Combining regular and irregular histograms by penalized likelihood

Abstract

A fully automatic procedure for the construction of histograms is proposed. It consists of constructing both a regular and an irregular histogram and then choosing between the two. For the regular histogram, only the number of bins has to be chosen. Irregular histograms can be constructed using a dynamic programming algorithm if the number of bins is known. To choose the number of bins, two different penalties motivated by recent work in model selection are proposed. A complete description of the algorithm and a proper tuning of the penalties is given. Finally, different versions of the procedure are compared to other existing proposals for a wide range of densities and sample sizes. In the simulations, the squared Hellinger risk of the procedure that chooses between regular and irregular histograms is always at most twice as large as the risk of the best of the other methods. The procedure is implemented in an R-Package.

Development and implementation in the Monte Carlo code PENELOPE of a new virtual source model for radiotherapy photon beams and portal image calculation

“…The following algorithm which solves the (-L 1 ) minimization problem is based on a linear programming method (Rozenholc et al, 2010). We can distinguish three main steps:…”

Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

“…We start by constructing a uniform partition with intervals N that maximizes the following penalized log-likelihood function [42] …”

ThebenchdenPackage: Benchmark Densities for Nonparametric Density Estimation

“…It should be rich enough to choose a subset of interesting cases for most applications, depending on the goal of estimation and the methods under consideration. Additionally, the benchden package contains four histogram densities which we will not describe here, but see Rozenholc, Mildenberger, and Gather (2009) for details.…”

“…Our package benchden (Mildenberger, Weinert, and Tiemeyer 2012), which is described in this article, aims at closing this gap. It implements the set of 28 test bed densities first introduced by Berlinet and Devroye (1994) and since used in Devroye (1997) and Rozenholc, Mildenberger, and Gather (2010) in R. This set of 28 densities is sufficiently large to cover a wide variety of situations that are of interest for the comparison of different methods. Unlike the densities proposed by Marron and Wand (1992), which vary greatly in shape but are all normal mixtures, these densities also differ widely in their mathematical properties such as smoothness or tail behavior and even include some densities with infinite peaks that are not square-integrable.…”