On the Power Function of Tests of Randomness Based on Runs up and Down

Abstract

It is shown that various statistics based on the number of runs up and down have an asymptotic multivariate normal distribution under a number of different alternatives to randomness. The concept of likelihood ratio statistics is extended to give a method for deciding what function of these runs should be used, and it is shown that the asymptotic power of these tests depends only on the covariance matrix, calculated under the hypothesis of randomness, and the expected values, calculated under the alternative hypothesis. A general method is given for calculating these expected values when the observations are independent, and these calculations are carried through for a constant shift in location from one observation to the next and for normal and rectangular populations.