= Uniformly most powerful test for an exponential rate
For independent samples from an <exponential distribution>, a test of rate at most $\lambda_0$ against rate greater than $\lambda_0$ rejects for small sample sums. All simple-alternative <likelihood ratios> against $\lambda_0$ decrease with that sum; its <gamma distribution> calibrates the size at $\lambda_0$. Monotonicity of the rejection probability in the rate proves that this is a <uniformly most powerful test> for the composite hypotheses.
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