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Biased regression, Mean squared error, Multicollinearity, Ridge regression, Liu estimator,

In this paper we introduce a class of estimators which includes the ordinary least squares (OLS), the principal components regression (PCR) and the Liu estimator (1). In particular, we show that our new estimator is superior, in the scalar meansquared error (mse) sense, to the Liu estimator, to the OLS estimator and to the PCR estimator.

The problem of multicollinearity and outliers in the dataset can strongly distort ordinary least squares estimates and lead to unreliable results. We propose a new Robust Liu-type M-estimator to cope with this combined problem of multicollinearity and outliers in the y-direction. Our new estimator has advantages over two parameter Liu-type estimator, Ridge-type M-estimator and Mestimator. Furthermore, we give a numerical example and a simulation study to illustrate some of the theoretical results.

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Several alternative methods for derivation of the restricted ridge regression estimator (RRRE) are provided. Theoretical comparison and relationship of RRRE with related methods for regression with the multicollinearity problem are described. We also find inter-connections among RRRE, ordinary ridge regression estimator (ORRE), restricted least squares estimator (RLSE), modified ridge regression estimator (MRRE) and restricted modified generalized ridge estimator (RMGRE). Finally, numerical comparison, in addition to theoretical derivation, is also conducted with a Monte Carlo simulation and a real data example.

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