Multivariate expectile trimming and the BExPlot

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble some similarities with the quantiles, and just like them, expectiles are indexed by a level α. In the present paper, we introduce and discuss the main properties of the expectile multivariate trimmed regions, a nested family of sets, whose instance with trimming level α is built up by all points whose univariate projections lie between the expectiles of levels α and 1 − α of the projected dataset. Such trimming level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We study here the convergence of the sample expectile trimmed regions to the population ones and the uniform consistency of the sample expectile depth. We also provide efficient algorithms for determining the extreme points of the expectile regions as well as for computing the depth of a point in R2. These routines are based on circular sequence constructions. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.This research was partially supported by the Spanish Ministry of Science and Innovation under grant ECO2015-66593-P

Data depth and multiple output regression, the distorted M-quantiles approach

For a univariate distribution, its M-quantiles are obtained as solutions to asymmetric minimization problems dealing with the distance of a random variable to a fixed point. The asymmetry refers to the different weights for the values of the random variable at either side of the fixed point. We focus on M-quantiles whose associated losses are given in terms of a power. In this setting, the classical quantiles are obtained for the first power, while the expectiles correspond to quadratic losses. The M-quantiles considered here are computed over distorted distributions, which allows to tune the weight awarded to the more central or peripheral parts of the distribution. These distorted M-quantiles are used in the multivariate setting to introduce novel families of central regions and their associated depth functions, which are further extended to the multiple output regression setting in the form of conditional regression regions and conditional depths

Expectile depth: Theory and computation for bivariate datasets

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble the quantiles, and just like them, expectiles are indexed by a level α in the unit interval. In the present paper, we introduce and discuss the main properties of the (multivariate) expectile regions, a nested family of sets, whose instance with level 0 < α ≤ 1/2 is built up by all points whose univariate projections lie between the expectiles of levels α and 1 − α of the projected dataset. Such level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We propose here algorithms for determining all the extreme points of the bivariate expectile regions as well as for computing the depth of a point in the plane. We also study the convergence of the sample expectile regions to the population ones and the uniform consistency of the sample expectile depth. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.This research was partially supported by the Spanish Ministry of Science and Innovation under grant ECO2015-66593-P

Los estadísticos dan guerra

Audiovisuales. Concurso Stat Wars. Disponible en http://www.youtube.com/watch?v=1a6J9KXyoQo .El pasado 13 de noviembre, en el marco de la Semana de la Ciencia, el departamento organizó la competición Stats Wars, una batalla estadística que congregó a mil asistentes, casi todos estudiantes de secundaria y bachillerato, en el auditorio de Leganés. El día 29, la universidad acogerá la jornada de Estadística de Leganés, que contará con otro concurso, Stats&Google, para estudiantes de grado.Contiene: ¿Es el año de la Estadística? Probablemente, sí (p.6) .-- Ser estadístico importa / Rosa Lillo (p.7) .-- Stat Wars,: que empiece el espectáculo (pp. .8-9) .-- La importancia de la estadística en lo cotidiano / Ignacio Cascos (p.9)

Models for expected returns with statistical factors

In this paper we propose factor-models assembled out of three new factors and evaluate them on European Equities. The new factors are built from statistical measurements on stock prices, in particular, coefficient of variation, skewness and kurtosis. Data come from Reuters, correspond to nearly 2000 EU companies and span from Jan-2008 to Feb-2018. Regarding methodology, we propose a non-parametric resampling procedure that accounts for time dependency in order to test the validity of the model and the significance of the parameters involved. We compare our bootstrap- based inferential results with classical proposals (based on F-statistics). Methods under assessment are Time-series regression, Cross-Sectional regression and the Fama-MacBeth procedure. The main findings indicate that the two factors that better improve the CAPM-model with regard to the adjusted R2 in the time-series regressions are the skewness and the cofficient of variation. For this reason, a model including those two factors together with the market is thoroughly studied.Research partially supported by ECO2015-66593-

Shuffle, cut, and learn: Crypto Go, a card game for teaching cryptography

This article belongs to the Special Issue Mathematical Modeling and Simulation in Science and Engineering Education II.Cryptography is the mathematical core of information security. It serves both as a source of hard computational problems and as precise language allowing for the formalization of sound security models. While dealing with the mathematical foundations of cybersecurity is only possible in specialized courses (tertiary level and beyond), it is essential to promote the role of mathematics in this field at early educational stages. With this in mind, we introduce Crypto Go, a physical card game that may be used both as a dissemination and as an educational tool. The game is carefully devised in order to entertain and stimulate players, while boosting their understanding on how basic cryptographic tools work and interplay. To get a preliminary assessment of our design, we collected data from a series of test workshops, which engaged over two hundred players from different ages and educational backgrounds. This basic evaluation indeed confirms that Crypto Go significantly improves students' motivation and has a positive impact in their perception and understanding of the field.The printouts of Crypto Go decks, and some of the experimental workshops described in this paper have been financially supported by several institutions: Instituto Nacional de Ciberseguridad (INCIBE; contract 2018/00520/001), Fundación Madri+d (Science Week), and Universidad Carlos III de Madrid (Technological Fridays). M.I.G.V.'s work is funded by the NATO Science for Peace and Security Programme, grant number G5448 and by MINECO under Grant MTM2016-77213-R

On the combination of depth-based ranks

The depth of a multivariate observation assesses its degree of centrality with respect to a probability distribution, and thus it can be interpreted as a measurement of the fit of the observation wrt the distribution. If such depth is transformed into a (depth-based) rank, then we obtain a kind of p-value of a goodness-of-fit test run on a single observation. For a sample of observations, the goal is to combine their ranks in order to decide whether they were taken from some prescribed distribution. From the meta-analysis literature, it is well known that there does not exist a combination procedure for such p-values (or ranks) that outperforms the remaining ones in all possible scenarios. Here we explore several combination procedures of the depth-based ranks and analyse their behaviour in the detection of some given shifts from a prescribed distribution.This work was started while Ignacio Montes was with the Department of Statistics of the Universidad Carlos III de Madrid. We acknowledge the financial support by projects ECO2015-66593 and TIN2014-59543-P

A stochastic order for random vectors and random sets based on the Aumann expectation

We de1ne a new stochastic order for random vectors in terms of the inclusion relation for the Aumann expectation of certain random sets. We derive some properties of this order, relate it with other well-known multivariate stochastic convex orders, give a geometrical interpretation in terms of the lift zonoid, build a probability metric based on it and extend it to compare random compact sets. Furthermore, a new characterisation of the linear convex order for random vectors is described and the corresponding concept is extended to compare random compact sets

How to explain the cross-section of equity returns through Common Principal Components

In this paper we propose a procedure to obtain and test multifactor models based on statistical and financial factors. In order to select the factors included in the model,as well as the construction of the portfolios, we use a multivariate technique called Common Principal Components. A block-bootstrap methodology is developed to assess the validity of the model and the significance of the parameters involved. Data come from Reuters, correspond to nearly 1250 EU companies, and span from October 2009 to October 2019. We also compare our bootstrap-based inferential results with those obtained via classical testing proposals. Methods under assessment are time-series regression and cross-sectional regression. The main findings indicate that the multifactor model proposed improves the Capital Asset Pricing Model with regard to the adjusted-R2 in the time-series regressions. Cross-section regression results reveal that Market and a factor related to Momentum and mean of stocks' returns have positive risk premia for the analysed period. Finally, we also observe that tests based onblock-bootstrap statistics are more conservative with the none than classical procedures

A stochastic order for random vectors and random sets based on the Aumann expectation

We define a new stochastic order for random vectors in terms of the inclusion relation for the Aumann expectation of certain random sets. We derive some properties of this order, relate it with other well-known multivariate stochastic convex orders, give a geometrical interpretation in terms of the lift zonoid, build a probability metric based on it and extend it to compare random compact sets. Furthermore, a new characterisation of the linear convex order for random vectors is described and the corresponding concept is extended to compare random compact sets.Stochastic order Convex order Random set Aumann expectation Lift zonoid