The Least Trimmed Squares Estimate in Nonlinear Eegression

Yeh-ling Chen, Arnold Stromberg, and Mai Zhou

Abstract:

Estimating high breakdown regression estimators such as the least trimmed squares (LTS) in nonlinear regression would involve an unmanageable computational load if no appropriate algorithm were provided. By the definition of the LTS in this article, for a nonlinear model with sample size 30 would require finding the least squares fit to 1.4542268e+8 subsets of the data. In this article, the proposed algorithm requires six starting subsets of half the data, thus substantially reduces the computational load, and also retains the high breakdown property of the nonlinear LTS regression. The most important property of an estimate is the consistency. In addition to the proposed algorithm, the strong consistency of the LTS estimate in nonlinear regression is also proved. When compared to Stromberg's (1993) algorithm for finding the LTS estimate, the proposed algorithm appears to be more reliable as the variability in the data increases for the three different nonlinear models used in simulations. The case presented applies the LTS estimates to Carroll and Ruppert's transformation-both-sides (TBS) model. The result of the case study in this article may differ from Carroll and Ruppert's, but the contribution to robust estimation is easy to be recognized.

Keywords: Least Median of Squares (LMS), Least Trimmed Squares (LTS), Feasible Solution Algorithm (FSA), Six Half Algorithm (SHA), Multistage Algorithm (MSA), Transformation-Both-Sides (TBS)

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