- We describe here a methodology that applies to any statistical test, and illustrated in the context of assessing independence between successive observations in a data set.
- Our test was first discussed in section 2.3 of a previous article entitled New Tests of Randomness and Independence for Sequences of Observations, and available here.
- Finally, rather than testing for independence in successive observations (say, a time series) one can look at the square of the observed autocorrelations of lag-1, lag-2 and so on, up to lag-k (say k = 10).
- The absence of autocorrelations does not imply independence, but this test is easier to perform than a full independence test.
- We want to test whether successive observations are independent or not, that is, whether x1, x2, …, xn-1 and x2, x3, …, xn are independent or not.
- The idea behind the test is intuitive: if q(α, β) is statistically different from zero for one or more of the randomly chosen (α, β)’s, then successive observations can not possibly be independent, in other words, xk and xk+1 are not independent, much less correlated.
- It applies to any statistical test of hypotheses, not just for testing independence.
- It is thus easy to test for independence and to benchmark various statistical tests: the larger b, the closer we are to serial, pairwise independence.
- With a pseudo-random number generator, one can generate a time series consisting of independently and identically distributed deviates, with a uniform distribution on [0, 1], to check the distribution of S (or T) and its expectation, in case of true independence, and compare it with values of S (or T) computed on the artificial data, using various values of b.

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