Gamma two-sample problem. Let X1,... Xm; Y1,..., Yn be independent samples from gamma distributions (g1, b1), (g2, b2), respectively.

(i) If g1, g2 are known, there exists a UMP unbiased test of H : b2 = b1 against one- and two-sided alternatives, which can be based on a beta distribution.

[Some applications and generalizations are discussed in Lentner and Buehler

(1963).]

(ii) If g1, g2 are unknown, show that a UMP unbiased test of H continues to exist, and describe its general form.

(iii) If b2 = b1 = b (unknown), there exists a UMP unbiased test of g2 = g1 against one- and two-sided alternatives; describe its general form.

[(i): If Yi(i = 1, 2) are independent (gi, b), then Y1 + Y2 is (g1 + g2,

b) and Y1/(Y1 + Y2) has a beta distribution.]