A Note on the Test for the Location Parameter of an Exponential Distribution
BY KEI TAKEUCHI New York University and University of Tokyo Suppose that X is distributed according to an exponential distribution with density (1) f(x) = 0-lexp [-0-'(x )] if x > y, = 0 otherwise where both 0 and y are unknown parameters. Let Y be another positive random variable independent of X and distributed according to a continuous distribution with scale parameter 0, and density g(y/0)/0, where g is a known function. Consider the hypothesis 7 = 7o, and let the procedure be such that the hypothesis is rejected if and only if (2) (X-yo)/Y ? a or (Xyo)/Y > b where 0 < a < b < oo . Since (X yo)/Y is independent of 0 under the hypothesis, a and b can be determined so that Pr {a < (X-7yo)/Y < b lI7o} = 1-(a for all 0. Then the following theorem holds true. THEOREM 1. For the alternative y < 7yo, the power of the test (2) above is given by (3) P(y) = 1 (1 a) exp [-0-(7o 7)] i.e. it is independent of the distribution of Y, and also of a or b. PROOF.