On asymptotic distribution of Koenker-Bassett estimator of the parameter of linear regression model with strongly dependent noise

Abstract

Asymptotic properties of Koenker - Bassett estimators of linear regression model parameters with discrete observation time and random noise being nonlinear local transformation of Gaussian stationary time series with singular spectrum are studied. The goal of the work lies in obtaining the requirements to regression function and time series that simulates the random noise, under which the Koenker - Bassett estimators of regression model parameters are asymptotically normal. Linear regression model with discrete observation time and bounded open convex parametric set is the object of the studying. Asymptotic normality of unknown parameters Koenker - Bassett estimators are obtained. For getting these results complicated concepts of time series theory and time series statistics have been used, namely: local transformation of Gaussian stationary time series, stationary time series with singular spectral density, spectral measure of regression function, admissibility of singular spectral density of stationary time series in relation to this measure, expansions by Chebyshev - Hermite polynomials of the transformed Gaussian time series values and it‘s covariances, central limit theorem for weighted sums of the values of such a local transformation.