Asymptotic Relative Efficiencies of Distribution-Free Tests of Randomness Against Normal Alternatives

S EVERAL writers, notably Hotelling and Pabst 5, have explicitly assumed that the relative efficiency of two test statistics is to be measured by their estimating efficiencies. While this seems reasonable, it is by no means obvious, since if the two tests are consistent, the ratio of their powers against any fixed alternative hypothesis must tend to unity with increasing sample size n, and it may easily be shown that for any n, the less efficient estimator may provide a more powerful test (Sundrum 14). Pitman 11 has proposed a measure of the asymptotic relative efficiency of consistent tests. Given that the two statistics, t1 and t2, have normal limit distributions with variances of order n-1, and that certain general regularity conditions are satisfied, he considered a limiting process in which the alternative hypothesis Hi differs from the null hypothesis Ho by a quantity of order n-12, so that as n increases, Hi tends to Ho. Under these conditions, he showed that the reciprocal of the ratio of sample sizes required to attain equal power against the same alternative was, in the limit,