Exponential densities. Let X1,..., Xn be a sample from a distribution with exponential density a−1e−(x−b)/a for x ≥ b.

(i) For testing a = 1 there exists a UMP unbiased test given by the acceptance region C1 ≤ 2

[xi − min(x1,..., xn)] ≤ C2, where the test statistic has a χ2-distribution with 2n − 2 degrees of freedom when α = 1, and C1,C2 are determined by

 C2 C1

χ2 2n−2(y) dy =

 C2 C1

χ2 2n(y) dy = 1 − α.

(ii) For testing b = 0 there exists a UMP unbiased test given by the acceptance region 0 ≤ n min(x1,..., xn)

[xi − min(xi,..., xn)] ≤ C.

When b = 0, the test statistic has probability density p(u) = n − 1

(1 + u)n , u ≥ 0.

[These distributions for varying b do not constitute an exponential family, and Theorem 4.4.1 is therefore not directly applicable. For (i), one can restrict attention to the ordered variables X(1) < ··· < X(n), since these are sufficient for a and

b, and transform to new variables Z1 = nX(1), Zi = (n − i + 1)[X(i) − X(i−1)] for i = 2,..., n, as in Problem 2.15. When a = 1, Z1 is a complete sufficient statistic for

b, and the test is therefore obtained by considering the conditional problem given z1. Since n i=2 Zi is independent of Z1, the conditional UMP unbiased test has the acceptance region C1 ≤ n i=2 Zi ≤ C2 for each z1, and the result follows.
For (ii), when b = 0, n i=1 Zi , is a complete sufficient statistic for

a, and the test is therefore obtained by considering the conditional problem given n i=1 zi . The remainder of the argument uses the fact that Z1/
n i=1 Zi is independent of n i=1 Zi , when b = 0, and otherwise is similar to that used to prove Theorem 5.1.1.]