= Most powerful test for a Laplace location shift
{title2=$X>c_\alpha$}
For two unit-scale <Laplace distributions> with locations $\theta_0<\theta_1$, the <likelihood ratio> is nondecreasing in $X$, with constant tails and an increasing middle interval. Therefore an upper-tail <statistical test> of exact <size of a statistical test> $\alpha$ is a <most powerful test>. Its cutoff is $c_\alpha=\theta_0-\log(2\alpha)$ for $\alpha\leq1/2$, and $c_\alpha=\theta_0+\log(2(1-\alpha))$ otherwise. Under location $\theta_1$, its <statistical power> is $\tfrac12e^{\theta_1-c_\alpha}$ if $c_\alpha\geq\theta_1$, and $1-\tfrac12e^{c_\alpha-\theta_1}$ otherwise. Constant <likelihood ratio> tails can make the <most powerful test> nonunique.
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