 
Summary: ELSEVIER ('ompt,tational Statistics & 1)ata Analysis 25 (1997}377 398
COMPUTATIONAL
STATISTICS
&DATAANALYSIS
Length modified ridge regression
Magne Aldrin*
Norwegian Computing Center. P.O. Box 114, 0314 OSI.O 3 Blindern. N~rway
Received 1 April 1996: revised 1 November 1996
Abstract
Biased regression methods may improve considerably on ordinary least squares regression with few
or noisy data, or when the predictor variables are highly collinear. In the present work, I present
a new, biased method that modifies the ordinary least squares estimate by adjusting each element of
the estimated coefficient vector. The adjusting factors are found by minimizing a measure of
prediction error. However, the optimal adjusting factors depend on the unknown coefficient vector as
well as the variance of the noise, so in practice these are replaccd by preliminary estimates. The final
estimate of the coefficient vector has the same direction as the preliminary estimate, but the length is
modified. Ridge regression is used as the principal method to find the preliminary estimate, and the
method is called length moditied ridge regression. In addition, length modified principal components
regression is considered. The prediction performance of the methods arc compared to other regression
methods (ridge, James Stein, partial least squares, principal components and variable subset selec
