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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1901-1911

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2008.117

\vspace{3pc} \normalsize \centerline{\large\bf ON THE
IDENTIFIABILITY OF ADDITIVE INDEX MODELS}
%\vspace{2pt}
%\centerline{\large\bf MEAN CONCENTRATION OF CONTAMINANTS}
\vspace{.25cm}
\author{Ming Yuan}
\vspace{.4cm} \centerline{\it Georgia Institute of Technology}
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\begin{quotation}
\noindent {\it Abstract:} In this paper, we investigate the
identifiability of the additive index model, also known as
projection pursuit regression. Although a flexible regression tool,
additive index models can be hard to interpret in practice due to a
lack of identifiability. As noted by \cite{10}, %Horowitz (1998),
``it is an open question whether there are identifying restrictions
that yield useful forms'', in reference to additive index models
that differ from the known structured nonparametric regression
models such as additive models and single index models. We provide
an affirmative answer to this question: the additive index model is
identifiable as long as the projection indices are linearly
independent and there is at most one quadratic ridge function.
Furthermore, we show that when there are multiple quadratic ridge
functions, the identifiability can still be ensured for the
non-quadratic ridge functions and their corresponding projection
indices, whereas it is not possible to identify the quadratic ridge
functions. Such an identifiability result enables us to check if a
more restrictive nonparametric model such as the additive model can
be adopted as opposed to the more general additive index model.\par
\vspace{9pt} \noindent {\it Key words and phrases:} Additive index
model, additive model, identifiability, projection pursuit
regression, single index model.
\par
\end{quotation}

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