Spectral analysis with replicated time series

Abstract

A doubly stochastic process {x(b,t);b∊B,t∊Z} is considered, with (B,β,Pβ) being a probability space so that for each b, {X(b,t);t ∊ Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = EB[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b1,…,br from (B,β,Pβ). For each b1∊ B, the processes X(b1,t) are observed at the same times t=1,…,N. Thus, r time series (x(b1,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.