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FUZZY WEIGHTED SCALED COEFFICIENTS

IN SEMI-PARAMETRIC MODEL

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

^{1} *Department of Statistics, Tamkang University,
Tamsui, Taipei,
Taiwan 25137, R.O.C.*

^{2} *Graduate 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}(*alpha*+*beta* *X*_{i};*epsilon*_{i}), *i*=1, 2, ...., *n*.
Here *alpha* is a constant, *beta* is a × *p* row vector,
*X*_{i} is a *p* × 1 column vector of explanatory variables,
*epsilon*_{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 *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.

**Source**
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