Comparing autocorrelation structures of multiple time series via the maximum distance between two groups of time series

In this paper, we propose some tests in the time domain to check the equality of autocorrelation structure of multiple (M > 2) independent stationary time series. To develop the tests, multiple time series are divided into two groups in different ways. Given that k time series are in one group and M - k time series are in the other group, test statistics T-M,T- k, k = 1, 2,..., [M/2] are the maximum Mahalanobis difference between 2 groups of time series over all possible ways of such grouping. The asymptotic distributions under the null are derived. An extensive simulation is conducted to check the finite sample properties of the test statistics. Suggestion is given on the selection of test statistics under different situations based on the simulation. The paper provides some options to compare multiple time series when the length of time series is relatively short and M is relatively large. An application of the tests on vibrational data analysis is discussed.