Variances of the Foster–Stuart round-trip test statistics

The Foster–Stuart test is often used in the analysis of time series to determine trends. The test is based on calculating the lower and upper time series records x 1 , …, x n . Unlike other tests of randomness, tests based on records are not invariant under a reversal of the direction of the time variable. To construct invariant round-trip tests, it is necessary to count the records in both forward and backward time variables. Thus far, the construction of this test has been impossible because the variances of test statistics were not known when the null hypothesis was true. Variances for Foster–Stuart round-trip test statistics D and S have been found in the present paper. The test was designed to identify positive and negative trends in means and variances of x 1 , …, x n time series. Dispersions of S and D test statistics have been found under the H 0 randomness null hypothesis. Asymptotic approximations were obtained for dispersions. Thus, it has turned out to be possible to construct a full-fledged invariant test for two-sided alternatives. The Foster–Stuart round-trip test application example has been reviewed.