  The problem we concentrate on is as follows: given (1) a convex compact set
$X$ in ${\mathbb{R}}^n$, an affine mapping $x\mapsto A(x)$, a parametric family
$\{p_{\mu}(\cdot)\}$ of probability densities and (2) $N$ i.i.d. observations
of the random variable $\omega$, distributed with the density $p_{A(x)}(\cdot)$
for some (unknown) $x\in X$, estimate the value $g^Tx$ of a given linear form
at $x$. For several families $\{p_{\mu}(\cdot)\}$ with no additional
assumptions on $X$ and $A$, we develop computationally efficient estimation
routines which are minimax optimal, within an absolute constant factor. We then
apply these routines to recovering $x$ itself in the Euclidean norm.
