Penalized smoothing of discrete distributions with sparse observations

Abstract

It happens quite often that we are faced with a sparse number of observations over a finite number of cells and we are interested in the estimation of the cell probabilities. The simple histogram produces approximations with the zero value for too many cells. Some polynomial smoothers have been proposed to circumvent this problem which show good properties in the analysis of such sparse situations but have the drawback of producing negative values. We propose a penalized polynomial smoothing for this problem. The estimators that are proposed in this paper are always positive and a simulation study show a very good behaviour with respect to the natural error criterias: mean squared sum of errors, sparse sup and the sup-norm. Our estimator perform specially well for sparse observations. Nevertheless, when the number of observations increases the proposed estimators still show good performance