Estimation of the dominating frequency for stationary and nonstationary fractional autoregressive models

This paper was motivated by the investigation of certain physiological series for premature infants. The question was whether the series exhibit periodic fluctuations with a certain dominating period. The observed series are nonstationary and/or have long-range dependence. The assumed model is a Gaussian process X-t whose mth difference Y-t = (1 - B)X-m(t) is stationary with a spectral density f that may have a pole (or a zero) at the origin. the problem addressed in this paper is the estimation of the frequency omega(max) where f achieves the largest local maximum in the open interval (0, pi). The process X-t is assumed to belong to a class of parametric models, characterized by a parameter vector theta, defined in Beran (1995). An estimator of omega(max) is proposed and its asymptotic distribution is derived, with theta being estimated by maximum likelihood. In particular, m and a fractional differencing parameter that models long memory are estimated from the data. Model choice is also incorporated. Thus, within the proposed framework, a data driven procedure is obtained that can be applied in situations where the primary interest is in estimating a dominating frequency. A simulation study illustrates the finite sample properties of the method. In particular, for short series, estimation of omega(max) is difficult, if the local maximum occurs close to the origin. The results are illustrated by two of the data examples that motivated this research.