Modified Almost Unbiased Liu Estimator in Linear Regression Model
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Modified Almost Unbiased Liu Estimator in Linear Regression Model Sivarajah Arumairajan1,2 · Pushpakanthie Wijekoon3
Received: 29 December 2015 / Revised: 23 September 2016 / Accepted: 13 May 2017 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2017
Abstract In this paper, we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator (AULE) and ridge estimator (RE) in a linear regression model when multicollinearity presents among the independent variables. Necessary and sufficient conditions for the proposed estimator over the ordinary least square estimator, RE, AULE and Liu estimator (LE) in the mean squared error matrix sense are derived, and the optimal biasing parameters are obtained. To illustrate the theoretical findings, a Monte Carlo simulation study is carried out and a numerical example is used. Keywords Multicollinearity · Ridge estimator · Almost unbiased Liu estimator · Liu estimator · Modified almost unbiased Liu estimator · Mean squared error matrix Mathematics Subject Classification 62J07 · 62F03
1 Introduction We consider the multiple linear regression model
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Sivarajah Arumairajan [email protected] Pushpakanthie Wijekoon [email protected]
1
Postgraduate Institute of Science, University of Peradeniya, Peradeniya, Sri Lanka
2
Department of Mathematics and Statistics, Faculty of Science, University of Jaffna, Jaffna, Sri Lanka
3
Department of Statistics and Computer Science, Faculty of Science, University of Peradeniya, Peradeniya, Sri Lanka
123
S. Arumairajan, P. Wijekoon
y = Xβ + ε, ε ∼ N 0, σ 2 I ,
(1.1)
where y is an n × 1 observable random vector, X is an n × p known design matrix of rank p, β is a p × 1 vector of unknown parameters and ε is an n × 1 vector of disturbances. The ordinary least square estimator of β is given by βˆOLSE = S −1 X y,
(1.2)
where S = X X . In the literature, there are several biased estimators proposed to overcome multicollinearity instead of OLSE. The well known one among these estimators is the ridge estimator (RE) proposed by Hoerl and Kennard [8]. Followed by Hoerl and Kennard [8], the almost unbiased ridge estimator (AURE) [15], the Liu estimator (LE) [11], almost unbiased Liu estimator (AULE) [1], stochastic restricted ordinary ridge estimator (SRORE) [4] and generalized optimal estimator (GOE) [5] have been proposed. Recently, Arumairajan and Wijekoon [3] proposed a new estimator named as generalized unrestricted estimator (GURE) to represent the RE, LE, AURE and AULE as follows: βˆGURE = A(i) βˆOLSE ,
(1.3)
where A(i) is a positive definite matrix. If A(i) = Wk in (1.3), the estimator GURE becomes the RE which is given by βˆRE (k) = Wk βˆOLSE ,
(1.4)
−1 for k ≥ 0. where Wk = I + k S −1 If A(i) = Fd in (1.3), the estimator GURE becomes the LE which is defined as βˆLE (d) = Fd βˆOLSE ,
(1.5)
where Fd = (S + I )−1 (S + d I ). If A(i) = Td in (1.3), the estimator GURE becomes the AULE which is defi
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