Abstract
Wu (2013) proposed an estimator, principal component Liu-type estimator, to overcome multicollinearity. This estimator is a general estimator which includes ordinary least squares estimator, principal component regression estimator, ridge estimator, Liu estimator, Liu-type estimator,r-kclass estimator, andr-dclass estimator. In this paper, firstly we use a new method to propose the principal component Liu-type estimator; then we study the superior of the new estimator by using the scalar mean squares error criterion. Finally, we give a numerical example to show the theoretical results.
Highlights
Consider the multiple linear regression model y = Xβ + ε, (1)where y is an n × 1 vector of observation, X is an n × p known matrix of rank p, β is a p × 1 vector of unknown parameters, and ε is an n×1 vector of disturbances with expectation E(ε) = 0 and variance-covariance matrix Cov(ε) = σ2In.According to the Gauss-Markov theorem, the classical ordinary least squares estimator (OLSE) is obtained as follows: β = (XX)−1Xy. (2)The OLSE has been regarded as the best estimator for a long time
Firstly we use a new method to propose the principal component Liu-type estimator; we study the superior of the new estimator by using the scalar mean squares error criterion
Kacıranlar and Sakallıoglu [7] introduced the r-d class estimator which is the combination of the Liu estimator (LE) and the PCRE, which is defined as follows: βr (d) = Tr(TrXXTr + Ir)−1 (TrXy + dTrβr), (13)
Summary
This estimator can be obtained by solving the following problem: min {(y − Xβ) (y − Xβ) + (β − dβ) (β − dβ)}. Huang et al [4] introduced a Liu-type estimator which includes the OLSE, RE, and LE, defined as follows:. Kacıranlar and Sakallıoglu [7] introduced the r-d class estimator which is the combination of the LE and the PCRE, which is defined as follows: βr (d) = Tr(TrXXTr + Ir)−1 (TrXy + dTrβr) ,. Wu [12] proposed the principal component Liu-type estimator (PCTTE), which is defined as βr (k, d) = Tr(TrXXTr + kIr)−1 (TrXy + dTrβr) , (14) k > 0, 0 < d < 1. Firstly we use a new method to propose the principal component Liu-type estimator. We give a numerical example to illustrate the theoretical results
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