IMPROVEMENT ON THE TEST IN THE EXPONENTIAL FAMILY
Zheng Zhongguo · 1993
Let x_1,…, 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 time t=r, where r is a stopping time. The observed data set is Z_n=(x_(1),…, 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 H_0: θ≤θ_0 against H_1: θθ_0, we use the total time of experiment S_r=sub from i=1 to k x_(i) +(n-k)r (x) as the test statistic. We reject H_0 as large value of S_r is observed. In this paper for a given stopping time r we construct a stopping time ro so that the resulting test is a best improvement of the one, which is based on r, in the sense that the power functions of the two tests are the same but the test based on r_0 is the one which has the minimnm total experiment time.