Estimation of the dominating frequency for stationary and nonstationary fractional autoregressive models (Q2742778)

Let \(X_t\) be a Gaussian process whose m-th difference \(Y_t=(1-B)^mX(t)\) is stationary for a suitably chosen \(m\in \{0,1,2,\dots\}\), where \(B\) is the backshift operator defined by \(B^k X_t=X_{t-k}\). The main problem considered in this paper is the estimation of the frequency \(\omega_{\max}\) where the spectral density \(f\) of the stochastic process \(Y_t\) achieves the largest local maximum in the open interval \((0,\pi)\) in the case when \(m\) and \(f\) may be infinite or zero at the origin. The process \(X_t\) is assumed to belong to a class of parametric models, characterized by a parameter vector \(\omega_{\max}\) introduced by \textit{J. Beran} [J. R. Stat. Soc., Ser. B 57, No. 4, 659-672 (1995; Zbl 0827.62088)].NEWLINENEWLINENEWLINEAn estimator of \(\omega_{\max}\) is proposed and its asymptotic distribution is derived with \(\theta\) being estimated by maximum likelihood. In particular, \(m\) and the fractional differencing parameter that models long memory are estimated from the data. A simulation study illustrates good agreement between asymptotic results and finite sample behavior. For short series the simulation also demonstrates the difficulty of estimating of \(\omega_{\max}\) if it is relatively close to the origin. Finally, the proposed method is applied to medical examples that motivated this research.