(Received May 23, 1994; revised April 26, 1995)
Abstract. In general, the regressor variables are stochastic, Duan and Li (1987, J. Econometrics, 35, 25-35), Li and Duan (1989, Ann. Statist., 17, 1009-1052) have been shown that under very general design conditions, the least squares method can still be useful in estimating the scaled regression coefficients of the semi-parametric model Yi = Q1(alpha+beta Xi;epsiloni), i=1, 2, ...., n. Here alpha is a constant, beta is a × p row vector, Xi is a p × 1 column vector of explanatory variables, epsiloni is an unobserved random error and Q1 is an arbitrary unknown function. When the data set (Xi, Yi), i=1, 2, ...., n, contains one or several outliers, the least squares method can not provide a consistent estimator of the scaled coefficients beta. Therefore, we suggest the ``fuzzy" weighted least squares method to estimate the scaled coefficients beta for the data set with one or several outliers. It will be shown that the proposed ``fuzzy" weighted least squares estimators are \sqrt(n)-consistent and asymptotically normal under very general design condition. Consistent measurement of the precision for the estimator is also given. Moreover, a limited Monte Carlo simulation and an example are used to study the practical performance of the procedures.
Key words and phrases: Least squares estimator, semi-parametric model, outlier, asymptotic normality, fuzzy weighted least squares estimator, Monte Carlo simulation.
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