TITLE: Approximating a probability distribution by a mixture
QUESTION [5 upvotes]: Let us consider a probability distribution $(g_n)_{n \in \mathbb{N}}$ which we want to approximate by a mixture of $(f_n(\lambda))_{n \in \mathbb{N}}$ where $\lambda \in \mathbb{R}$ is a parameter.
Are there known techniques that allow one to find the mixture minimizing the $L^1$ norm:
\begin{equation}
\min_{p} \sum_{n=0}^{\infty} \left|g_n - \int \rm{d} \lambda \;  p(\lambda) f_n(\lambda) \right|
\end{equation} 
where $p(\lambda)$ is a normalized probability distribution?
The motivation of this problem is linked to experimental physics: ideally one would like to generate an experimental process characterized by the probability distribution $g$ but this is really not practical. What is really easy, however, is to generate an experimental process with the distribution $f(\lambda)$ where $\lambda$ is a tunable parameter.
Therefore, the goal is to approximate $g$ as closely as possible with such a mixture of $f(\lambda)$, where the distance between the two distribution is computed with the $L^1$ norm, that is, I want to minimize the variation distance between the two distributions.
In the specific problem I consider, $f(\lambda)$ is a Poisson distribution with parameter $\lambda \geq 0$, but I really am interested in a general method to approach this problem-
Any pointer to the relevant literature would be greatly appreciated.
Thanks a lot!

REPLY [2 votes]: perhaps for starters you could take $p$ to be supported on a finite number of points. then the constraints on $p$ become simple linear inequalities and you have a [convex - perhaps even linear] programming problem. 
there has been attention in the statistical literature to fitting models using least absolute deviations, rather than least squares. [in the simplest case, the minimizer of
$$\sum_{i=1}^n |x_i - a|$$
is the sample median - rather than the sample mean one gets for $a$ using least squares.] 
you could see if references in the monograph by yadolah dodge [L$_1$ statistical procedures and related topics, ims lecture notes - monograph series vol 31 1997] give anything useful.