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 di erent 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, di erent 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.