12. Exponential densities. Let XI" ' " x" be a sample from a distribution with exponential density a -I e -(x -hl/u for x b.

(i) For testing a = 1 there exists a UMP unbiased test given by the acceptance region CI s 2E[x; - min(xI""'Xn ) ] s C2 , where the test statistic has a X2-distribution with 2n - 2 degrees of freedom when a = 1, and CI, C2 are determined by lC2X~n_2(Y) dy = lC2X~n(Y) dy = 1 -

a. C1 C1 (ii) For testing b = ° there exists a UMP unbiased test given by the acceptance region nmin(XI" "'Xn ) When b = 0, the test statistic has probability density n-l p( u) = (1 + ur ' [These distributions for varying b do not constitute an exponential family, and Theorem 3 of Chapter 4 is therefore not directly applicable. (i): One can restrict attention to the ordered variables X(I) < . . . < X(nl' since these are sufficient for a and

b, and transform to new variables Z\ = nX(\1' Z; = (n - i + 1)[X(i) - Xli-I)] for i = 2, .. . , n, as in Problem 14 of Chapter 2. When a = 1, 2 1 is a complete sufficient statistic for

b, and the test is therefore obtained by considering the conditional problem given ZI' Since E7_2Z; is independent of ZI' the conditional UMP unbiased test has the acceptance region CI s E7_2Z; s C2 for each ZI, and the result follows. (ii): When b = 0, E7_1 Z; is a complete sufficient statistic for

a, and the test is therefore obtained by considering the conditional problem given E7_lz; , The remainder of the argument uses the fact that Zt/1:7_ IZ; is independent of E;'_\ Z; when b = 0, and otherwise is similar to that used to prove Theorem 1.]