
Accession Number : AD0248660
Title : CENTRAL LIMIT THEOREM AND CONSISTENCY IN LINEAR REGRESSION
Corporate Author : NORTH CAROLINA UNIV AT CHAPEL HILL DEPT OF STATISTICS
Personal Author(s) : EICKER, FRIEDHELM
Report Date : DEC 1960
Pagination or Media Count : 1
Abstract : Several asymptotic properties of the least squares estimators for the parameters in the linear regression model with nonidentical, independent errors are derived with regard to infinitely increasing sample size. The notion of convergence of a sequence of random variables b on a set F is introduced and applied, i.e., the b depend functionally on an arbitrarily chosen sequence of random variables all of which are elements of the set F, and they converge for (over) each such sequence. If F is any set of random variables with zero means and bounded variances which contains at least one normal variable, then there exists a simple necessary and sufficient condition for the regression matrix X such that the estimators are consistent on F. For asymptotic normality of essentially these estimators on any subset G in the set of all zeromean random variables whose variances exist, necessary and sufficient conditions are given both for X and the set G simultaneously to be fulfilled. This is the central limit theorem for the linear regression model. For the case of unknown error variances, a statistic is constructed, and sufficient conditions for X and G are given to assure its asymptotic normality on G. One important aspect of the theory developed is its nonparametric character and the width of the range of admitted error distributions. Some examples and illustrations are given. (Author)
Descriptors : *MATRICES(MATHEMATICS), *SAMPLING, ERRORS, LEAST SQUARES METHOD.
Distribution Statement : APPROVED FOR PUBLIC RELEASE