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This paper focuses on the analysis of open-ended questions answered in different languages. Closed-ended questions, called contextual variables, are asked to all respondents in order to understand the relationships between the free and the closed responses among the different samples since the latter assumably affect the word choices. We have developed "Multiple Factor Analysis on Generalized Aggregated Lexical Tables" (MFA-GALT) to jointly study the open-ended responses in different languages through the relationships between the choice of words and the variables that drive this choice. MFA-GALT studies if variability among words is structured in the same way by variability among variables, and inversely, from one sample to another. An application on an international satisfaction survey shows the easy-to-interpret results that are proposed., Comment: 18 pages

Comparisons of different treatments or production processes are the goals of a significant fraction of applied research. Unsurprisingly, two-sample problems play a main role in Statistics through natural questions such as `Is the the new treatment significantly better than the old?'. However, this is only partially answered by some of the usual statistical tools for this task. More importantly, often practitioners are not aware of the real meaning behind these statistical procedures. We analyze these troubles from the point of view of the order between distributions, the stochastic order, showing evidence of the limitations of the usual approaches, paying special attention to the classical comparison of means under the normal model. We discuss the unfeasibility of statistically proving stochastic dominance, but show that it is possible, instead, to gather statistical evidence to conclude that slightly relaxed versions of stochastic dominance hold.

We develop a general theory to address a consensus-based combination of estimations in a parallelized or distributed estimation setting. 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 distributions on R^d with finite second order moments. We include general existence and consistency results as well as characterizations of barycenters of probabilities that belong to (non necessarily elliptical) location and scatter familes. On these families, the effective computation of barycenters and distances can be addressed through a consistent iterative algorithm. Since, once a shape has been chosen, these computations just depend on the locations and scatters, the theory can be applied to cover with great generality a wide consensus approach for location and scatter estimation or for obtaining confidence regions., Comment: 27 pages, 3 fogures

Let $\mathcal{P}_{2,ac}$ be the set of Borel probabilities on $\mathbb{R}^d$ with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of finding the barycenter (or Fr\'echet mean) of a finite set of probabilities $\nu_1,\ldots,\nu_k \in \mathcal{P}_{2,ac}$ with respect to the $L_2-$Wasserstein metric. For this task we introduce an operator on $\mathcal{P}_{2,ac}$ related to the optimal transport maps pushing forward any $\mu \in \mathcal{P}_{2,ac}$ to $\nu_1,\ldots,\nu_k$. Under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter. The procedure allows effective computation of barycenters in any location-scatter family, including the Gaussian case. In such cases the barycenter must belong to the family, thus it is characterized by its mean and covariance matrix. While its mean is just the weighted mean of the means of the probabilities, the covariance matrix is characterized in terms of their covariance matrices $\Sigma_1,\dots,\Sigma_k$ through a nonlinear matrix equation. The performance of the iterative procedure in this case is illustrated through numerical simulations, which show fast convergence towards the barycenter., Comment: 18 pages, 2 figures

An algorithm for determining stationary periods for time series of random sea waves is proposed in this work. This is a problem in which changes between stationary sea states are usually slow and segmentation procedures based on change-point detection frequently give poor results. The method is based on a new procedure for time series clustering, built on the use of the total variation distance between normalized spectra as a measure of dissimilarity. The oscillatory behavior of the series is thus considered the central characteristic for classification purposes. The proposed algorithm is compared to several other methods which are also based on features extracted from the original series and the results show that its performance is comparable to the best methods available and in some tests it performs better. This clustering method may be of independent interest.

Stochastic ordering among distributions has been considered in a variety of scenarios. Economic studies often involve research about the ordering of investment strategies or social welfare. However, as noted in the literature, stochastic orderings are often a too strong assumption which is not supported by the data even in cases in which the researcher tends to believe that a certain variable is somehow smaller than other. Instead of considering this rigid model of stochastic order we propose to look at a more flexible version in which two distributions are said to satisfy an approximate stochastic order relation if they are slightly contaminated versions of distributions which do satisfy the stochastic ordering. The minimal level of contamination that makes this approximate model hold can be used as a measure of the deviation of the original distributions from the exact stochastic order model. Our approach is based on the use of trimmings of probability measures. We discuss the connection between them and the approximate stochastic order model and provide theoretical support for its use in data analysis. We also provide simulation results.

We say that two probabilities are similar at level $\alpha$ if they are contaminated versions (up to an $\alpha$ fraction) 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 overfitting 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., Comment: Published in at http://dx.doi.org/10.3150/11-BEJ351 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)