On Chernoff-Savage Statistics and Sequential Rank Tests

Abstract

In this paper, we shall represent a generalized Chernoff-Savage statistic as the sum of i.i.d. random variables plus a remainder term and analyze the order of magnitude of the remainder term. While Chernoff and Savage have proved that the remainder term, when suitably normalized, converges to O in probability, we obtain a stronger form of convergence in this paper. Our result gives an invariance principle and a law of the iterated logarithm for generalized Chernoff-Savage statistics. We also use our result to obtain asymptotic approximations for the stopping rules of certain sequential rank tests.