Estimation of the population spectrum with replicated time series

Abstract

A random sample of r objects from a certain population is considered, measuring for each of them and at the same time points a stationary process Xi(t) whose spectral distribution is absolutely continuous. Each spectral density function fi(ω) may be considered as a realization of a stationary process R(ω), for which a population spectrum f(ω) is defined as E[R(ω)]. Thus, r time series are available to estimate the population spectrum f(ω). It is well known that when a single time series is analysed, the periodogram is a poor estimate of the spectral density. However, when using replicated time series, the average periodogram behaves adequately as an estimate of the population spectrum if the number of objects is large. The asymptotic properties of the average periodogram are analysed and confidence intervals for the population spectrum are constructed. An alternative bootstrap method is proposed for the estimation of the population spectrum and the asymptotic validity, in the sense of Bickel and Freedman, is proved when the number of objects is large. Replicated time series simulated from a moving average process with random coefficients and confidence intervals are constructed for the population spectrum using the bootstrap approach. These intervals are compared with the intervals obtained by means of the asymptotic properties of the average periodogram.