Xi'an Institute of Surveying and Mapping
I. ROBUST ESTIMATION
Within the last four years, much attention has been paid to parameter estimation for correlated observations and Lp estimation, in the robust estimation field. A set of self-contained theory system on robust estimation is researched based on equivalent variance-covariance. A new robust estimator for correlated observations (RECO) by bi-factor equivalent weight elements was developed, which keeps the symmetry and the correlation of the observations unchanged[35,37].
The robust estimators of parameters and variance factor with minimum mean square error are derived, all parameters of which are determined according to statistics.
The parameter estimation problem when outliers and ill-conditioning exist simultaneously is researched. A class of new estimators, shrunken type robust estimators, are proposed by grafting the biased estimation techniques philosophy into the robust estimator, and their properties are discussed. The un-biasness for the robust estimation of the location parameters is discussed, based on the hypothesis that the weight function is an anti-symmetrical function. The unbiased estimation of posterior variance and the optimal robust estimate have been studied. It is demonstrated that for stochastic error model and mean shift error model, they have the same estimate formula.
Robust Kalman filtering model and its application in GPS monitoring networks are studied. The Robust Kalman filtering model based on the rules of the influences of the outliers on the state vectors is derived. The theory and the method as well as the robustness of the parameter estimation based on the principle of information spread are researched. Givens-Gentleman orthogonal transformation is applied to robust estimation. The formulas of posterior root mean square error and the covariance matrix of the parameter estimates have been derived. The calculation of equivalent weights is discussed. The numerical stability is analyzed for extended Givens-Gentleman orthogonal transformation.
Lp estimator is an influential class of robust estimation which has been widely studied in China. The p-norm distribution is a distributive class which includes the most frequently used distributions such as Laplace, normal and rectangular ones. The p-norm distribution can be represented by the linear combination of Laplace distribution and normal distribution or by the linear combination of normal distribution and rectangular distribution approximately. The approximate distribution has the same first four-order moments as the original p-norm distribution. Because every density function used in the approximate formulas has a simple form, using the approximate density function to replace the p-norm ones will simplify the problems of p-norm distributed data processing obviously. Based on the theory of the M-estimation for the location parameters, the influence functions of the Lp estimates of the parameters (one dimension and multi-dimension) are derived, the robustness of Lp estimation when the observational errors obey the contaminated distribution is analyzed, and the asymptotic variance and efficiency of the <