Weak Convergence of the Empirical Process of Residuals in Linear Models with Many Parameters

When fitting, by least squares, a linear model (with an intercept term) with p parameters to n data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided p31og2(p)/n → 0 for any error density having finite variance and a bo...

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Bibliographic Details
Published in:The Annals of statistics 2001-06, Vol.29 (3), p.748-762
Main Authors: Chen, Gemai, Lockhart, Richard A.
Format: Article
Language:English
Subjects:
62E20
62J99
Covariance
empirical processes
Error rates
Estimators
Exact sciences and technology
Experimental design
goodness-of-fit
Integers
Least squares
Linear inference, regression
Linear models
Linear regression
Mathematical functions
Mathematical vectors
Mathematics
Point estimators
Probability and statistics
regression
Regression and Design
Residual
Sciences and techniques of general use
Statistics
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Summary:When fitting, by least squares, a linear model (with an intercept term) with p parameters to n data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided p31og2(p)/n → 0 for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1009210688