75. Gamma two-sample problem . Let Xl"'" Xm ; YI , ... , y" be independent samples from gamma distributions I'(g, bl)' r(g2' b2 ) respectively. (i) If gl' g2 are known, there exists a UMP unbiased test of H : b2 = bl 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 gl' g2 are unknown, show that a UMP unbiased test of H continues to exist, and describe its general form. (iii) If b2 = bl .= b (unknown), there exists a UMP unbiased test of g2 = gl against one- and two-sided alternatives; describe its general form. [(i): If Y; (i = 1,2) are independent r(g;, b), then YI + Y2 is r(g, + g2'

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