Modern Stochastics: Theory and Applications

A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.

The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of X, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in A is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of X.

We obtain limit theorems for extreme residuals in linear regression model in the case of minimax estimation of parameters.

We consider the Berkson model of logistic regression with Gaussian and homoscedastic error in regressor. The measurement error variance can be either known or unknown. We deal with both functional and structural cases. Sufficient conditions for identifiability of regression coefficients are presented.