Eustasio del Barrio

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.

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.

Similarity of samples and trimming

We say that two probabilities are similar at levelif they are contaminated versions (up to anfraction) of the same common probability. We show how this model is related to minimal distances between sets of trimmed probabilities. Empirical versions turn out to present an over- fitting effect in the sense that trimming beyond the similarity level results in trimmed samples that are closer than expected to each other. We show how this can be combined with a bootstrap approach to assess similarity from two data samples.

Asymptotic stability of the bootstrap sample mean

The asymptotic distribution of the bootstrap sample mean depends on the resampling intensity. This paper explores the sensitivity of that distribution against different resampling intensities. It is generally assumed that small resampling sizes make the bootstrap work. However, we will show that the bootstrap mean can only be highly unstable for small resampling intensities. Our setup considers resampling from a triangular array of row-wise independent and identically distributed random variables satisfying the Central Limit Theorem.

Wide consensus aggregation in the Wasserstein space. Application to location-scatter families

We introduce a general theory for a consensus-based combination of estimations of probability measures. Potential applications include parallelized or distributed sampling schemes as well as variations on aggregation from resampling techniques like boosting or bagging. Taking into account the possibility of very discrepant estimations, instead of a full consensus we consider a “wide consensus” procedure. The approach is based on the consideration of trimmed barycenters in the Wasserstein space of probability measures. We provide general existence and consistency results as well as suitable properties of these robustified Fréchet means. In order to get quick applicability, we also include characterizations of barycenters of probabilities that belong to (non necessarily elliptical) location and scatter families. For these families we provide an iterative algorithm for the effective computation of trimmed barycenters, based on a consistent algorithm for computing barycenters, guarantying applicability in a wide setting of statistical problems. AMS Subject Classification: Primary: 60B05, 62F35, Secondary 62H12.

The empirical cost of optimal incomplete transportation

We consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the d-dimensional unit cube [0,1]d and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. We show that the empirical cost of optimal incomplete transportation vanishes at rate OP(n−1/d), where n denotes the sample size. In dimension d≥3 the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate.

An Optimal Transportation Approach for Assessing Almost Stochastic Order

When stochastic dominance does not hold, we can improve agreement to stochastic order by suitably trimming both distributions. In this work we consider the \(L_2\)–Wasserstein distance, \(\mathscr {W}_2\), to stochastic order of these trimmed versions. Our characterization for that distance naturally leads to consider a \(\mathscr {W}_2\)-based index of disagreement with stochastic order, \(\varepsilon _{\mathscr {W}_2}(F,G)\). We provide asymptotic results allowing to test \(H_0: \varepsilon _{\mathscr {W}_2}(F,G)\ge \varepsilon _0\) versus \(H_a: \varepsilon _{\mathscr {W}_2}(F,G)<\varepsilon _0\), that, under rejection, would give statistical guarantee of almost stochastic dominance. We include a simulation study showing a good performance of the index under the normal model.