Keh-Shin Lii

Spectral estimation of continuous-time stationary processes from random sampling

Abstract Let X = {X(t), − ∞ φ X (λ) of φX(λ) based on the discrete-time observation {X(τk), τk} are considered. Asymptotic expressions for the bias and covariance of φ X (λ) are derived. A multivariate central limit theorem is established for the spectral estimators φ X (λ) . Under mild conditions, it is shown that the bias is independent of the statistics of the sampling point process {τk} and that there exist sampling point processes such that the asymptotic variance is uniformly smaller than that of a Poisson sampling scheme for all spectral densities φX(λ) and all frequencies λ.

An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes

An approximate maximum likelihood procedure is proposed for the estimation of parameters in possibly nonminimum phase (noninvertible) moving average processes driven by independent and identically distributed non-Gaussian noise. Under appropriate conditions, parameter estimates that are solutions of likelihood-like equations are consistent and are asymptotically normal. A simulation study for MA(2) processes illustrates the estimation procedure.

Identification and estimation of non-Gaussian ARMA processes

A method to identify and estimate non-Gaussian autoregressive moving average (ARMA) processes which uses bispectral analysis and the Pade approximation is presented. It is shown that the method will consistently identify the order of the ARMA model and estimate the parameters of the model. Various asymptotic distributions are given to facilitate the model identification and parameter estimation. A few examples are presented to illustrate the effectiveness of the method. The procedure is modified to handle the case when there is additive Gaussian noise. The modified procedure is asymptotically consistent in the estimation of orders and parameters of the ARMA model when Gaussian noise is present. >

Asymptotic normality of cumulant spectral estimates

The paper considers higher-order cumulant spectral estimates obtained by directly Fourier transforming weighted cumulant estimates. Such estimates computationally are different from those based on the finite Fourier transform. These estimates can be looked at continuously as well as directly on submanifolds. The estimates of cumulants are based on unbiased moment estimates. Asymptotic normality is obtained for these estimates and is based on a strong mixing condition and only a finite number of cumulant summability conditions.

Bispectra and Energy Transfer in Grid-Generated Turbulence

Publisher Summary Spectral energy transfer has been the subject of turbulence research for many years. Typically, a hypothesis relating the net energy transfer spectrum T(k) to the energy spectrum E ( k ) has been proposed by appeal to intuitive physical arguments, which is then solved for the spectrum E ( k , t ). Numerous transfer functions T(k ) have been postulated, and a survey of many of these net energy transfer relations is given by Monin and Yaglom (1975) or by Hinze (1975). This chapter describes the relationship between the measurable bispectra and the one-dimensional energy transfer terms. The chapter presents arguments to show which bispectra may be expected to vanish in the presence of an isotropic turbulent flow field. It also highlights the computational symmetries for the relevant bispectra and cross bispectra.

Nonminimum phase non-Gaussian deconvolution

A procedure for deconvolution of nonminimum phase non-Guassian time series based on the estimation of higher order (greater than two) spectra is given. This can be applied to the analysis of seismograms. The procedure allows estimation of the wavelet. Knowledge of cumulant spectra of order greater than two allows estimation of the phase of the wavelet. In this way one has access to information not available in the ordinary second-order deconvolution procedures. Computational details of the method for estimating the phase of the wavelet are given. There are simulated illustrative examples. One of the examples is based on an actual reflectivity series from a sonic well log. The method is effective asymptotically in the nonminimum phase non-Gaussian context where the Wiener-Levinson procedure does not apply.

Estimation and deconvolution when the transfer function has zeros

The problem of estimation of the transfer function and deconvolution of a linear process is considered. This paper specifically deals with the case when the transfer function has zeros on the unit circle or equivalently the spectral density function has zeros. It is shown that if the zeros are finitely many and are of finite order then we can still consistently estimate the transfer function without the minimum phase assumption when the process is non-Gaussian. Statistical properties of the estimate are given. Convergence of the deconvolution is also given. It is shown that if the transfer function vanishes on an interval, then, essentially, we cannot identify the transfer function. Two simple simulated examples are given to illustrate the procedures.

Nonlinear systems and higher-order statistics with applications

Abstract Linear processes have been used extensively to model various systems. Most of the theories are developed under the Gaussian assumption. With the rapid availability of vast and inexpensive computing power, models which are non-Gaussian, nonlinear, or even nonstationary are being investigated with increasing intensity. Statistical tools used in such investigations usually involve higher-order statistics in addition to the usual first- and second-order statistics which characterize a Gaussian process. In this paper we illustrate the various ways higher-order statistics are used in the analysis of various nonlinear models. When the process under study is known to obey a set of nonlinear equations then the equations can be manipulated to yield relationships which give insight of the process with physical interpretations. The Navier-Stokes equation is used to illustrate the nonlinear interaction of different wave numbers in a homogeneous velocity field. When modeling a system with unknown structure it is necessary to decide whether linear or nonlinear models should be used. Bispectral density function is often used to test the linearity. A hybrid method is used to illustrate the estimation of bispectral density function of a continuous-time stationary process sampled by a random point process. When there are competing models to be used to model a process it is necessary to decide which model is most compatible with the available data. A set of specific nonlinear models are used to demonstrate the use of higher-order statistics to discriminate among competing linear and nonlinear models.

Model fitting for continuous-time stationary processes from discrete-time data

Abstract Let X = { X ( t ), −∞ t X (λ; θ), where θ is a vector of unknown parameters. Let { τ k } be a stationary point process on the real line which is independent of X . The identifiability and the estimation of θ from the discrete-time observation { X ( τ k ), τ k } are considered. The consistency of appropriate estimates θ T as the time T a ∞ is extablished and a central limit theorem for θ T is given.

Wavelet spectral density estimation under irregular sampling

It has become increasingly accepted that wavelet based estimation techniques are generally better adapted to function estimates having large variations or, for want of a better term, roughness. We consider a class of nonlinear wavelet estimators for the spectral density function of a zero-mean, stationary, not necessarily Gaussian continuous-time stochastic process, which is sampled at irregularly spaced intervals. A stationary point process is used to model the sampling method. We investigate the bias as well as covariance properties of these alias-free estimators. Simulation examples are presented to illustrate the salient features of this procedure.