Exact solutions in log-concave maximum likelihood estimation

We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, maximum likelihood estimate is exponential a piecewise linear function determined by finitely many quantities, namely values, or heights, at data points. explore in what sense exact solutions to this problem possible. First, we show heights given generically transcendental. For cell one dimension, estimator expressed closed form using generalized W-Lambert function. Even more, finding log-concave equivalent solving collection polynomial-exponential systems special form. case two equations, very little known about these systems. As an alternative, use Smale's alpha-theory refine approximate numerical and certify estimation.