Thoralf Mildenberger

Combining regular and irregular histograms by penalized likelihood

A new 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. To choose the number of bins in the irregular histogram, two different penalties motivated by recent work in model selection are proposed. A 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 new procedure is always at most twice as large as the risk of the best of the other methods. The procedure is implemented in the R-Package histogram available from CRAN.

Residual-based localization and quantification of peaks in X-ray diffractograms

We consider data consisting of photon counts of diffracted x-ray radiation as a function of the angle of diffraction. The problem is to determine the positions, powers and shapes of the relevant peaks. An additional difficulty is that the power of the peaks is to be measured from a baseline which itself must be identified. Most methods of de-noising data of this kind do not explicitly take into account the modality of the final estimate. The residual-based procedure we propose uses the so-called taut string method, which minimizes the number of peaks subject to a tube constraint on the integrated data. The baseline is identified by combining the result of the taut string with an estimate of the first derivative of the baseline obtained using a weighted smoothing spline. Finally, each individual peak is expressed as the finite sum of kernels chosen from a parametric family.

A note on the geometry of the multiresolution criterion

Several recent developments in nonparametric regression are based on the concept of data approximation: They aim at finding the simplest model that is an adequate approximation to the data. Approximations are regarded as adequate iff the residuals ?look like noise?. This is usually checked with the so-called multiresolution criterion. We show that this criterion is related to a special norm (the ?multiresolution norm?), and point out some important differences between this norm and the p-norms often used to measure the size of residuals. We also treat an important approximation problem with regard to this norm that can be solved using linear programming. Finally, we give sharp upper and lower bounds for the multiresolution norm in terms of p-norms.

Constructing irregular histograms by penalized likelihood

We propose a fully automatic procedure for the construction of irregular histograms. For a given number of bins, the maximum likelihood histogram is known to be the result of a dynamic programming algorithm. To choose the number of bins, we propose two different penalties motivated by recent work in model selection by Castellan [6] and Massart [26]. We give a complete description of the algorithm and a proper tuning of the penalties. Finally, we compare our procedure to other existing proposals for a wide range of different densities and sample sizes.

A geometric interpretation of the multiresolution criterion in nonparametric regression

A recent approach to choosing the amount of smoothing in nonparametric regression is to select the simplest estimate for which the residuals ‘look like white noise’. This can be checked with the so-called multiresolution criterion, which Davies and Kovac [P.L. Davies and A. Kovac, Local extremes, runs, strings and multiresolutions (with discussion and rejoinder), Ann. Stat. 29 (2001), pp. 1–65.] introduced in connection with their taut-string procedure. It has also been used in several other nonparametric procedures such as spline smoothing or piecewise constant regression. We show that this criterion is related to a norm, the multiresolution norm (MR-norm). We point out some important differences between this norm and p-norms. The MR-norm is not invariant w.r.t. sign changes and permutations, and this makes it useful for detecting runs of residuals of the same sign. We also give sharp upper and lower bounds for the MR-norm in terms of p-norms.

The benchden package

June 26, 2007 Type Package Title 28 benchmark densities from Berlinet/Devroye (1994) Version 1.0.0 Date 2007-06-26 Author Thoralf Mildenberger, Henrike Weinert, Sebastian Tiemeyer Maintainer Thoralf Mildenberger Description Full implementation of the 28 distributions introduced as benchmarks for nonparametric density estimation by Berlinet and Devroye (1994). Includes densities, cdfs, quantile functions and generators for samples. License GPL version 2 or newer