3. UMP test for exponential densities. Let XI' ... ' Xn be a sample from the exponential distribution E(

a, b) of Chapter 1, Problem 18, and let ,\(1) = min( XI, .. . , Xn ) . (i) Determine the UMP test for testing H: a = ao against K : a :I: ao when b is assumed known. (ii) The power of any MP level-a test of H : a = ao against K: a = al < ao is given by /l*(al ) = 1 - (1 - a)e-n(aO-a1)/b. (iii) For the problem of part (i), when b is unknown, the power of any level a test which rejects when X(I) - ao E[ X; - X(1)] s CI or ;;:: C2 against any alternative (ai'

b) with al < ao is equal to fJ*( al) of part (ii) (independent of the particular choice of CI and C2 ) . (iv) The test of part (iii) is a UMP level-a test of H: a = ao against K : a * ao (b unknown). (v) Determine the UMP test for testing H : a = ao, b = bo against the alternatives a < ao, b < boo (vi) Explain the (very unusual) existence in this case of a UMP test in the presence of a nuisance parameter [part (iv)] and for a hypothesis specifying two parameters [part (v)]. [(i): the variables 1'; = e"x.r» are a sample from the uniform distribution on (0,

e- a / b ) .] Note. For more general versions of parts (iiHiv) see Takeuchi (1969) and Kabe and Laurent (1981).