Numerical enhancements for robust Rényi decomposable minimum distance estimators

Abstract

Different numerical aspects of Rényi pseudo-distance estimators are studied. These estimators are based on the minimization of information-theoretic divergences between empirical and hypothetical probability distributions. They are not classical distances, because the symmetry or triangle inequality does not hold. Robust properties of the minimum Rényi pseudodistance estimators are required by various applications in mathematical modeling, physics, or material science. Therefore we model the distribution of contaminated data as a mixture of the true distributions P and error distribution Q under different contamination level ε. We focus on the estimators for relatively small data samples or very sparse and scattered data with high variance, which appears mostly in high energy physics (signal and sparse background). In this case, the strict minimization leads to delta functions and it is impossible to obtain satisfactory numerical results. A way of adjusting the Rényi minimum distance estimators to these conditions is proposed. This so called ‘blurring’ is created as a convolution of Rényi distance with averaging Gaussian mask. Simultaneously, the effect of the input parameter alpha to the robustness is presented based on Monte-Carlo simulations for Gaussian model. Thus the Rényi distance is ready to be used in divergence decision trees for the signal versus background separations, e.g. in high energy physics NOvA or DUNE experiments at Fermilab.