Testing discrete-valued time series for whiteness

We consider the problem of testing a univariate discrete-valued time series for whiteness in the sequency domain, using Walsh–Fourier analysis. We show that the distribution of the lag window estimator of the Walsh spectral density is a scaled chi-square distribution, where the scale and degrees of freedom, both depend on the bandwidth of the smoothing window associated with the estimator. The definition of the bandwidth is extended from the frequency to the sequency domain. To address our problem, we propose three tests: the first one is based on the cumulative Walsh periodogram, and is shown to converge to a Brownian bridge. The second test is based on applying the Cramer–von Mises functional to an estimate of the Walsh spectral density, and is shown to converge to a Normal distribution, while the last test is based on a distance to whiteness, and is shown to have an approximate scaled chi-square distribution. Simulations are reported on the performance of the tests. Finally, we apply the proposed tests to the brain functional connectivity of schizophrenic patients.