Asymptotic Power of Certain Test Criteria (Based on First and Second Differences) for Serial Correlation between Successive Observations
Von Neumann [12] proposed the ratio' w2 = 62/82 for testing the randomness of the sequence against the alternatives of serial correlation or trend. W = d/s, which has the advantage of being simpler for computation, was suggested by Kamat [8] for the same purpose. Similarly, we can construct two more ratio criteria, W2 = 62/s2 and W2 = d2/s, by using the second order differences. And, finally, using both first and second successive differences, it is possible to construct a third type of ratio criteria, u2 = a2/62 and U = d2/d, which may also be used for detecting serial correlation or trend in successive observations of the sequence. (See Tintner [11].) Under the hypothesis of randomness, when the xi are independent normal (,u~, o), the distributions of the ratio statistics w2, w2, W and W2 are known, either in exact or approximate form (Von Neumann [12], Kamat [8] and [9].) Under the same hypothesis, Kamat [9] has shown that they are all asymptotically normal. (See also Anderson [1], Hsu [6], and Dixon [3].) The distributions, however, do not appear to have been discussed under any alternative hypotheses of non-randomness.