Generalized Probability Density Function Estimation via Convex Optimization

A longstanding problem in statistics pertains to the estimation of probability density functions of continuous random variables from a finite set of their samples. In this paper, we propose a new parametric probability density function estimator based on convex programming. Our formulation decomposes the unknown distribution as a Gaussian penalty function plus an error function, which is then expanded by multi-scale wavelet functions (specifically frames) such as B-Spline and Mexican Hat wavelets. To recover the wavelet coefficients in the error function, a convex quadratic program is formulated which takes into account the positivity of the probability density function-through a linear constraint. The proposed decomposition model is shown to facilitate an accurate estimation of the probability density functions of interest.