On the Whittle estimator for linear random noise spectral density parameter in continuous-time nonlinear regression models

Abstract

A continuous-time nonlinear regression model with LГқvy-driven linear noise process is considered. Sufficient conditions of consistency and asymptotic normality of the Whittle estimator for the parameter of the noise spectral density are obtained in the paper. IntroductionThe paper is focused on such an important aspect of the study of regression models with correlated observations as an estimation of random noise functional characteristics. When considering this problem the regression function unknown parameter becomes nuisance and complicates the analysis of noise. To neutralise its presence, we must estimate the parameter and then build estimators, say, of spectral density parameter of a stationary random noise using residuals, that is the difference between the values of the observed process and fitted regression function.So, in the first step we employ the least squares estimator (LSE) for unknown parameter of nonlinear regression, because of its relative simplicity. Asymptotic properties of the LSE in nonlinear regression model were studied by many authors. Numerous results on the subject can be found in monograph by Ivanov and Leonenko [34], Ivanov [32].

Asymptotic normality of the residual correlogram in the continuous-time nonlinear regression model

“…For the stationary noise it can be estimation of the noise spectral density or covariance function. Asymptotic properties of the Whittle and Ibragimov estimators of spectral density parameters in the continuous time nonlinear regression model were considered in Ivanov and Prykhod'ko [16,15], Ivanov et al [17]. Exponential bounds for the probabilities of large deviations of the stationary Gaussian noise covariance function in the similar regression model are obtained in Ivanov et al [11].…”