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Linear regression by functional least squares

Published online by Cambridge University Press:  14 July 2016

Abstract

The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.

Type
Part 5 — Statistical Theory
Copyright
Copyright © 1982 Applied Probability Trust 

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References

Andrews, D.F. (1974) A robust method for multiple linear regression. Technometrics 16, 523531.CrossRefGoogle Scholar
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H., and Tukey, J.W. (1972) Robust Estimates of Location. Princeton University Press, Princeton, N.J. Google Scholar
Chambers, R.L. and Heathcote, C.R. (1981) On the estimation of slope and identification of outliers in linear regression. Biometrika , 68, 2134.Google Scholar
Daniel, C. and Wood, F.S. (1971) Fitting Equations to Data. Wiley, New York.Google Scholar
Gould, A.L. (1969) A regression technique for angular variates. Biometrics 25, 683700.Google Scholar
Huber, P.J. (1973) Robust regression: asymptotics, conjectures and Monte Carlo. Ann. Statist. 1, 799821.Google Scholar
Mardia, K.V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar