Juan A. Cuesta-Albertos

The random Tukey depth

The computation of the Tukey depth, also called halfspace depth, is very demanding, even in low dimensional spaces, because it requires that all possible one-dimensional projections be considered. A random depth which approximates the Tukey depth is proposed. It only takes into account a finite number of one-dimensional projections which are chosen at random. Thus, this random depth requires a reasonable computation time even in high dimensional spaces. Moreover, it is easily extended to cover the functional framework. Some simulations indicating how many projections should be considered depending on the kind of problem, sample size and dimension of the sample space among others are presented. It is noteworthy that the random depth, based on a very low number of projections, obtains results very similar to those obtained with the Tukey depth.

DD-Classifier: Nonparametric Classification Procedure Based on DD-Plot

Using the DD-plot (depth vs. depth plot), we introduce a new nonparametric classification algorithm and call it DD-classifier. The algorithm is completely nonparametric, and it requires no prior knowledge of the underlying distributions or the form of the separating curve. Thus, it can be applied to a wide range of classification problems. The algorithm is completely data driven and its classification outcome can be easily visualized in a two-dimensional plot regardless of the dimension of the data. Moreover, it has the advantage of bypassing the estimation of underlying parameters such as means and scales, which is often required by the existing classification procedures. We study the asymptotic properties of the DD-classifier and its misclassification rate. Specifically, we show that DD-classifier is asymptotically equivalent to the Bayes rule under suitable conditions, and it can achieve Bayes error for a family broader than elliptical distributions. The performance of the classifier is also examined using simulated and real datasets. Overall, the DD-classifier performs well across a broad range of settings, and compares favorably with existing classifiers. It can also be robust against outliers or contamination.

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramér-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.

Convergence rates in nonparametric estimation of level sets

A level set of type {f[less-than-or-equals, slant]c} (where f is a density on and c is a positive value) can be estimated by its empirical version , where denotes a nonparametric (kernel) density estimator. We analyze, from two different points of view, the asymptotic behavior of the probability content of . Our results are motivated by applications in cluster analysis and outlier detection. Although the mathematical treatment is quite different in both cases, the conclusions are basically coincident. Roughly speaking, we show that the convergence rates are at most of type n-1/(d+2). For the univariate case d=1 this would be in the same spirit of the classical cube-root results found in some nonparametric setups.

The random projection method in goodness of fit for functional data

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.

Impartial trimmed k-means for functional data

A robust cluster procedure for functional data is introduced. It is based on the notion of impartial trimming. Existence and consistency results are obtained. Furthermore, a feasible algorithm is proposed and implemented in a real data example, where patterns of electrical power consumers are observed.

On the unconditional strong law of large numbers for the bootstrap mean

We first analyze some results by Athreya (1983) and Csorgo (1992). Then, by taking into account the different rates of convergence of the resampling size, we give new, simple proofs of those results. We provide examples that show that the sizes of resampling required by our results to ensure a.s. convergence are not far from being optimal.

Uniqueness and approximate computation of optimal incomplete transportation plans

For a given trimming level 2 (0,1) an trimmed version, P , of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according a positive weight function, f 1 1 , in the way P (B) = R B f(x)P(dx). If P,Q are probability measures on an euclidean space, we consider the optimization problem of obtaining the best L2 Wasserstein approximation between say a fixed probability and trimmed versions of the other, say trimmed versions of both probabilities. These best trimmed approximations naturally lead to new perspectives in the theory of Mass Transportation, where a part of the mass could be not necessarily transported. Since optimal transportation plans are not easily computable, we provide theoretical support for Monte-Carlo approximations, through a general consistency result. As a remarkable and unexpected additional result, with important implications for future work, we obtain the uniqueness of the optimal solution. Notice that such solution involves an optimal map T transporting some trimmed version P of P to some other Q of Q, thus for any point x in the support of P the weight function associated to P allows to partially or completely avoid the consideration of x in the transport. Our results show that in fact only the non-trimmed points (verifying f(x) = 1 1 ) are transported, while the partially trimmed points (verifying 0 < f(x) < 1 1 ) must remain untransported by T.

Trimmed Comparison of Distributions

This article introduces an analysis of similarity of distributions based on the L2-Wasserstein distance between trimmed distributions. Our main innovation is the use of the impartial trimming methodology, already considered in robust statistics, which we adapt to this setup. Instead of simply removing data at the tails to provide some robustness to the similarity analysis, we develop a data-driven trimming method aimed at maximizing similarity between distributions. Dissimilarity is then measured in terms of the distance between the optimally trimmed distributions. We provide illustrative examples showing the improvements over previous approaches and give the relevant asymptotic results to justify the use of this methodology in applications.

A characterization for the solution of the Monge-Kantorovich mass transference problem

Let P and Q be probabilities on the Borel [sigma]-algebra in a metric space (M, d). We prove that if the support of Q is finite and P verifies a certain continuity condition, then all the solutions of the Monge-Kantorovich mass transference problem between P and Q can be written as (X, H(X)) where X is any random element with distribution P and H only depends on P and Q.