# The Limit Theorems for Extreme Residuals in Nonlinear Regression Model with Gaussian Stationary Noise

In this paper non-linear regression model with Gaussian stationary random noise and continuous time is considered. The behavior of normalized in some way maximum residuals and maximum of residuals absolute values in which its the least squares estimator is substituted instead of unknown parameter of regression function. The convergence of distribution of these normalized maximum to double exponent law is proved which follows from the assumption of random noise normality.

# Moments Asymptotic Expansion of the Least Squares Estimator of the Vector-Parameter of Nonlinear Regression with Correlated Observations

A nonlinear regression model with continuous time and mean square continuous separable measurable Gaussian stationary random noise with zero mean and integrable covariance function is considered. Parameter estimation in the models of such kind is an important problem of statistics of random processes. In this paper, the first terms of asymptotic expansions of the bias vector and covariance matrix of the least square estimator of nonlinear regression function vector parameter are obtained.

# The Limit Theorems for Extreme Residuals in Linear Regression Model with Gaussian Stationary Noise

We consider linear regression model with continuous time and strongly dependent stationary Gaussian random noise. The behavior of normalized in some way extreme residuals, that are the maximum differences, or their absolute values, between the observations and the values of the regression function where instead of unknown parameter the least squares estimator is substituted. For linear regression model the conditions of weak convergence of normalized extreme residuals to double exponent curve are obtained which follows from the assumption of normality of random noise.

# Periodogram Estimator Properties of the Parameters of the Modulate almost Periodic Signal

The problem of detection of hidden periodicities is considered in the paper. In the capacity of useful signal model the modulated almost periodic signal is taken observed on the background of random noise being the local functional of Gaussian strongly dependent stationary process. For estimation of unknown amplitude and angular frequency of modulated signal periodogram estimators are chosen. Sufficient conditions on consistency and asymptotic normality of the estimators are obtained. The exact form of limiting normal distribution is found.

# Asymptotic Properties of Estimator of Linear Regression Parameter in Case of Long-Range Dependent Regressors

The paper considers linear regression model with long-range/weak dependent random noise and time dependent regressors which are observed with long-range dependent errors. Parameter estimation of these models is an important problem of statistics of random processes. We choose widely used least squares estimator for the estimation. The aim of this work is to prove consistency and asymptotic normality of least squares estimator of the regression model.

# Periodogram Estimator Properties of the Parameters of the Regression Model with Strongly Dependent Noise

The problem of detection of hidden periodicities is considered in the paper. In the capacity of useful signal model the harmonic oscillation observed on the background of random noise, that is a local functional of Gaussian strongly dependent stationary process is taken. For estimation of unknown angular frequency and amplitude of harmonic oscillation periodogram estimator is chosen, for which sufficient conditions of asymptotic normality are obtained and limit normal distribution is found.

# [mu]--admissibility of spectral density of strongly dependent random noise in nonlinear regression models

In this paper, we highlight the obtained sufficient conditions, under which the limiting normal distribution covariance matrix of the least squares estimator of the nonlinear regression model parameter with strongly dependent stationary random noise can be represented as an integral of discontinuous and unbounded spectral density of the noise by the regression function spectral measure.

# The estimator consistency of least squares parameters of a sum of harmonic oscillations in the models with strongly dependent noise

In this paper, we obtain sufficient conditions the estimator weak consistency of least squares amplitudes and angular frequencies of a sum of harmonic oscillations observed on the random noise background. We expect that the noise is a local functional of the Gaussian stationary strongly dependent process.

# Consistency of the least moduli estimator in the nonlinear regression model

In this paper, we obtain the sufficient conditions of the least moduli estimator consistency of a parameter of the nonlinear regression model with continuous time and strong dependent Gaussian stationary noise.

# On the unique M-estimates of nonlinear regression model parameters

Through experiments conducted, we uncover the conditions, under which M-estimate of the model parameter in the nonlinear regression model with continuous time and strong/weak dependent random noise is a unique solution of the system (in a definite asymptotic sense).