引用本文: 姜兆英, 刘国林, 于胜文. 病态方程基于Liu估计的一种迭代估计新方法[J]. 武汉大学学报 ( 信息科学版), 2017, 42(8): 1172-1178.
JIANG Zhaoying, LIU Guolin, YU Shengwen. A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations[J]. Geomatics and Information Science of Wuhan University, 2017, 42(8): 1172-1178.
 Citation: JIANG Zhaoying, LIU Guolin, YU Shengwen. A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations[J]. Geomatics and Information Science of Wuhan University, 2017, 42(8): 1172-1178.

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

• 摘要: 当线性回归模型的设计矩阵病态时，最小二乘（least square，LS）估值方差大且不稳定，已不是一种优良估计。为了减弱病态性，许多有偏估计法如岭估计、主成分估计、Liu估计等被提出。基于Liu估计，引入迭代的思想，提出了一种新的有偏估计法—迭代估计法。借助对称正定矩阵的谱分解，将迭代公式转化为便于解算的解析表达式，并证明迭代公式在修正因子d∈-1，1是收敛的。基于Liu估计中修正因子d的确定方法，在均方误差最小的情况下给出最优修正因子d的确定公式。最后，分别利用LS估计、岭估计、Liu估计和提出的迭代估计对两个算例进行计算并给出实验结果。在第一个算例中，对观测向量添加不同的扰动，结果表明迭代估计法具有更强的抗干扰能力；第二个算例的结果表明，迭代估计法所得结果更接近于真值，即迭代估计法在均方误差意义下优于LS估计、岭估计和Liu估计。

Abstract: 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.

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