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\footnotesize
\noindent Statistica Sinica {\bf 16}(2006), 195-211

\vspace{3pc} \normalsize
\centerline{\large\bf DENSITY ESTIMATION WITH NORMAL MEASUREMENT }
\vspace{2pt}
\centerline{\large\bf ERROR WITH UNKNOWN VARIANCE}
\vspace{.25cm}
\author{Alexander Meister}
\vspace{.4cm}
\centerline{\it Universit\"at Stuttgart}
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\begin{quotation}
\noindent {\it Abstract:}
This paper deals with the problem of estimating a
density based on observations which are contaminated by a normally distributed
error whose variance is unknown. In the case of a completely unknown error
variance, the impossibility of a uniformly consistent estimation is shown; however,
a semi-uniformly consistent estimator is constructed under nonparametric smoothness conditions
on the target density, and its rates are studied. If, in contrast, the error variance can be located
in a known compact interval, we derive uniform consistency for this estimator which achieves
nearly optimal rates. Simulations show the practical merit of the estimator.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Deconvolution, errors-in-variables, inversion
problems, nonparametric estimation, reconstruction.\par
\end{quotation}

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