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\begin{document}

\section{Problem}

\begin{itemize}
\item  $D = (x_i)_{1\leq i\leq n}$
\item $(x_i)_{1\leq i\leq n}$ sampled in an i.i.d fashion from $\mu$
\end{itemize}

There is a function $F$ and you are interested in estimating the value
$F(\mu)$. We assume that you have an estimation scheme $\tilde{F}$,
which given a set of data points $S$ returns an estimation
$\tilde{F}(S)$ which is optimal in some sense. Your also given a
revenue function $R$ which is a decreasing function of the estimation
error. Then the value $V$ of the databse is defined by:
\begin{displaymath}
  V(D) = \max_{S\subseteq D} R\left(| F(\mu) - \tilde{F} |\right)
\end{displaymath}

\begin{example}
  
\end{example}

\begin{fact}

  \begin{itemize}
  \item If $R$ is decreasing then $V$ is increasing in the size of $D$.
  \item If $R$ is concave then $V$ is supermodular.
  \end{itemize}
\end{fact}

\begin{proof}
  
\end{proof}

\end{document}