A. V. Ivanov

Statistical Analysis of Random Fields

1. Elements of the Theory of Random Fields.- 1.1 Basic concepts and notation.- 1.2 Homogeneous and isotropic random fields.- 1.3 Spectral properties of higher order moments of random fields.- 1.4 Some properties of the uniform distribution.- 1.5 Variances of integrals of random fields.- 1.6 Weak dependence conditions for random fields.- 1.7 A central limit theorem.- 1.8 Moment inequalities.- 1.9 Invariance principle.- 2. Limit Theorems for Functionals of Gaussian Fields.- 2.1 Variances of integrals of local Gaussian functionals.- 2.2 Reduction conditions for strongly dependent random fields.- 2.3 Central limit theorem for non-linear transformations of Gaussian fields.- 2.4 Approximation for distribution of geometric functional of Gaussian fields.- 2.5 Reduction conditions for weighted functionals.- 2.6 Reduction conditions for functionals depending on a parameter.- 2.7 Reduction conditions for measures of excess over a moving level.- 2.8 Reduction conditions for characteristics of the excess over a radial surface.- 2.9 Multiple stochastic integrals.- 2.10 Conditions for attraction of functionals of homogeneous isotropic Gaussian fields to semi-stable processes.- 3. Estimation of Mathematical Expectation.- 3.1 Asymptotic properties of the least squares estimators for linear regression coefficients.- 3.2 Consistency of the least squares estimate under non-linear parametrization.- 3.3 Asymptotic expansion of least squares estimators.- 3.4 Asymptotic normality and convergence of moments for least squares estimators.- 3.5 Consistency of the least moduli estimators.- 3.6 Asymptotic normality of the least moduli estimators.- 4. Estimation of the Correlation Function.- 4.1 Definition of estimators.- 4.2 Consistency.- 4.3 Asymptotic normality.- 4.4 Asymptotic normality. The case of a homogeneous isotropic field.- 4.5 Estimation by means of several independent sample functions.- 4.6 Confidence intervals.- References.- Comments.

Asymptotic theory of nonlinear regression

Introduction. 1. Consistency. 2. Approximation by a Normal Distribution. 3. Asymptotic Expansions Related to the Least Squares Estimator. 4. Geometric Properties of Asymptotic Expansions. Appendix: I: Subsidiary Facts. II: List of Principal Notations. Commentary. Bibliography. Index.

Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra

The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.

Estimation of harmonic component in regression with cyclically dependent errors

This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.

The Berry-Esseen inequality for the distribution of the least square estimate

A nonlinear regression modelxt=gt(θ0)+ εt,t⩾1, is considered. Under a number of conditions on its elements εt and gt(θ0) it is proved that the distribution of the normalized least square estimate of the parameter θ0 converges uniformly on the real axis to the standard normal law at least as quickly as a quantity of the order T−1/2 as T → ∞, where T is the size of the sample, by which the estimate is formed.

Limit theorems for the maximal residuals in linear and nonlinear regression models

Limit theorems for the maximal residuals in linear and nonlinear regression models are obtained in the paper. An application of the main result for constructing a regression model adequacy test is given. Classical results of the regression analysis describe various properties of residual sums of squares errors (see, for example, [1, 2]). In other words, the classical results deal with the properties of the sum of squares of deviations between observations and a regression function where the least squares estimator is substituted in place of the unknown parameter. The current paper deals with the asymptotic behavior of the maximal residual. We prove the convergence of distributions of the normalized appropriately maximal residual to a limit distribution of the normalized maximum as the size of a sample grows to infinity of identically distributed errors of observations. We consider both cases, the linear and nonlinear regression models. We apply the asymptotic results to construct some regression model adequacy test (see Section 3). 1. Linear regression model Consider the following linear regression model

Consistency of the least squares estimator of the amplitudes and angular frequencies of a sum of harmonic oscillations in models with long-range dependence

The strong consistency of least squares estimators of unknown amplitudes and angular frequencies of the sum of harmonic oscillations observed in a strongly dependent Gaussian stationary noise is proved in the paper. 1. Let a stochastic process (1) X(t) = g(t, θ) + ε(t), t ∈ [0, T ], be observed, where g(t, θ) = N ∑ k=1 (Ak cosφ o kt+B o k sinφ o kt) , (2) θ = ( θ 1, θ o 2, θ o 3, . . . , θ o 3N−2, θ o 3N−1, θ o 3N ) = (A1, B o 1 , φ o 1, . . . , A o N , B o N , φ o N ) , (3) (Ak) 2 + (B k) 2 > 0, k = 1, . . . , N ; here ε(t), t ∈ R, is a stochastic process defined on a complete probability space (Ω, ,P) and satisfying the following condition. A1. ε(t), t ∈ R, is a real, measurable, mean square continuous, stationary, Gaussian, zero-mean stochastic process. We also assume that at least one of the following two conditions holds. A2. The correlation function of the process ε(t), t ∈ R, is such that (4) E ε(t)ε(0) = B(t) = L (|t|) |t|α , 0 < α < 1, where L(t) is a function slowly varying at infinity and B(0) = 1. A3. (5) B(t) = cosκt (1 + t2)α/2 , 0 < α < 1, κ ∈ [0,∞). The statistical estimation of unknown amplitudes and angular frequencies (3) of a sum of harmonic oscillations (2) observed in a random noise ε(t) is a probabilistic setting of the problem of detection of hidden periodicities. Investigations of this problem as well as of its deterministic counterpart (ε(t) ≡ 0) are initiated by Lagrange. Many applications of this problem in numerous scientific fields are also known (see [1]). 2000 Mathematics Subject Classification. Primary 62J02; Secondary 62J99.

Asymptotic Expansions Related to the Least Squares Estimator

In this Chapter we find the a.e. of the moments of the l.s.e. ( hat theta _n ) and the a.e. of the distributions of a series of functional of the l.s.e. used in mathematical statistics. In this Chapter the assumptions of Chapter 2 about smoothness of the regression functions g(j, θ) are kept: for each j there exist derivatives with respect to the variables θ = (θ1,...,θ q ) up to some order k ≥ 4 inclusive that are continuous in Θ c , where Θ ⊆ ℝ q is an open convex set, The assumption of Section 10 about the normalisation n 1/21 q instead of d n θ is also used.

Asymptotic Properties of Koenker–Bassett Estimator in Regression Model with Long-Range Dependence

This article deals with asymptotic distribution of Koenker-Bassett estimator in continuous time regression model with long-range dependent noise. It is proved that the asymptotic distribution of the normed estimator coincides with asymptotic distribution of the integral of an indicator random process generated by the random noise weighted by regression function gradient. A theorem is formulated also on asymptotic normality of the last integral.

Geometric Properties of Asymptotic Expansions

The linear theory of estimation by the method of least squares uses the language of algebra and plane geometry. In the non-linear theory planes yield place to surfaces and inference acquires a local character. Therefore the natural geometrical language in non-linear regression analysis is the language of differential geometry and tensor calculus. Nowadays an intensive geometric reinterpretation of the basic concepts of mathematical statistics is made. One of the goals pursued in this consists in the move from geometric invariants of statistical matters, alloted a geometric structure, to invariant statistical inference.