Robust weights and designs for biased regression models: Least squares and generalized M-estimation

• Published in: Journal of statistical planning and inference Vol. 83; no. 2; pp. 395 - 412
• Main Author: Wiens, Douglas P.
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 01.02.2000; New York,NY Elsevier Science; Amsterdam
• Subjects: A-optimality; Bounded influence M-estimation; D-optimality; Exact sciences and technology; Experimental design; Least median of squares; Legendre polynomial; Linear inference, regression; Mathematics; Multiple regression; Optimal design; Parametric inference; Polynomial regression; Probability and statistics; Q-optimality; Robustness; Sciences and techniques of general use; Statistics; Weighted least squares; Weighted least squares; secondary 62J05; Legendre polynomial; Multiple regression; A-optimality; Optimal design; Bounded influence M-estimation; D-optimality; Q-optimality; Robustness; primary 62K05, 62F35; Polynomial regression; Least median of squares; Error estimation; Linear regression; Estimator robustness; Covariance matrix; Mean square error; Statistical method; Statistical regression; Function minimization; Least squares method; Experimental design; Point estimation; Optimal design(statistics); M estimation; Biased estimation
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/S0378-3758(99)00102-0

We consider an ‘approximately linear’ regression model, in which the mean response consists of a linear combination of fixed regressors and an unknown additive contaminant. Only the linear component can be modelled by the experimenter. We assume that the experimenter chooses design points and then estimates the regression parameters by weighted least squares or by generalized M-estimation. For this situation we exhibit designs and weights which minimize scalar-valued functions of the covariance matrix of the regression estimates, subject to a requirement of unbiasedness. We report the results of a simulation study in which these designs/weights result in significant improvements, with respect to both bias and mean squared error, when compared to some common competitors.