Strong Consistency Of Least Squares Estimates Research Articles

Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables

ABSTRACT In this paper, we establish some results on strong consistency for the least square estimator in a simple linear errors-in-variables regression models based on widely orthant dependent errors. The results obtained in the paper generalize or improve the corresponding ones for dependent random variables. We also carry out some simulation studies to verify the validity of our theoretical results.

On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r⩽ 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.

Strong Law of Large Numbers with Operator Normalizations for Martingales and Sums of Orthogonal Random Vectors

We establish the strong law of large numbers with operator normalizations for vector martingales and sums of orthogonal random vectors. We describe its applications to the investigation of the strong consistency of least-squares estimators in a linear regression and the asymptotic behavior of multidimensional autoregression processes.

Asymptotic Properties of Least-Squares Estimates in Stochastic Regression Models

Strong consistency of least-squares estimates in stochastic regression models is established under the assumption that the underlying model can be reparametrized so that the new design vectors are weakly correlated. An application to fixed-width interval estimation in stochastic approximation schemes is also discussed.

Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters

This paper establishes several almost sure asymptotic properties of general autoregressive processes. By making use of these properties, we obtain a proof of the strong consistency of the least-squares estimates of the parameters of the process without any assumption on the roots of the characteristic polynomial.

An inequality for a sum of quadratic forms with applications to probability theory

Let {x t } be a sequence of p -component vectors, and let A t =∑ T s =1 x t x ′ t with A n nonsingular for some n ⩾ p . It is shown that ∑ t t = q +1 x ′ t A -2 t x t ⩽ trA -1 q for n ⩽ q < T . An application of this proposition is the convergence of a certain martingale with probability 1. A matrix version of Kronecker's lemma then leads to strong consistency of least-squares estimates under a certain condition.

Strong consistency of least-squares estimates in regression models

A general theorem on the limiting behavior of certain weighted sums of i.i.d. random variables is obtained. This theorem is then applied to prove the strong consistency of least-squares estimates in linear and nonlinear regression models with i.i.d. errors under minimal assumptions on the design and weak moment conditions on the errors.