  We show a statistical version of Taylor's theorem and apply this result to
non-parametric density estimation from truncated samples, which is a classical
challenge in Statistics \cite{woodroofe1985estimating, stute1993almost}. The
single-dimensional version of our theorem has the following implication: "For
any distribution $P$ on $[0, 1]$ with a smooth log-density function, given
samples from the conditional distribution of $P$ on $[a, a + \varepsilon]
\subset [0, 1]$, we can efficiently identify an approximation to $P$ over the
\emph{whole} interval $[0, 1]$, with quality of approximation that improves
with the smoothness of $P$."
  To the best of knowledge, our result is the first in the area of
non-parametric density estimation from truncated samples, which works under the
hard truncation model, where the samples outside some survival set $S$ are
never observed, and applies to multiple dimensions. In contrast, previous works
assume single dimensional data where each sample has a different survival set
$S$ so that samples from the whole support will ultimately be collected.
