Dispersion of the test statistic of round-trip criteria Foster—Stuart

Abstract

The criterion Foster—Stuart in the analysis of time series to determine trends is often used. The criterion is based on the calculation of the lower and upper time series records x1, . . . ,xn. 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 is necessary to count the records in both forward and backward time variable. Thus far the construction of such test was impossible. When the null hypothesis is true the variances of test statistics were not known. Foster—Stewart round-trip criteria D and S dispersion statistics has been found in this article. This criteria was designed for x1, . . . ,xn time series mean and dispersion positive and negative trend receiving. Assuming H0 randomness null-hypothesis criteria variances statistics S and D was found. Asymptotic approximations was obtained for dispersions. This is possible to construct an invariant test for two-sided alternatives. Foster—Stewart round-trip criteria application example is reviewed. Refs 8. Tables 3.