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Spanish Ministry of Science and Innovation [MTM2017-83506-C2-2-P, MTM-PID2019-104486GB-I0 0]

A new concept of depth for central regions is introduced. The proposed depth notion assesses how well an interval fits a given univariate distribution as its zonoid region of level 1/2, and it is extended to the multivariate setting by means of a projection argument. Since central regions capture information about location, scatter, and dependency among several variables, the new depth evaluated on an empirical zonoid region quantifies the degree of similarity (in terms of the features captured by central regions) of the corresponding sample with respect to some reference distribution. Applications to statistical process control and the joint monitoring of multivariate and interval-valued data in terms of location and scale are presented.

A novel notion of depth for the origin and scale of a distribution with left-bounded support, like those modelling lifetime data, is presented. This origin-scale depth, evaluated on parameter estimates, serves as plotting statistic of a Shewhart-type chart for rational samples of a fixed size for which three variations are considered: a nonparametric version, built without distributional assumptions, and parametric and semiparametric versions, built for shifted exponential processes (paradigmatic model governed by origin and scale parameters). An EWMA scheme based on the rank induced by the origin-scale depth is also presented and, together with the Shewhart-type chart, compared with other existing monitoring proposals. The new proposals are competitive when the reference distribution is a shifted exponential one, while outperform them for other reference distributions.

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.

In this paper, we propose multifactor models for the pan-European Equity Market using a block-bootstrap method and compare the results with those of traditional inferential techniques. The new factors are built from statistical measurements on stock prices&mdash, in particular, coefficient of variation, skewness, and kurtosis. Data come from Reuters, correspond to nearly 2000 EU companies, and span from January 2008 to February 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&ndash, MacBeth procedure. The main findings indicate that the two factors that better improve the Capital Asset Pricing Model with regard to the adjusted R2 in the time-series regressions are the skewness and the coefficient of variation. For this reason, a model including those two factors together with the market is thoroughly studied. We also observe that our block-bootstrap methodology seems to be more conservative with the null of the GRS test than classical procedures.

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&rsquo, motivation and has a positive impact in their perception and understanding of the field.

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.

Control charts are designed to monitor on-going production processes by tracking subsequent samples of the production using some statistic of a quality characteristic. We propose to track the parameter depths of estimates of a parameter by means of depth (D) charts, or the associated depth-based ranks by means of r-charts. More precisely, given a general parameter (e.g. mean, standard deviation or pair given by mean and standard deviation) and some historical data of the production, the parameter depth of an estimate of the parameter on new samples of the production with regard to the historical data is computed. The process is considered to be out-of-control when the depth of the estimate of the parameter falls below some given threshold (control limit). Some control limits of specific D-charts are obtained under the assumption of normality of the quality characteristic. (C) 2017 Elsevier Inc. All rights reserved. The authors gratefully acknowledge the support of the Spanish Ministry of Science and Innovation under Grants ECO2015-66593 and MTM2015-63971-P and the Principado de Asturias under grant FC-15-GRUPIN14-101.

Bands of vector-valued functions \(f:T\mapsto \mathbb {R}^d\) are defined by considering convex hulls generated by their values concatenated at m different values of the argument. The obtained m-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for \(m=1\)) to the convex hull in the functional space if the number m of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set-valued portfolios. This situation includes the classicalKabanov’s transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov’s exchange cone model, it is shown how the selection riskmeasure relates to the set-valued riskmeasures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ.Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).

We describe several simulation algorithms that yield random probability distributions with given values of risk measures. In case of vanilla risk measures, the algorithms involve combining and transforming random cumulative distribution functions or random Lorenz curves obtained by simulating rather general random probability distributions on the unit interval. A new algorithm based on the simulation of a weighted barycentres array is suggested to generate random probability distributions with a given value of the spectral risk measure.

Under some mild conditions on probability distribution P, if lim(n) P-n = P weakly then the sequence of zonoid depth functions with respect to P-n converges uniformly to the zonoid depth function with respect to P. (C) 2015 Elsevier Inc. All rights reserved. Research partially supported by grants MTM2011-22993 and ECO2011-25706 of the Spanish Ministry of Science and Innovation and grant FC-15-GRUPIN14-101 of the Principado de Asturias.

Consider any kind of parameter for a probability distribution and a fixed distribution. We study the subsets of the parameter space constituted by all the parameters of the probabilities in the α-trimming of the fixed distribution. These sets will be referred to as parameter trimmed regions. They are composed of all parameter candidates whose degree of suitability as such a parameter for the distribution is, at least, a specific value α.In particular, we analyze location, scale, and location-scale parameters and study the properties of the trimmed regions induced by them. Several specific examples of parameter trimmed regions are studied. Among them, we should mention the zonoid trimmed regions obtained when the chosen parameter is the mean value and the location-scale regions of a univariate distribution obtained when the parameter is the pair given by the mean and the standard deviation.

This paper aims to quantify the proximity of the distribution of a random vector in ℝd to the distribution of a random set also in ℝd. The latter is characterized by a capacity functional, so we introduce the notion of proximity function induced by a capacity functional as a mapping from the space of the probabilities on the Borel σ-algebra on ℝd, which assigns to each probability its degree of proximity to the capacity functional. Two particular proximity functions are considered and their properties are thoroughly analyzed, placing particular emphasis on their asymptotic behavior. An example where proximity functions are used to rank probabilities with respect to a capacity functional is discussed.

We define a new family of central regions with respect to a probability measure. They are induced by a set or a family of sets of functions and we name them integral trimmed regions. The halfspace trimming and the zonoid trimming are particular cases of integral trimmed regions. We focus our work on the derivation of properties of such integral trimmed regions from conditions satisfied by the generating classes of functions. Further we show that, under mild conditions, the population integral trimmed region of a given depth can be characterized in terms of certain regions based on empirical distributions.

In this paper we introduce an interval-valued inequality index for random intervals based on a convex function. We show that if this function does not grow faster than x p , then the inequality index is continuous on the space of random intervals with finite p-th moment. A bound for the distance between the inequality indices of two random intervals is also constructed. An example is presented to motivate and illustrate the developments in this paper.

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.

We construct new risk functionals and stochastic orderings based on integrals using discontinuous distortion functions. We prove a result on the subadditivity of the risk functional. As to the stochastic order we give a characterization of this order which allows us to construct spectral risk measures that are consistent with that order.

We study the Generalized Lorenz curves of the minima of samples from a random variable. These Lorenz curves can be used to compare distributions in terms of their variability and define coherent risk measures. The dual Lorenz curves of the maxima of samples from the random variable are also considered.

The sequence of $\alpha$-trimmings of empirical probabilities is shown to converge, in the Painlev\'{e}--Kuratowski sense, on the class of probability measures endowed with the weak topology, to the $\alpha$-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ109 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship., Comment: 26 pages. Substantially revised version with a number of new results added

We study the properties of the zonoid hull of the probability distribution of a random vector with finite first moment. The zonoid hull was first considered in [1] as a convex body whose inclusion relation characterizes the symmetric stochastic order. A family of trimmed regions, namely the zonoid hull trimmed regions, is obtained from the zonoid hull, the same way zonoid trimmed regions are obtained from the lift zonoid. Further a depth-type function induced by these trimmed regions is also considered.