Shape parameter of a gamma distribution. Let X1,..., Xn be a sample from the gamma distribution (g,

b) defined in Problem 3.36.

(i) There exist UMP unbiased tests of H : g ≤ g0 against g > g0 and of H : g = g0 against g = g0, and their rejection regions are based on W = /(Xi/X¯).

(ii) There exist uniformly most accurate confidence intervals for g based on W.

[Shorack (1972).]

Notes.

(1) The null distribution of W is discussed in Bain and Engelhardt (1975), Glaser

(1976), and Engelhardt and Bain (1978).

(2) For g = 1, (g,

b) reduces to an exponential distribution, and (i) becomes the UMP unbiased test for testing that a distribution is exponential against the alternative that it is gamma with g > 1 or with g = 1.
(3) An alternative treatment of this and some of the following problems is given by Bar-Lev and Reiser (1982).