SPECTRAL DENSITY ESTIMATION FROM NONLINEARLY OBSERVED DATA

Abstract. The estimation of the spectral density function of a stationary Gaussian process at the input of an instantaneous nonlinearity is considered when the nonlinearity is known and a finite set of observations of the output process is given. A class of spectral estimates is considered and their quadratic-mean consistency is established; precise asymptotic expressions for their bias and covariance are derived and their asymptotic normality is obtained.     

NEAREST-NEIGHBOUR METHODS FOR TIME SERIES ANALYSIS

Abstract. The nearest-neighbour method, because of its intuitively appealing nature and competitive theoretical properties, deserves consideration in time-series applications akin to attention it has received lately in the i.i.d. case. Here it is shown that as a nonparametric regression device, like the kernel method, under the G ₂ mixing assumption, it converges in quadratic mean at the Stone-optimal rate. In the closing sections, our methodology is extended to a broader pattern-recognition context, and applied to hydrologic data.     

SPECTRAL ESTIMATION AND DECONVOLUTION FOR A LINEAR TIME SERIES MODEL

Abstract. A vector linear time series model is observed as the sum of a convolution of an unknown signal and an additive noise process. The main objective is the estimation or deconvolution of the signal when the spectra of the signal and noise processes are unknown. We prove the strong consistency of a class of nonparametric spectral estimators derived by maximizing a particular Gaussian likelihood function. We also study the mean square convergence of the finite-sample deconvolution estimators as a function of the sample length T , the filter length M and the spectral bandwidth B_T = L_T/T .     

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

ON THE SELECTION OF RANDOM SAMPLING SCHEMES FOR THE SPECTRAL ESTIMATION OF CONTINUOUS TIME PROCESSES

Abstract. Let X={X(t), -∝<t<∝} be a continuous time stationary process with spectral density φ_x(Λ) and {φ_k} be a stationary point process independent of X. The problem of selecting the sampling point process {τ_k} for the estimation of φ_x(Λ) based on the discrete time observations {X(τ_k), τ_k} is considered. Sufficient conditions are established for the admittability of a broad class of delayed renewal point processes. Examples are given for delayed renewal point processes {τ_k} which have a mixture of Gamma densities for their inter-arrival times. Guidelines are given for the selection of specific point processes in the estimation of broadband and narrowband spectral density functions.