A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

In many cases an optimum or computationally convenient test of a simple hypothesis H_0against a simple alternative H_1may be given in the following form. Reject H_0if S_n = \sum^n_{j=1} X_j \leqq k,where X_1, X_2, \cdots, X_nare nindependent observations of a chance variable Xwhose distribution depends on the true hypothesis and where kis some appropriate number. In particular the likelihood ratio test for fixed sample...