Normality Of Least Squares Estimator Research Articles

Asymptotic Normality of Estimators in Heteroscedastic Semi-Parametric Model with Strong Mixing Errors

Consider the heteroscedastic semi-parametric model y i = x i β +g(t i ) + σ i e i (1 ≤ i ≤ n), where , the design points (x i , t i , u i ) are known and non random, g(·) and f(·) are unknown functions defined on closed interval [0, 1], and the random errors e i are assumed to be a sequence of stationary α-mixing random variables with mean zero. Under appropriate conditions, we study the asymptotic normality of least-squares estimator and two weighted least-squares estimators of β. Also, the asymptotic normality of the estimators of g(·) and f(·) is considered. Finite sample behavior of the estimators is investigated via simulations as well.

Strong consistency and asymptotic normality of least squares estimators for PGARCH and [formula omitted] models

This paper deals with the probabilistic structure and the asymptotic properties of parameters least squares estimates ( L S E ) for periodic GARCH ( P G A R C H ) and for P A R M A – P G A R C H models. In this class of models, the parameters are allowed to switch between different regimes. Firstly, we give necessary and sufficient conditions ensuring the existence of stationary solutions (in a periodic sense) and for the existence of moments of any order. Secondly, a least squares estimation approach for estimating P G A R C H and P A R M A – P G A R C H models are discussed. The strong consistency and the asymptotic normality of the estimators are studied given mild regularity conditions, requiring strict stationarity and the finiteness of moments of some order for the errors term.

Consistency and asymptotic normality of least squares estimators in generalized STAR models

Space–time autoregressive (STAR) models, introduced by Cliff and Ord [Spatial autocorrelation (1973) Pioneer, London] are successfully applied in many areas of science, particularly when there is prior information about spatial dependence. These models have significantly fewer parameters than vector autoregressive models, where all information about spatial and time dependence is deduced from the data. A more flexible class of models, generalized STAR models, has been introduced in Borovkovaet al. [Proc. 17th Int. Workshop Stat. Model. (2002), Chania, Greece] where the model parameters are allowed to vary per location. This paper establishes strong consistency and asymptotic normality of the least squares estimator in generalized STAR models. These results are obtained under minimal conditions on the sequence of innovations, which are assumed to form a martingale difference array. We investigate the quality of the normal approximation for finite samples by means of a numerical simulation study, and apply a generalized STAR model to a multivariate time series of monthly tea production in west Java, Indonesia.

Local asymptotics for polynomial spline regression

In this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem. The polynomial spline spaces we consider include univariate splines, tensor product splines, and bivariate or multivariate splines on triangulations. We establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation. The asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities, error distributions and regression functions. The bias is controlled by the minimum $L_\infty$ norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate. The control of bias relies on the stability in $L_\infty$ norm of $L_2$ projections onto polynomial spline spaces. Asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory. In addition, a preliminary analysis of additive models is provided.

A proof of asymptotic normality for some VARX models

Here we present a proof of the asymptotic normality of least squares estimates for stable multivariate autoregressive models excited by a deterministic second order input signal.

Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems

Strong consistency and asymptotic normality of least squares estimates in stochastic regression models are established under certain weak assumptions on the stochastic regressors and errors. We discuss applications of these results to interval estimation of the regression parameters and to recursive on-line identification and control schemes for linear dynamic systems.

Consistency and asymptotic normality of least squares estimates used in linear systems identification

Least squares estimation of the parameters of a single input-single output linear autonomous system is considered where both plant noise and observation noise are present. It is shown that under fairly general conditions the estimates converge almost surely to the true system parameters and that the estimates are asymptotically normal.

Asymptotic normality of least squares estimates of the regression coefficients of homogeneous and isotropic random fields

Asymptotic normality of least squares estimates of the regression coefficients of homogeneous and isotropic random fields