In nonparametric statistics one abandons the requirement that a probability density function belongs to a statistical model. Given a sample of points, we want to find the best log-concave distribution that fits this data. This problem was originally studied by statisticians, who found that the optimal solutions have probability density functions whose logarithm is piecewise-linear and used numerical methods to approximate the pdf. In this work we use algebraic and combinatorial methods to provide exact solutions to this problem. At the same time we use tools from algebraic geometry to test if the solutions provided by statistical software can be certified to be correct.

How many regions of linearity does the depicted tent function have?

Algebraic methods can find wide use in parameter estimation from statistical moments. In this work we use tools from elimination theory, the extended Prony method, and numerical algebra to reconstruct parameters from mixtures of local Diracs. Computational algebra mixes successfully with applications in statistics and signal processing.

The picture shows the moment hypersurface of a first order local Dirac mixture, i.e. the variety parametrized by the moments. It is the tangent to the twisted cubic, in the same way that regular mixing gives rise to secants.