摘要
Let x<sub>1</sub>,…, x<sub>n</sub> 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<sub>n</sub>=(x<sub>1</sub>,…, x<sub>k</sub>, r（x）),where x<sub>i</sub>（i=1,…, n） is the order statistic of x<sub>i</sub>, i=1,…, n and k=k（x） is the num-ber of the observed data. Suppose that the distribution family of x<sub>i</sub>, i=1,…, n is i.i.d.exponential with life expectation θ】0. For testing the hypothesis H<sub>0</sub>: θ≤θ<sub>0</sub> againstH<sub>1</sub>: θ】θ<sub>0</sub>, we use the total time of experiment S<sub>r</sub>=sub from i=1 to k x<sub>i</sub> +（n-k）r （x） as the teststatistic. We reject H<sub>0</sub> as large value of S<sub>r</sub> 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 r<sub>0</sub> is the one which has the minimnm total experiment time.
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 timet=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 θ>0. For testing the hypothesis H_0: θ≤θ_0 againstH_1: θ>θ_0, 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 H_0 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 r_0 is the one which has the minimnm total experiment time.
基金
Supported by the NSFC and the Doctoral Programm Foundation of China.
