Elias Masry

Nonparametric Estimation and Identification of Nonlinear ARCH Time Series Strong Convergence and Asymptotic Normality: Strong Convergence and Asymptotic Normality

We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.

Multivariate regression estimation local polynomial fitting for time series

We consider the estimation of the multivariate regression function m(x1, ..., xd) = E[[psi](Yd)X1 = x1, ..., Xd = xd], and its partial derivatives, for stationary random processes Yi, Xi using local higher-order polynomial fitting. Particular cases of [psi] yield estimation of the conditional mean, conditional moments and conditional distributions. Joint asymptotic normality is established for estimates of the regression function and its partial derivatives for strongly mixing and [varrho]-mixing processes. Expressions for the bias and variance/covariance matrix (of the asymptotically normal distribution) for these estimators are given.

The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion

The wavelet transform of random processes with wide-sense stationary increments is shown to be a wide-sense stationary process whose correlation function and spectral distribution are determined. The second-order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained. Applications to the spectral analysis and to the synthesis of fractional Brownian motion are given. >

Recursive probability density estimation for weakly dependent stationary processes

Recursive estimation of the univariate probability density function f(x) for stationary processes \{X_{j}\} is considered. Quadratic-mean convergence and asymptotic normality for density estimators f_{n}(x) are established for strong mixing and for asymptotically uncorrelated processes \{X_{j}\} . Recent results for nonrecursive density estimators are extended to the recursive case.

Alias-free sampling: An alternative conceptualization and its applications

A new concept of alias-free sampling of continuous-time processes X = \{ X(t), -\infty is introduced. The new concept is shown to be distinct from the traditional concept [1]-[3]. various criteria for a given sampling scheme \{t_{n}\} to be alias-free in the new sense are developed. The relationship of the new definition to the question of estimating the spectral density function \Phi(\lambda) of the continuous-time process X from its samples \{X(t_{t})\} is discussed.

Closed-Form Analytical Results for the Rejection of Narrow-Band Interference in PN Spread-Spectrum Systems--Part II: Linear Interpolation Filters

The performance of PN spread-spectrum communication systems in the presence of narrow-band interference is studied when linear interpolation filters are employed for the estimation and subsequent suppression of the interference. Closed-form analytical expressions for the systems performance are established for a broad class of interference processes. The advantages of linear interpolation filters over predictive filters with identical number of taps are examined analytically and some unexpected results are obtained. The analytical results are illustrated by examples.

Probability density estimation from sampled data

For broad classes of deterministic and random sampling schemes \{t_{k}\} we establish the consistency and asymptotic expressions for the bias and covariance of discrete-time estimates f_{n}(x) for the marginal probability density function f(x) of continuous-time processes X(t) . The effect of the sampling scheme and the sampling rate on the performance of the estimates is studied. The results are established for continuous-time processes X(t) satisfying various asymptotic independence-uncorrelatedness conditions.

Additive nonlinear ARX time series and projection estimates

We propose projections as means of identifying and estimating the components (endogenous and exogenous) of an additive nonlinear ARX model. The estimates are nonparametric in nature and involve averaging of kernel-type estimates. Such estimates have recently been treated informally in a univariate time series situation. Here we extend the scope to nonlinear ARX models and present a rigorous theory, including the derivation of asymptotic normality for the projection estimates under a precise set of regularity conditions.

Poisson sampling and spectral estimation of continuous-time processes

A class of spectral estimates of continuous-time stationary stochastic processes X(t) from a finite number of observations \{X(t_{n})\}^{N}_{n}=l taken at Poisson sampling instants \{t_{n}\} is considered. The asymptotic bias and covariance of the estimates are derived, and the influence of the spectral windows and the sampling rate on the performance of the estimates is discussed. The estimates are shown to be consistent under mild smoothness conditions on the spectral density. Comparison is made with a related class of spectral estimates suggested in [15] where the number of observations is {\em random}. It is shown that the periodograms of the two classes have distinct statistics.

Memory-universal prediction of stationary random processes

We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process (Xi)/sub i=-/spl infin///sup /spl infin//. The best mean-square predictor of X/sub 0/ is its conditional mean given the entire infinite past (X/sub i/)/sub i=-/spl infin///sup -1/. Given a sequence of observations X/sub 1/, X/sub 2/, X/sub N/, we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension, for example, neural networks and Legendre polynomials. The proposed estimators select both the model memory and the model dimension, in a data-driven fashion, by minimizing certain complexity regularized least squares criteria. When the underlying predictor function has a finite memory, we establish that the proposed estimators are memory-universal: the proposed estimators, which do not know the true memory, deliver the same statistical performance (rates of integrated mean-squared error) as that delivered by estimators that know the true memory. Furthermore, when the underlying predictor function does not have a finite memory, we establish that the estimator based on Legendre polynomials is consistent.