A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations

In the presence of the design matrix's collinearity (which is equivalent to ill-conditioning) in the linear regression model, the least squares (LS) estimator has large variances and its solution is rather unstable, so the LS estimator is not the precise estimation any more. In order to weaken the ill-conditioning, many biased estimator methods are introduced, such as ridge estimator, the principal components estimator, the Liuestimator and so on. In this paper, based on the famous Liuestimator, we present a new biased estimator which is called a biased iterative estimator method. With the aid of spectral decomposition of the symmetric and positive matrix, the iterative formula is converted to a simple analytical expression conveniently for calculating. And the iterative formula is proved to be convergent in the condition of modified parameterd∈-1, 1. Following the deter mination method of modified parameter in the Liuestimator, we give a formula of the optimal modified parameter to minimize the mean squared error (MSE). Finally, we use the proposed biased iterative estimator, LS estimator, ridge estimator and the Liuestimator to calculate two numerical examples and give their experimental results. In the first example, we respectively add different perturbations to the observation vector. The simulation results show that compared with other three methods, the biased iterative estimator is more stable under the perturbation. Comparison results of the second example show that our new biased iterative estimator is more closed to the real value, that is superior, in the mean squared error sense, to the LS estimator, ridge estimator and the Liuestimator.