Fast DD-classification of functional data

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study

Price majorization and the inverse Lorenz function

The paper presents an approach to the measurement of economic disparity in several commodities. We introduce a special view on the usual Lorenz curve and extend this view to the multivariate situation: Given a vector of shares of the total endowments in each commodity, the multivariate inverse Lorenz function (ILF) indicates the maximum percentage of the population by which these shares or less are held. Its graph is the Lorenz hypersurface. Many properties of the ILF are studied and the equivalence of the pointwise ordering of ILFs and the price Lorenz order is established. We also study similar notions for distributions of absolute endowments. Finally, several disparity indices are suggested that are consistent with these orderings. --Multivariate Lorenz order,directional majorization,price Lorenz order,generalized Lorenz function,multivariate disparity indices

Can Markov-regime switching models improve power price forecasts? Evidence for German daily power prices

Nonlinear autoregressive Markov regime-switching models are intuitive and frequently proposed time series approaches for the modelling of electricity spot prices. In this paper such models are compared to an ordinary linear autoregressive model with regard to their forecast performance. The study is carried out using German daily spot prices from the European Energy Exchange in Leipzig. Four nonlinear models are used for the forecast study. The resultsof the study suggest that Markov regime-switching models provide better forecasts than linear models. --Electricity spot prices,Markov regime-switching,forecasting

An exact algorithm for weighted-mean trimmed regions in any dimension

Trimmed regions are a powerful tool of multivariate data analysis. They describe a probability distribution in Euclidean d-space regarding location, dispersion, and shape, and they order multivariate data with respect to their centrality. Dyckerhoff and Mosler (201x) have introduced the class of weighted-mean trimmed regions, which possess attractive properties regarding continuity, subadditivity, and monotonicity. We present an exact algorithm to compute the weighted-mean trimmed regions of a given data cloud in arbitrary dimension d. These trimmed regions are convex polytopes in Rd. To calculate them, the algorithm builds on methods from computational geometry. A characterization of a region's facets is used, and information about the adjacency of the facets is extracted from the data. A key problem consists in ordering the facets. It is solved by the introduction of a tree-based order. The algorithm has been programmed in C++ and is available as an R package. --central regions,data depth,multivariate data analysis,convex polytope,computational geometry,algorithm,C++, R

Representative endowments and uniform Gini orderings of multi-attribute welfare

For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Spectral social evaluation functions are used in the multivariate setting, and Gini dominance orderings are introduced that are uniform in attribute weights. Classes of spectral evaluators are considered that are ordered by their aversion to inequality. Then a set-valued representative endowment is defined that characterizes $d$-dimensioned welfare. It consists of all points above the lower border of a convex compact in $R^d$, while the pointwise ordering of such endowments corresponds to uniform Gini dominance. An application is given to the welfare of 28 European countries. Properties of uniform Gini dominance are derived, including relations to other orderings of $d$-variate distributions such as convex and dependence orderings. The multi-dimensioned representative endowment can be efficiently calculated from data; in a sampling context, it consistently estimates its population version.Comment: 19 pages. A previous version was titled "Representative endowments and uniform Gini orderings of multi-attribute inequality

Fast computation of Tukey trimmed regions and median in dimension p>2p>2

Given data in $\mathbb{R}^{p}$, a Tukey $\kappa$-trimmed region is the set of all points that have at least Tukey depth $\kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension $p > 2$. We construct two novel algorithms to compute a Tukey $\kappa$-trimmed region, a na\"{i}ve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the na\"{i}ve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median

Checking for orthant orderings between discrete multivariate distributions: An algorithm

We consider four orthant stochastic orderings between random vectors X and Y that have finitely discrete probability distributions in IRk. For each of the orderings conditions have been developed that are necessary and sufficient for dominance of Y over X. We present an algorithm that checks these conditions in an efficient way by operating on a semilattice generated by the support of the two distributions. In particular, the algorithm can be used to compute multivariate Smirnov statistics. --Multivariate stochastic orders,decision under risk,comparison of empirical distribution functions

A power comparison of homogeneity tests in mixtures of exponentials

The empirical power of several test procedures is studied which test for homogeneity against mixtures of exponential distributions. --