FUZZY WEIGHTED SCALED COEFFICIENTS IN SEMI-PARAMETRIC MODEL

JONG-WUU WU^{1}, JIAHN-BANG JANG^{2} AND TZONG-RU TSAI^{2}

1Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan 25137, R.O.C.

2Graduate School of Statistics, National Chengchi University, 64, 2nd Section, Chi-nan Rd., Taipei, Taiwan 11623, R.O.C.

(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 Y_{i} = Q_{1}(α + βX_{i}; ϵ_{i}), i = 1, 2, …, n. Here α is a constant, β is a 1 × p row vector, X_{i} is a p × 1 column vector of explanatory variables, ϵ_{i} is an unobserved random error and Q_{1} is an arbitrary unknown function. When the data set (X_{i}, Y_{i}), i = 1, 2, …, n, contains one or several outliers, the least squares method can not provide a consistent estimator of the scaled coefficients β. Therefore, we suggest the “fuzzy” weighted least squares method to estimate the scaled coefficients β 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.