Double-exponential distribution. Let X1, …, Xn be a sample from the double-exponential distribution with density 1 2 e−|x−θ|

. The LMP test for testing

θ ≤ 0 against θ > 0 is the sign test, provided the level is of the form

α = 1 2n m

k=0



n k



, so that the level-α sign test is nonrandomized.

[Let Rk (k = 0,..., n) be the subset of the sample space in which k of the X’s are positive and n − k are negative. Let 0 ≤ k < l < n, and let Sk , Sl be subsets of Rk , Rl such that P0(Sk ) = P0(Sl) = 0. Then it follows from a consideration of Pθ (Sk )

and P0(Sl) for small θ that there exists  such that Pθ (Sk ) < Pθ (Sl) for 0 <θ<.

Suppose now that the rejection region of a nonrandomized test of θ = 0 against θ > 0 does not consist of the upper tail of a sign test. Then it can be converted into a sign test of the same size by a finite number of steps, each of which consists in replacing an Sk by an Sl with k < l, and each of which therefore increases the power for θ

sufficiently small.]