9. Double-exponential distribution. Let Xl" ' " X; be a sample from the double-exponential distribution with density }e- 1x - O/. The LMP test for testing 8 S 0 against 8 > 0 is the sign test, provided the level is of the form 1 m a=-2:(n) 2n k k' -0 so that the level-a 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 S k < I < n, and let Sk' S, be subsets of Rk, R, such that PO(Sk) = Po(S,) "* O. Then it follows from a consideration of Po (Sk) and Po (S,) for small 8 that there exists t:. such that Po(Sd < Po(S,) for 0 < 8 < t:.. Suppose now that the rejection region of a nonrandomized test of 8 = 0 against 8 > 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 S, with k < I, and each of which therefore increases the power for 8 sufficiently small.]