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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, June 6, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{ Peter Hall}\\
Australian National University\\
Visiting UC Davis
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  \textbf{ On Deconvolution with Repeated Measurements   }
            
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\noindent
In many statistical inverse problems it is necessary to suppose that 
the transformation that is inverted is known.  Although this 
assumption might seem unrealistic, the problem is often insoluble 
without it.  However, if additional data are available then it is 
possible to estimate consistently the unknown error density. Data are 
seldom available directly on the transformation, but repeated, or 
replicated, measurements increasingly are becoming available. Such 
data consist of ``intrinsic'' values that are measured several times, 
with errors that are generally independent.  Working in this setting 
we treat the nonparametric deconvolution problems of density 
estimation with observation errors, and regression with errors in 
variables.  We show that, even if the number of repeated measurements 
is quite small, it is possible for modified kernel estimators to 
achieve the same, optimal level of performance they would if the 
error distribution were known.

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