On the Independence of Tests of Randomness and Other Hypotheses
A common problem in statistical inference is "Do the observations xi, xn come from a population with a prescribed mean value?" When the observations are normally distributed, a solution to this problem is to use the "t" test. Notice that it is not assumed that the observations represent independently and identically distributed random variables. In this paper it will be pointed out that for random samples, i.e., independently and identically distributed random variables, many of the non-parametric tests of randomness are independent of "symmetric" tests of hypotheses. This result can be used to test both the "random" and "parametric" parts of a hypothesis with procedures having known significant levels. As an application one might wish to test the hypothesis that x(t) is an observation on a Wiener Process [1]. If the null hypothesis is true, then the quantities x(ia) -x((i-1)5) (i-1, * * , N) form a random sample from a normal distribution with mean zero and variance proportional to S. To test this hypothesis one must test for both randomness and normality. The test for randomness would depend on the alternatives of interest; perhaps rank correlation [2] would be found suitable. The test of normality could be performed using the classical chi-square goodness-of-fit test with one parameter estimated. If this were the test program, the rank correlation test and the chi-square test would be independent under the null hypothesis.