Exact Solutions in LogConcave Maximum Likelihood Estimation
Abstract
We study probability density functions that are logconcave. Despite the space of all such densities being infinitedimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized WLambert function. Even more, we show that finding the logconcave maximum likelihood estimate is equivalent to solving a collection of polynomialexponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's alphatheory to refine approximate numerical solutions and to certify solutions to logconcave density estimation.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.04840
 Bibcode:
 2020arXiv200304840G
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Combinatorics;
 Mathematics  Optimization and Control;
 62R01;
 62G05;
 62G07
 EPrint:
 29 pages, 5 figures