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\footnotesize
\noindent Statistica Sinica {\bf 15}(2005), 1069-1079

\vspace{3pc} \normalsize
\centerline{\large\bf ASYMPTOTIC DISTRIBUTIONS FOR DIMENSION}
\vspace{2pt}
\centerline{\large\bf  REDUCTION IN THE SIR-II METHOD}
\vspace{.25cm}
\author{Xiangrong Yin and Lynne Seymour}
\vspace{.4cm}
\centerline{\it The University of Georgia}
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\begin{quotation}
\noindent {\it Abstract:}
The asymptotic distribution of the test statistic for testing the dimensionality
in the SIR-II method
is derived and shown to be a linear combination of $\chi^2$ random
variables under weak assumptions. This statistic is based on Li's (1991)
sequential test statistic
for sliced inverse regression (SIR). Also presented is a simulation study of
the result.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Asymptotic distributions, Central subspaces,
Dimension-reduction subspaces, Regression graphics.\par
\end{quotation}

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