Scientific Production

Vertical Barrier Models (VBM) are a family of imprecise probability models that generalise a number of well known distortion/neighbourhood models (such as the Pari-Mutuel Model, the Linear-Vacuous Model, and others) while still being relatively simple. Several of their properties were established in previous works; in this paper we explore, in a finite framework, further facets of these models: their interpretation as neighbourhood models, the structure of their credal set in terms of maximum number of its extreme points, the result of merging operations with VBMs, the properties of their mass function, the conditions for VBMs to be belief functions or maxitive measures and the approximation of other models by VBMs.

The Lilliefors test of normality is a popular and easy-to-explain method for testing whether a sample comes from a normal distribution. Unfortunately, since it relies on the sample mean and sample standard deviation for estimating the parameters of the normal distribution, the Lilliefors test is quite sensitive to the presence of outliers. Contrarily to what could be expected, the substitution of the estimators of location and scale by robust alternatives still does not suffice for obtaining a robust method when either the number of outliers or the sample size is large. In this paper, we propose an actual robust alternative relying on classical and data-driven trimming techniques. The presented test depends on the choice of a subsetting technique, of which we explore three possibilities, and of one parameter, which models the robustness of the test in the presence of outliers. As expected, the choice of parameter is a delicate issue since the gain in robustness comes at the price of a reduced power against some types of alternatives.

We focus on the notion of negative conglomerability. This is far less known than its counterpart, conglomerability. Both relate to the combination of conditional and unconditional information, with the latter taking in particular a foundational role in the special case of infinite partitions of the possibility space. The two notions look superficially very similar and are even equivalent in the case of precise probabilistic models. In the present paper, we do a thorough technical study of their relations with other main concepts in the literature, such as marginal extension and dilation, both in the precise and imprecise case. Moreover, we discuss why they are somewhat surprisingly different from the prescriptive point of view, in that conglomerability has a rationality stance that its negative counterpart has not.

We consider a decision making problem under imprecision, where the probabilistic information is given in terms of a set of probability measures, and where finding the optimal alternative(s) may be difficult. To ease the computation, we propose to transform the initial model into another one that (i) belongs to some subclass with better mathematical properties, such as supermodularity or complete monotonicity; (ii) is at least as informative as the original model, while being as close as possible to it. We show that the problem can be approached in terms of linear or quadratic programming and that it can be connected with the one of determining the incenter of a credal set. Finally, we compare the solutions of a decision making problem with the initial and the transformed models and illustrate how our approach can be applied in a decision making problem under severe uncertainty.

The law of iterated expectation tells us how to combine hierarchical pieces of information when our uncertainty is modelled by means of probability measures. It has been extended to the imprecise case through Walley’s marginal extension theorem for coherent lower previsions. In this paper, we investigate the extent to which a similar result can be established for other imprecise probability models that are either more general (choice functions) or more particular (possibility measures, belief functions) than coherent lower previsions. By doing this, we also establish links with other results established in the literature in the context of imprecise versions of Jeffrey’s rule.

The present paper presents some results that allow us to perform with High Relative Accuracy linear algebra operations with correlation and covariance matrices of Gaussian Markov Random Fields over graphs of paths. Some numerical experiments are carried out showing the computational benefits of this approach.

Gaussian Markov Random Fields are a popular statistical model that has been used successfully in many fields of application. Recent work has studied conditions under which the covariance matrix of a Gaussian Markov Random Field over a graph of paths is totally positive. In such case, many linear algebra operations concerning the covariance matrix can be performed with High Relative Accuracy (the relative error is of order of machine precision). Unfortunately, classical estimators of the covariance matrix do not necessarily yield a totally positive matrix, even when the population covariance matrix is totally positive. Essentially, this inconvenience prevents the available High Relative Accuracy methods to be used with real-life data. Here, we present a method for the estimation of the covariance matrix of a Gaussian Markov Random Field over a graph of paths assuring the estimated covariance matrix (or its inverse) is totally positive.

In recent years, the construction and characterization of certain uninorms on bounded lattices have been exhaustively studied. In this paper, we study the structure of the set of elements incomparable with the neutral element, and provide a taxonomy of different types of bounded lattices. Moreover, we present different methods for constructing uninorms on some of these types of bounded lattices. The presented construction methods extend existing construction methods in the sense that the constructed uninorm may take a more general range of values on the set of elements incomparable with the neutral element.

A common approach used in robust statistics to robustify a probabilistic model is to distort a probability measure or to create a neighbourhood around it with a given radius and with respect to an appropriate distorting function. This approach establishes a clear connection with lower probabilities, also referred to as non-additive measures or capacities, which serve as tools to model uncertainty in a probability measure and are formally equivalent to normalised coalitional games. In this contribution, we take this idea a step further by analysing the problem of directly distorting a lower probability. For this purpose, we introduce in first place a formal definition of a generic distortion procedure and examine some desirable properties such a procedure may satisfy. Afterwards, we focus particularly on the distortion procedure based on the total variation distance and investigate the properties it satisfies. Finally, we demonstrate that the distortion of lower probabilities has a clear interpretation from the perspective of coalitional games, showing that the distortion based on the total variation distance aligns with a procedure commonly known as strong-δ-core, used to relax the constraints imposed by coalitions in order to ensure the non-emptiness of the core.

We consider the problem of aggregating belief models elicited by experts when these are expressed by means of coherent lower previsions. These constitute a framework general enough so as to include as particular cases not only probability measures but also the majority of models from imprecise probability theory. Although the aggregation problem has already been tackled in the literature, our contribution provides a unified view by gathering a number of rationality criteria and aggregation rules studied in different papers. Specifically, we consider six aggregation rules and twenty rationality criteria. We exhaustively analyse the relationships between the rules, the properties satisfied by each rule and the characterisations of the rules in terms of the criteria.

The Total Variation is a common distance between probability distributions that measures the maximum difference in probability among all events. When comparing non-additive measures, that can be represented by closed and convex sets of probability measures, the Total Variation distance can be extended in multiple ways. This paper explores three specific approaches in detail. The first approach considers the minimum Total Variation distance between the probability measures dominating the non-additive measures. The second approach replaces the minimum with the maximum of the Total Variation distances. The third approach modifies the first one by using the supremum distance instead of the Total Variation. We analyse the main properties of each approach and, in particular, apply them to distort non-additive measures. Finally, we demonstrate that the distortion of non-additive measures with the proposed extensions are closely connected to the strong and weak cores in coalition game theory.

Robust statistics aims to develop methods that ensure the obtained results remain stable regardless of the quality of the data. A natural approach to enhancing the robustness of a probabilistic model is to consider a neighbourhood around a probability measure with a given radius. These models require two tools: a function for comparing probability measures and the radius measuring the level of imprecision or robustness to be added to the model. With these tools, a neighbourhood or distortion model is defined as the closed ball centred at the probability measure. Many well-known robust models, such as the ϵ-contamination, pari-mutuel model, or constant odds ratio, can be incorporated into the broader framework of neighbourhood models. Additionally, distances between probability measures, such as the total variation or the Kolmogorov distances, determine a neighbourhood model. This paper delves into the neighbourhood models induced by the Euclidean distance applied to the probability mass functions and the Kullback-Leibler divergence. For investigating these models from the perspective of imprecise probability theory, we make use of common optimisation tools with restrictions. We conclude this study providing a comparison study among different neighbourhood models.

The process of measuring a continuous variable is typically subject to imprecision due to causes as varied as measurement errors or rounding, and it is the duty of practitioners to take account of this imprecision when performing different statistical inference tasks. In this paper, we address the problem of accounting for this imprecision in the context of confidence intervals for parameters of probability distributions. For such purpose, we formalize the notions of inner and outer confidence interval, both of which generalize the classical notion of confidence interval in the presence of imprecision. Different properties of both mathematical constructs are here studied and their explicit expressions are provided in five prominent cases in the field of Statistics.

The Lilliefors normality test is a classical extension of the Kolmogorov–Smirnov goodness-of-fit test tailored to assessing normality. A recent modification improves its robustness to outliers by introducing a subsetting function. In this paper, we propose an alternative approach that replaces subsetting functions with Ordered Weighted Averaging (OWA) functions and further generalizes the test to any location-scale family, not only the normal distribution. We conduct extensive experiments on the power of the resulting test for three representative location-scale families -normal, uniform and shifted-exponential- using different weight vectors to define the OWA functions. The results indicate that the best trade-off between robustness and statistical power is achieved by a well-known special class of OWA functions: the order statistics.

In many statistical applications, numerical computations with covariance matrices need to be performed. The error made when performing such numerical computations increases with the condition number of the covariance matrix, which is related to the number of variables and the strength of the correlation between the variables. In a recent work, a method for estimating the covariance matrix of a Gaussian Markov Random Field under a total positivity constraint was proposed. This estimation allows for performing many numerical computations with covariance matrices to high relative accuracy (the relative error is of the order of machine precision). However, the necessary conditions for this estimation method to produce a covariance matrix that is close to the population covariance matrix may be too demanding for real-life data. In this paper, we study a particular setting related to order statistics in which these necessary conditions are inherently satisfied. In addition to the theoretical study, an extensive discussion concerning many potential applications is addressed, and a real-life example of an application related to sports data is presented.

Random variables coupled by the dependence structure of comonotonicity posses the property of increasing or decreasing simultaneously. Moreover, comonotone random variables share a number of interesting properties. One of them is that, given two marginal probability distributions, there exists a joint comonotone model with the given marginals, such a joint model can be easily constructed, and it is unique. This paper explores comonotonicity within the context of uncertainty modelled using imprecise probability models. Specifically, we assume that the uncertainty about the marginal models is described in terms of coherent lower probabilities and we seek a comonotone lower probability with the given marginals, referred to as the comonotone extension. In this setting, we analyse whether a comonotone extension can be built (existence), how to build it (construction) and if it is unique (uniqueness). The potential lack of uniqueness leads us to examine the same questions in relation to the most conservative comonotone extension, known as the comonotone natural extension.

A random set, and in particular a random interval, can be regarded as the outcome of observing a random variable with imprecision. Several authors have extended the notions of expected value (Aumann expectation) and variance (Kruse variance) from random variables to random sets/intervals. In this paper, our aim is to follow this line of research and present potential extensions of the notion of the median of a random variable to random intervals. More precisely, four different definitions of the median of a random interval are proposed. The properties fulfilled and the relationships between these medians are analyzed and their explicit form is given under rather general circumstances (comonotonicity of the endpoints of the random interval). Finally, we study the extension of these definitions to quantiles and linear combinations of quantiles.

Distortion or neighbourhood models are popular approaches in robust statistics that allow making decisions under uncertainty that are less susceptible to errors in the elicitation process. They transform a given probability measure or create a neighbourhood around it, giving rise in either case to an imprecise model under quite general assumptions on the robustification procedure. Building on our earlier work on the generalisation of the Total Variation Model to the case where the starting point is a lower probability instead of a probability measure, we extend here Vertical Barrier Models, encompassing as particular cases the Linear Vacuous, Pari Mutuel and the aforementioned Total Variation Models. Moreover, the connections of these generalised distortion models with the discounting of credal sets and sets of almost desirable gambles, as well as their applications to coalitional game theory are also investigated.

Given a number of imprecise probability models, we aim at aggregating them into a joint one using an aggregation rule such as the conjunction, disjunction, convex mixture, Pareto, conjunction-disjunction or maximal consistent subsets rules. We focus on the problem of analysing if these operators are closed, in the sense that the output belongs to the same family as the inputs. Specifically, we analyse this problem for the family of comparative probabilities, 2-monotone capacities, probability intervals, belief functions, p-boxes and minitive measures.

The measurement and representation of a continuous variable necessarily involves some degree of rounding and inevitably results in a loss of information that is potentially dangerous for some statistical inference tasks. In the present paper, we study the particular case of normality tests and show that these tests are typically doomed when applied to rounded data if the sample size is large or the scale is too small in comparison to the digit to which the data is rounded. For such purpose, we formalize the problem by using random sets to model both the uncertainty associated with the realization of the random variable and the imprecision caused by rounding. We end by providing some guidelines of use for practitioners that need to apply normality tests with rounded data.

We generalise the idea of distortion models to the case where the starting model is an imprecise probability model instead of a precise probability. Specifically, we discuss the transformation of a lower probability or a credel set into a more imprecise model, and analyse a number of desirable properties any such transformation should satisfy. Then, we investigate in detail the extension of the total variation distortion model from this perspective.

Stochastic orderings are a commonly used tool in probability theory for comparing random variables or probability distributions. In a recent publication we showed that stochastic orderings are, to some extent, in correspondence with voting procedures. For instance, we demonstrated that the Borda count is equivalent to the comparison of expectations while the Condorcet method is equivalent to statistical preference. This contribution establishes as well a correspondence between stochastic orderings and methods used in the literature for ranking the elements of a poset. Specifically, we show that some well-known methods used in the literature for ranking the elements of a poset, namely the averaged ranking, mutual rank probabilities and the maximal method, are formally equivalent to comparing expectations, statistical preference and multivariate probabilistic preference, respectively.

Aggregation processes appear naturally in many fields of application. The formalization of such processes has been a core topic for researchers in the fuzzy set community for decades, mostly focusing on the aggregation of elements of a bounded poset (typically a bounded real interval). Recent work by the present authors has aimed at further generalizing aggregation theory so it can accommodate aggregation processes on more general structures such as multivariate data, ranking data and string data. In this work, the aggregation of circular data is explored and, in particular, two prominent examples of aggregation functions for circular data (namely the circular mean and the circular median) are presented within this revisited aggregation theory framework.

Distortion or neighbourhood models are tools within the imprecise probability theory that allow to robustify a probability measure. These are built by considering the closed ball around a probability measure with a given radius and using a distorting function to compare probability measures. These include well-known models such as the linear vacuous, pari-mutuel or total variation models. In this contribution we focus on the distortion models that arise from considering the Euclidean distance or the Kullback-Leibler divergence as distorting functions, and analyse their behaviour under different aggregation rules: conjunction, disjunction or convex mixtures.

Stochastic  orders are probabilistic tools used for comparing random quantities. Statistical preference is a stochastic ordering with two main features: firstly, the comparison is based on the joint distribution of the random variables, thus considering potential dependence between them. Secondly, it is accompanied by a winning probability, which measures the strength of the preference. This contribution proposes several ways to extend statistical preference for comparing (bivariate) random vectors. These proposals follow three different approaches: (i) aggregating the winning probabilities of each component, (ii) computing the winning probabilities between the aggregated random vectors, or (iii) defining a purely bivariate winning probability. Besides investigating these properties, the relationships between them and the componentwise median are studied.

We  consider the problem of aggregating information provided by experts when this information is expressed by means of coherent lower previsions. These constitute a framework general enough so as to include as particular cases not only probability measures but also the majority of models from the imprecise probability theory. Although the aggregation problem has already been tackled in the literature, our contribution provides a unified view by putting together a number of rationality criteria and aggregation rules studied in different papers. Specifically, we consider five aggregation rules, twelve rationality criteria and provide a detailed analysis of the properties satisfied by each rule.

In discrete mathematics, an interesting problem that has called the attention of many scholars is that of ranking the elements of a given partially ordered set. In this contribution, we propose reasonable properties that a method for ranking the elements of a poset may satisfy and we study the relationships between these properties. Interestingly, it is shown how a plethora of additional properties may be borrowed from the field of social choice theory and, in particular, from the problem of the aggregation of rankings. Finally, we present three prominent methods for ranking the elements of a poset (namely, the averaged rank method, the mutual rank probabilities and the maximal method) and analyse which among the proposed properties each of these three methods satisfies.

Skewness coefficients are measures for quantifying the degree of asymmetry of a random variable. Different authors have proposed several skewness coefficients, most of which are positioned within the axiomatic definition introduced by Oja. This contribution presents a general family of skewness coefficients based on OWA functions. After showing that this family fits within Oja’s axiomatic definition of a skewness coefficient, we present a sample version of this coefficient, study its asymptotic distribution, and use it for defining a goodness-of-fit test to a location-scale family.

Aggregation theory offers different tools for combining multiple sources of information into a single summary measure. Among these, Ordered Weighted Averaging (OWA) functions have recently been employed to define skewness coefficients, which quantify the degree of asymmetry of a random variable. Because skewness coefficients are location-scale invariant, the proposed OWA-based skewness coefficients can be used to conduct two-sample tests of goodness-of-fit to both a known and unknown location-scale family. In this contribution, we introduce the corresponding test statistics, derive their asymptotic distribution, and evaluate the performance of the associated tests through Monte Carlo simulation.

Ordered Weighted Averaging (OWA) functions are a prominent tool in aggregation theory and decision making for selecting a representative of a given list of values to be aggregated. From an apparently different field, skewness coefficients are often used in probability and statistics for quantifying the asymmetry of a probability distribution. Recently, a connection between both constructs has been made explicit by using OWA functions to define a family of skewness coefficients. In this work, we use this OWA-based family of skewness coefficients to develop a new method for constructing asymptotic confidence intervals for a skewness parameter within a location–scale–skewness family. The method is illustrated through a comprehensive case study for the (shifted) Weibull distribution. Theoretical results for this distribution together with an empirical study via Monte Carlo simulation are provided.

This work investigates the robustification of probability measures, on finite-dimensional possibility spaces, through the family of Weighted Total Variation distances. This collection is indexed by a set of vectors of weights, each of which grades the willingness to change the probability initially assigned to each singleton, and embraces the Total Variation distance when the weights are uniform. Thus, these distances allow to create a neighbourhood around a probability measure taking into account the importance of singletons due to application-specific needs. After motivating the interpretation and potential applications of a robustifying procedure with such characteristics, we demonstrate explicit formulae for the extreme points of the neighbourhood models and the coherent lower previsions that characterise them. Finally, we show that these neighbourhood models are 2-monotone, essentially, if and only if the weights are uniform, and provide insights into future research directions.

We study the complexity of the credal set determined by a comparative probability order on the singletons of a finite possibility space. We show that the bound established by Miranda and Destercke in [15] can be reduced under some conditions on the associated graph, and give sufficient conditions for this new bound to be attained. We focus in particular on the case where the order relation associated with the comparative probability is either bounded above or below, and discuss specific cases of these in detail. In addition we investigate the properties of the lower probability determined by a comparative probability ordering.

Starting from a finite number of lower probabilities, we study the problem of their aggregation into a global coherent lower prevision on gambles. We follow two approaches: on the one hand, we may first apply the natural extension to each lower probability, obtaining a lower prevision, and then combine these using an aggregation rule; or we may first apply an aggregation rule to the lower probabilities and then take the natural extension of the output. In this paper, we analyse whether the resulting models coincide. In addition, we study if any of the two approaches preserves the property of k-monotonicity.

We introduce linear variational models, which include as particular cases some of the most prominent distortion models in the literature, and investigate their interest in a decision making problem under uncertainty and imprecision. We compare five optimality criteria: -maximin, -maximax, E-admissibility, maximality and interval dominance, and provide an expression of the optimal alternatives when the credal set modelling the uncertainty is determined by a linear variational model. In addition, we measure the extent of the distortion that should be put in place for an alternative to be optimal.

En problemas de decisión es habitual asumir que unos expertos determinan una distribución de probabilidad que modele la incertidumbre. Para evitar la influencia de la opinión de los expertos y con el fin de obtener un modelo más robusto, una posibilidad es utilizar los llamados modelos de distorsión, basados en considerar un entorno centrado en una medida de probabilidad. En este trabajo se estudia la generalización de estos modelos a situaciones en las que el modelo original es una medida no aditiva, una probabilidad inferior o un conjunto de medidas de probabilidad. Se proponen varias aproximaciones al problema dependiendo de aspectos del modelo de partida y del procedimiento de distorsión considerado. Además, se analiza la relación con estudios realizados por S. Moral (Discounting imprecise probabilities), se investiga la agregación de modelos de este tipo en un modelo global en base a varias propiedades axiomáticas deseables y se establece una conexión con juegos cooperativos.

En muchos problemas de decisión bajo incertidumbre es difícil conocer con precisión la distribución de probabilidad. Una posible solución en esos casos es recurrir a herramientas de la teoría de probabilidades imprecisas, como por ejemplo las probabilidades inferiores, también conocidas como medidas no aditivas. Si bien estos modelos permiten representar la información disponible en la mayoría de los casos, tienen el inconveniente de ser más complejos computacionalmente, lo que dificulta su uso. Este problema se palía en parte cuando cumplen la condición de 2-monotonía (también llamada supermodularidad o convexidad). En este trabajo partimos de un problema de decisión cuya incertidumbre se modela con una probabilidad inferior y analizamos cómo varía la solución, con respecto a varios criterios, cuando ésta se reemplaza por una probabilidad inferior 2-monótona que esté “lo más cerca posible”, lo que en trabajos previos denominamos aproximaciones interiores o exteriores 2-monótonas.

En el contexto de la probabilidad subjetiva, la axiomatización de la teoría de la decisión de Savage consideró el fenómeno conocido como “sure thing”: si una transacción resulta deseable sea cual sea el resultado de un experimento que se realizará en el futuro, también debería de considerarse deseable en el momento actual. Este concepto se relaciona con el de conglomerabilidad de Bruno de Finetti, y a su vez con la combinación de información marginal y condicionada sobre un experimento. En este trabajo, se estudia el concepto dual de
conglomerabilidad negativa: si una transacción no es deseable sea cual sea el resultado de un experimento futuro, tampoco debería ser deseable ahora. Si bien ambos conceptos resultan equivalentes bajo información precisa, la equivalencia se pierde en el caso impreciso. En este trabajo, se estudian las consecuencias de la conglomerabilidad negativa, su relación con las nociones de extensión marginal y dilatación, y se concluye que hay una diferencia entre la conglomerabilidad y la conglomerabilidad negativa desde el punto de vista descriptivo.

We generalise the idea of distortion models to the case where the starting model is an imprecise probability model instead of a precise probability. Specifically, we discuss the transformation of a lower probability or a credel set into a more imprecise model, and analyse a number of desirable properties any such transformation should satisfy. Then, we investigate in detail the extension of the total variation distortion model from this perspective.

We generalise the idea of distortion models to the case where the starting model is an imprecise probability model instead of a precise probability. Specifically, we discuss the transformation of a lower probability or a credel set into a more imprecise model, and analyse a number of desirable properties any such transformation should satisfy. Then, we investigate in detail the extension of the total variation distortion model from this perspective.

Aggregation processes appear naturally in many fields of application. The formalization of such processes has been a core topic for researchers in the fuzzy set community for decades, mostly focusing on the aggregation of elements of a bounded poset (typically a bounded real interval). Recent work by the present authors has aimed at further generalizing aggregation theory so it can accommodate aggregation processes on more general structures such as multivariate data, ranking data and string data. In this work, the aggregation of circular data is explored and, in particular, two prominent examples of aggregation functions for circular data (namely the circular mean and the circular median) are presented within this revisited aggregation theory framework.

Aggregation functions are a common tool in Statistics for constructing data summaries. One of the most prominent types of data summaries are the so-called skewnesss coefficients, which measure the degree of asymmetry of a probability distribution. In most cases, these skewness coefficients may be used for defining a test of symmetry. In the present work, some experiments are carried out comparing the performance of the tests of symmetry associated with a popular family of skewness coefficients introduced by Hinkley.

Stochastic orders are probabilistic tools used for comparing random quantities. Statistical preference is a stochastic ordering with two main features: firstly, the comparison is based on the joint distribution of the random variables, thus considering potential dependence between them. Secondly, it is accompanied by a winning probability, which measures the strength of the preference. This contribution proposes several ways to extend statistical preference for comparing (bivariate) random vectors. These proposals follow three different approaches: (i) aggregating the winning probabilities of each component, (ii) computing the winning probabilities between the aggregated random vectors, or (iii) defining a purely bivariate winning probability. Besides investigating these properties, the relationships between them and the componentwise median are studied.

Distortion or neighbourhood models are tools within the imprecise probability theory that allow to robustify a probability measure. These are built by considering the closed ball around a probability measure with a given radius and using a distorting function to compare probability measures. These include well-known models such as the linear vacuous, pari-mutuel or total variation models. In this contribution we focus on the distortion models that arise from considering the Euclidean distance or the Kullback-Leibler divergence as distorting functions, and analyse their behaviour under different aggregation rules: conjunction, disjunction or convex mixtures.

We  consider the problem of aggregating information provided by experts when this information is expressed by means of coherent lower previsions. These constitute a framework general enough so as to include as particular cases not only probability measures but also the majority of models from the imprecise probability theory. Although the aggregation problem has already been tackled in the literature, our contribution provides a unified view by putting together a number of rationality criteria and aggregation rules studied in different papers. Specifically, we consider five aggregation rules, twelve rationality criteria and provide a detailed analysis of the properties satisfied by each rule.

En el contexto de estadística robusta, es habitual considerar entornos alrededor de una medida de probabilidad, con el fin de garantizar la solidez de las inferencias respecto a pequeños cambios en la distribución estimada. Sin embargo, en situaciones de información imprecisa o ambigua, la estimación de dicha distribución puede resultar compleja, y una alternativa de interés es partir de una medida no aditiva. Este tipo de medidas surgen también de manera natural en el contexto de juegos cooperativos.
En este trabajo, realizamos un estudio formal del procedimiento de distorsión de medidas no aditivas, prestando atención a las propiedades satisfechas por el modelo resultante y analizando en detalle el caso en el que la distorsión se realiza mediante una extensión de la distancia de la variación total. Se establece además una conexión entre este modelo y el strong-epsilon-core, un procedimiento habitual en juegos cooperativos para evitar que el core sea vacío.

Numerosos estudios de la literatura han analizado las diferencias por género en las actitudes frente al riesgo y la ambigüedad en un problema de decisión bajo incertidumbre. Sin embargo, las conclusiones de los mismos difieren en cuanto al diseño del experimento, las estimaciones de dichas actitudes y las herramientas estadísticas empleadas en el análisis. En este trabajo, presentamos los resultados de un experimento propio sobre el tema en el que se tiene en cuenta la escala de utilidades considerada y se miden las actitudes frente a la ambigüedad a través de primas de reserva.

In discrete mathematics, an interesting problem that has called the attention of many scholars is that of ranking the elements of a given partially ordered set. In this contribution, we propose reasonable properties that a method for ranking the elements of a poset may satisfy and we study the relationships between these properties. Interestingly, it is shown how a plethora of additional properties may be borrowed from the field of social choice theory and, in particular, from the problem of the aggregation of rankings. Finally, we present three prominent methods for ranking the elements of a poset (namely, the averaged rank method, the mutual rank probabilities and the maximal method) and analyse which among the proposed properties each of these three methods satisfies.

Skewness coefficients are measures for quantifying the degree of asymmetry of a random variable. Different authors have proposed several skewness coefficients, most of which are positioned within the axiomatic definition introduced by Oja. This contribution presents a general family of skewness coefficients based on OWA functions. After showing that this family fits within Oja’s axiomatic definition of a skewness coefficient, we present a sample version of this coefficient, study its asymptotic distribution, and use it for defining a goodness-of-fit test to a location-scale family.

We consider the problem of aggregating a number of imprecise probability models into a joint one, and compare four aggregation rules: conjunction, disjunction, mixture and Pareto. We investigate for which particular cases of imprecise probability models these operators are closed, meaning that the output belongs to the same family as the inputs. Specifically, we analyse this problem for comparative probability models, 2-monotone capacities, probability intervals, belief functions, p-boxes and minitive measures

Distortion or neighbourhood models are useful tools in the imprecise probability theory allowing to robustify a probabilistic model by considering a neighbourhood around a given probability measure. In this work, we tackle the more general problem of distorting a lower probability. This problem can be interesting when we believe that a given lower probability is too precise, or in coalitional game theory when the set of solutions is empty. Our main purpose is to investigate how the linear vacuous and pari mutuel models can be defined for the distortion of lower probabilities, and for this aim we address the problem in a more general manner: we extend the vertical barrier models, which include the linear vacuous and pari mutuel models, and investigate the properties they satisfy.

This paper studies the use of distortions as a tool for addressing conflicts between a finite number of uncertainty models, in the general case where these initial models may not be precise, in particular a lower probability. We propose to distort the associated credal sets until the global conflict is removed, and to use the conjunction aggregation rule in that moment. We investigate this procedure in the case where the distortion is made using the total variation distance, and compare its properties with other aggregation rules from the literature. In addition, we also explore an alternative where the distortion is tweaked so as to enlarge the credal sets only in the directions where conflict is present.

Lower probabilities are a common tool for modeling uncertainty in the presence of imprecision; however, the assessments given by a lower probability may sometimes be inconsistent or conflicting. For this reason, we consider the problem of measuring the conflict present in a lower probability, and compare five different measures: the distance to the closest model that avoids sure loss, that is coherent, or to its natural extension; the measure of conflict considered by Schervisch et al; and a measure based on making the initial model more imprecise and then measuring the distance of the distorted model with its natural extension. Our comparisons are made by means of a number of desirable properties, such as the invariance under transformations of the possibility space, the monotonicity of the measure, or convexity.

Aggregation theory offers different tools for combining multiple sources of information into a single summary measure. Among these, Ordered Weighted Averaging (OWA) functions have recently been employed to define skewness coefficients, which quantify the degree of asymmetry of a random variable. Because skewness coefficients are location-scale invariant, the proposed OWA-based skewness coefficients can be used to conduct two-sample tests of goodness-of-fit to both a known and unknown location-scale family. In this contribution, we introduce the corresponding test statistics, derive their asymptotic distribution, and evaluate the performance of the associated tests through Monte Carlo simulation.

Ordered Weighted Averaging (OWA) functions are a prominent tool in aggregation theory and decision making for selecting a representative of a given list of values to be aggregated. From an apparently different field, skewness coefficients are often used in probability and statistics for quantifying the asymmetry of a probability distribution. Recently, a connection between both constructs has been made explicit by using OWA functions to define a family of skewness coefficients. In this work, we use this OWA-based family of skewness coefficients to develop a new method for constructing asymptotic confidence intervals for a skewness parameter within a location–scale–skewness family. The method is illustrated through a comprehensive case study for the (shifted) Weibull distribution. Theoretical results for this distribution together with an empirical study via Monte Carlo simulation are provided.

This work investigates the robustification of probability measures, on finite-dimensional possibility spaces, through the family of Weighted Total Variation distances. This collection is indexed by a set of vectors of weights, each of which grades the willingness to change the probability initially assigned to each singleton, and embraces the Total Variation distance when the weights are uniform. Thus, these distances allow to create a neighbourhood around a probability measure taking into account the importance of singletons due to application-specific needs. After motivating the interpretation and potential applications of a robustifying procedure with such characteristics, we demonstrate explicit formulae for the extreme points of the neighbourhood models and the coherent lower previsions that characterise them. Finally, we show that these neighbourhood models are 2-monotone, essentially, if and only if the weights are uniform, and provide insights into future research directions.

We study the complexity of the credal set determined by a comparative probability order on the singletons of a nite possibility space. We show that the bound established by Miranda and Destercke can be reduced under some conditions on the associated graph, and give sucient conditions for this new bound to be attained. We focus in particular on the case where the order relation associated with the comparative probability is either bounded above or below, and discuss specic cases of these in detail. In addition we investigate the properties of the lower probability determined by a comparative probability ordering.

Starting from a finite number of lower probabilities, we study the problem of their aggregation into a global coherent lower prevision on gambles. We follow two approaches: on the one hand, we may first apply the natural extension to each lower probability, obtaining a lower prevision, and then combine these using an aggregation rule; or we may first apply an aggregation rule to the lower probabilities and then take the natural extension of the output. In this paper, we analyse whether the resulting models coincide. In addition, we study if any of the two approaches preserves the property of k-monotonicity.

We introduce linear variational models, which include as particular cases some of the most prominent distortion models in the literature, and investigate their interest in a decision making problem under uncertainty and imprecision. We compare five optimality criteria: $\Gamma$-maximin, $\Gamma$-maximax, $E$-admissibility, maximality and interval dominance, and provide an expression of the optimal alternatives when the credal set modelling the uncertainty is determined by a linear variational model. In addition, we measure the extent of the distortion that should be put in place for an alternative to be optimal.

The only systematic aggregation rules that, for any profile, commute with Bayesian conditioning on an event are the dictatorial ones. However, a recent work on dynamic rationality determined sufficient conditions for commutativity to be satisfied for specific classes of profiles and conditioning events, for any systematic rule. Following this path, and further focusing on a single given profile and conditioning event, the present work seeks and analyses sufficient conditions for a systematic rule to accomplish with the commutativity property in that specific case. In doing so, we characterise the systematic rules that guarantee the specific commutativity by means of a system of equations, find sufficient conditions embracing the mentioned results from the literature, show that none of the latter are necessary and give insights into future lines of research.

Since the introduction of fuzzy logic, a wide variety of aggregation functions have been studied by many researchers. Among the most relevant examples are t-norms, t-conorms, uninorms, and nullnorms. These aggregation functions have been investigated on different algebraic structures, such as intervals, lattices, and trellises. In this work, we focus on the study of aggregation functions defined on bounded lattices. In particular, we analyze the structural properties of certain classes of bounded lattices that allow the construction of specific uninorms. It is well known that any uninorm restricted to the subset of elements smaller than or equal to the neutral element acts as a t-norm and restricted to the subset of elements greater than or equal to the neutral element acts as a t-conorm. Building on this idea, we prove that, under appropriate conditions, there exist bounded lattices on which a uninorm can be defined that acts as a t-norm or a t-conorm when restricted to the subset of elements incomparable with the neutral element. Moreover, we show that by imposing stronger structural restrictions on the underlying bounded lattice, it is possible to construct a uninorm that acts as a proper uninorm when restricted to the subset of elements incomparable with the neutral element. This process can be iterated, leading to a hierarchical construction method for uninorms.

Skewness measures quantify the asymmetry of probability distributions and are essential for understanding their shape. In this work, we focus on a recently-introduced family of skewness coefficients based on Ordered Weighted Averaging (OWA) functions. We use these skewness coefficients to construct statistical tests for the convex order of van Zwet. An empirical study via Monte Carlo simulation of the behaviour of these statistical tests restricted to a specific location-scale-skewness family (the Weibull distribution) is provided.

Handling data subject to imprecision is a common problem in statistics. In particular, samples drawn from continuous random variables are usually affected by a rounding process, and applying statistical procedures to the rounded data may lead to incorrect decisions. Recently, we proposed a random set-based approach to model rounded data, using more realistic tools than probability theory to capture the lack of information. Following with this random set-based approach, in this contribution we investigate how skewness coefficients can be defined in this setting. In particular, we provide an explicit formula for Hinkley’s skewness coefficient for rounded data and apply this approach to normality tests with rounded data.

Building a neighbourhood around a given probability measure is a common approach to obtain a robust uncertainty model, and two prominent examples are those induced by the Total Variation or the Kolmogorov distances. In previous contributions, we generalised neighbourhood models to the case where the starting point is a lower probability instead of a probability measure. Here, we deal with the case where the initial model is a p-box, generalising the Kolmogorov model and investigating which properties of interest are satisfied by the procedure.

The assessments given by a lower probability may sometimes be inconsistent or conflicting. In this paper, we consider the problem of measuring this conflict and compare three different measures: the distance to the closest model that avoids sure loss, that is coherent, or to its natural extension. Our comparisons are made in terms of a number of desirable properties, such as the invariance under transformations of the possibility space, the monotonicity of the measure, or convexity.

La medición y representación de variables aleatorias absolutamente continuas implica necesariamente un proceso de redondeo que conlleva pérdida de información. Esta pérdida, aunque a menudo se considera menor, puede tener consecuencias relevantes en ciertos procedimientos estadísticos, como por ejemplo en pruebas de normalidad. En particular, dichas pruebas pueden volverse poco fiables cuando se aplican a datos redondeados, especialmente si el tamaño de la muestra es grande o si la escala de medición es baja en relación con el dígito al que se redondea. Para abordar este problema, se propone un marco teórico basado en conjuntos aleatorios, que permite modelar tanto la incertidumbre inherente a la variable aleatoria como la imprecisión introducida por el redondeo. Este marco teórico permite analizar la influencia del redondeo en los tests de normalidad y cuantificar la influencia que el redondeo tiene en la toma de decisiones.

En problemas de decisión en los que se tiene información parcial sobre el contexto del experimento, las preferencias del decisor o consecuencias de algunas de las alternativas, la Teoría clásica de Decisión puede resultar inadecuada. En este trabajo, presentamos un marco de decisión robustificado mediante la incorporación de modelos de probabilidades imprecisas y la extensión del criterio de la utilidad esperada. En particular, introducimos los Linear Variational Models que satisfacen varias propiedades deseables y generalizan a su vez a otros modelos relevantes de probabilidades imprecisas. Asimismo, caracterizamos las alternativas óptimas para los criterios $\Gamma$-maximin, $\Gamma$-maximax, E-admisibilidad, Maximalidad y Dominancia Intervalar, y damos una medida de distorsión requerida para asegurar la optimalidad de una alternativa.