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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, February 8, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Florentina Bunea}
\centerline{\sl Dept. of Statistics}
\centerline{\sl University of Washington}
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\centerline{\bf A Model Selection Approach to Partially Linear Regression}
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We extend the model selection approach proposed by Barron, Birg\'e and
Massart (1999) for nonparametric regression to partially linear regression.
 That is, we consider the model $Y = X'\beta + f(T) + W$, where $X$ belongs
to $R^q$,  $T$ belongs to $R, W$ is the error independent of $(X,T)$ and $f$ 
is a function of unknown smoothness. This model has  received considerable
attention in the  literature,  and it is mostly used in cases in which the
parameter of  interest is the linear component, 
whereas the variables appearing  in the nonlinear part are viewed as
confounders, hence $f$ is regarded as a nuisance parameter.  We study this
model in two different cases.

Case  A.
The number of covariates appearing in the linear part is a priori given,
say $q$.

Case  B.
We have available $q$ possible regressors for the linear part, but only an
unknown (possibly much smaller) subset of them are relevant for $Y$, hence
we would like to {\it select} it.

We propose a penalized least squares approach and, in both cases, we obtain
finite sample upper bounds for the risk of the estimator and, as a
consequence, the consistency of the estimators  of $\beta$ and $f$ at the
optimal nonparametric rate. In {\bf Case  B}, for $f \equiv 0$, we derive
rates of convergence for estimators of $\beta$ corresponding to BIC and
AIC. We also discuss the distributional properties of  $\hat{\beta}$ in
Case  A.


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