Nonparametric time series analysis for periodically correlated processes

Abstract

Correlation functions of continuous-time periodically correlated processes can be represented by a Fourier series with coefficient functions. It is shown that the usual estimator for stationary covariances, formed from a single sample path of the process, can be simply modified to provide a consistent (in quadratic mean) estimator for any of the coefficient functions resulting from the aforementioned representation. It is shown that, if the process is Gaussian and B/sub k/( tau ) is a Fourier integral with respect to a density function g/sub k/( lambda ), a two-dimensional periodogram, formed from a single sample function, can be smoothed along a line of constant difference frequency to provide a consistent estimator for g/sub k/( lambda ). This natural extension of the well-known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>