Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series B, Pt. 3, pp. 443--459

A NEW BIASED ESTIMATOR IN LINEAR REGRESSION AND A DETAILED ANALYSIS OF THE WIDELY-ANALYSED DATASET ON PORTLAND CEMENT

By

SELAHATTIN KAÇIRANLAR, SADULLAH SAKALLIOGLU, FIKRI AKDENIZ * University of Çukurova, Adana, Turkey*

GEORGE P. H. STYAN, *McGill University, Montréeal, Québec, Canada *

and

HANS JOACHIM WERNER, *University of Bonn, Bonn, Germany*

*SUMMARY.* This paper deals with the standard multiple
linear regression model (*y*,Xb ,s^{2}*I*), where the model
matrix *X* is assumed to be of full column rank. We introduce a new
biased estimator for b and discuss its properties in some
detail. In particular, we show that our new estimator is superior,
in the scalar mean-squared error sense, to both the usual restricted
least-squares estimator and to the new biased estimator introduced
by Liu (1993). We illustrate our findings with a numerical example
based on the widely-analysed dataset on Portland cement, cf. Woods,
Steinour and Starke (1932), Hald (1952, pp. 635--652), Piepel and
Redgate (1998).

*AMS (1991) subject classification.*62J05, 62J07.

*Key words and phrases. *Anti-quirk;
biased estimator; ill-conditioning; least squares estimator; Liu
estimator; mean-squared error; multicollinearity; Portland cement
data; quirk; restricted least squares; ridge regression estimator.