IMPROVEMENT ON THE TEST IN THE EXPONENTIAL FAMILY

摘要

Let x₁,…, x_n be the life spans of n items. Suppose that we start the experi-ment at the time t=0 for all the items simultaneuously aud stop the experiment at the timet=r, where r is a stopping time. The observed data set is Z_n=(x₁,…, x_k, r（x）),where x_i（i=1,…, n） is the order statistic of x_i, i=1,…, n and k=k（x） is the num-ber of the observed data. Suppose that the distribution family of x_i, i=1,…, n is i.i.d.exponential with life expectation θ0. For testing the hypothesis H₀: θ≤θ₀ againstH₁: θθ₀, we use the total time of experiment S_r=sub from i=1 to k x_i +（n-k）r （x） as the teststatistic. We reject H₀ as large value of S_r is observed. In this paper for a given stoppingtime r we construct a stopping time ro so that the resulting test is a best improvement ofthe one, which is based on r, in the sense that the power functions of the two tests are thesame but the test based on r₀ is the one which has the minimnm total experiment time.