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\noindent Statistica Sinica {\bf 13}(2003), 881-902

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\centerline{\large\bf RATES OF CONVERGENCE AND ADAPTATION OVER}
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\centerline{\large\bf  BESOV SPACES UNDER POINTWISE RISK}
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\author{T. Tony Cai}
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\centerline{\it University of Pennsylvania}
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\begin{quotation}
\noindent {\it Abstract:}
Function estimation over the Besov spaces under pointwise $\ell^r$
($1\leq r < \infty$) risks is considered. Minimax rates of convergence
are derived using a constrained  risk inequality and wavelets.
Adaptation under pointwise risks is also considered. Sharp lower
bounds on the cost of adaptation are obtained and are shown to be
attainable by a wavelet estimator. The results
demonstrate important differences between the minimax properties under
 pointwise and global risk measures. The minimax rates and
adaptation for estimating derivatives under  pointwise risks are also
presented. A general $\ell^r$-risk oracle inequality  is
developed for the proofs of the main results.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Adaptability, adaptive estimation, Besov spaces, constrained risk
inequality, minimax estimation, nonparametric functional estimation,
oracle inequality, rate of convergence, wavelets, white noise model. \par
\end{quotation}
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