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\noindent Statistica Sinica {\bf 13}(2003), 143-162
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\centerline{\large\bf USING LINEAR SMOOTHERS TO ASSESS THE}
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\centerline{\large\bf STRUCTURAL DIMENSION OF REGRESSIONS}
\vspace{.25cm}
\centerline{E. Bura}
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\centerline{\it The George Washington University}
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\begin{quotation}
\noindent {\it Abstract:}
Sliced Inverse Regression (Li (1991)) is a simple nonparametric estimation
method for the {\it structural dimension} of a regression, that is, for the
dimension of the linear subspace spanned by projections of the multidimensional
regressor vector $\bX$ that contains part or all of the modelling information
about the regression of a random variable $Y$ on $\bX$. In this paper, the
nonparametric estimation method is extended to include the family of linear
smoothers. No restrictions are placed on the distribution of the regressors
except for the {\it linearity condition} and existence of second moments. An
asymptotic chi-square test for dimension is obtained. Theoretical results are
illustrated with a small comparative simulation study.\par
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\noindent {\it Key words and phrases:}
Asymptotic chi-square test, dimension reduction, inverse regression.\par
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