Biased Estimation in Gauss-Markov Model not of Full Rank

Gui Qingming, Guo Jianfeng, Yang Yuanxi, Sui Lifen

The biased estimation problem in Gauss-Markov model not of full rank is considered when some multicollinearities exist among the columns of the design matrix in this paper. We propose a class of biased estimators by grafting the generalized shrunken least squared (GSLS) estimation technique (Gui and Liu 2000) philosophy into the MNLS estimator, and establish some important properties. Many well-known biased estimator in Gauss-Markov model with full rank, e.g. ordinary ridge estimator, principal components estimator, combining ridge and principal components estimator, combined principal components estimator, single-parametric principal components estimator, root-root estimator etc. are extended to Gauss-Markov model not of full rank. A numerical example in free net adjustment is presented to illustrate that these biased estimators are better than the MNLS estimator when some multicollinearities exist.

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