On confidence bands for time series problems in the time and frequency domains

Abstract

The construction of (asymptotic) simultaneous confidence bands for some time series problems is studied, typically for the sample autocorrelogram and windowed spectral density estimate. The following approaches are explored: (1) To use the close-form results available in the literature; (2) To use the asymptotic independence of the sample quantities to derive new procedures; (3) To resort to inequalities. As expected, the bands turn out being wider than those frequently encountered in the literature, based on point-by-point confidence intervals. Numerical values of the necessary constants are given for selected values of the joint confidence coefficient and various numbers of sample quantities. The use of confidence sets of non-uniform width is also briefly explored. Monte Carlo simulations are presented, for problems in the time and in the frequency domains.