda}\exp\{-(\theta_1-\theta_0)S\}$, where $S=\sum_iX_i$. This decreases in $S$, so the <Neyman-Pearson lemma> gives one common lower-tail ordering for all alternatives.
Let $G_a$ be the unit-rate <gamma distribution> <cumulative distribution function> with shape $a$. The <uniformly most powerful test> of exact <test size> $\alpha\in(0,1)$ rejects when $S<c_\alpha$, where $c_\alpha=G_{n\lambda}^{-1}(\alpha)/\theta_0$. Its <power function> is displayed above. If $n\lambda=1$, the sum has an <exponential distribution>, so $c_\alpha=-\log(1-\alpha)/\theta_0$ and $\pi(\theta)=1-(1-\alpha)^{\theta/\theta_0}$. Monotonicity of $G_a(\theta c_\alpha)$ also makes this test valid and uniformly most powerful for the larger <null hypothesis> $\theta\leq\theta_0$.
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