On the Independence of Tests of Randomness and Other Hypotheses
I. Richard Savage · Journal of the American Statistical Association · 1957
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 variables. In this paper it will be pointed out that for samples, i.e., independently and identically distributed 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 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.