Angela Blanco-Fernández

Estimation of a flexible simple linear model for interval data based on set arithmetic

The estimation of a simple linear regression model when both the independent and dependent variable are interval valued is addressed. The regression model is defined by using the interval arithmetic, it considers the possibility of interval-valued disturbances, and it is less restrictive than existing models. After the theoretical formalization, the least-squares (LS) estimation of the linear model with respect to a suitable distance in the space of intervals is developed. The LS approach leads to a constrained minimization problem that is solved analytically. The strong consistency of the obtained estimators is proven. The estimation procedure is reinforced by a real-life application and some simulation studies.

A distance-based statistical analysis of fuzzy number-valued data

Abstract Real-life data associated with experimental outcomes are not always real-valued. In particular, opinions, perceptions, ratings, etc., are often assumed to be vague in nature, especially when they come from human valuations. Fuzzy numbers have extensively been considered to provide us with a convenient tool to express these vague data. In analyzing fuzzy data from a statistical perspective one finds two key obstacles, namely, the nonlinearity associated with the usual arithmetic with fuzzy data and the lack of suitable models and limit results for the distribution of fuzzy-valued statistics. These obstacles can be frequently bypassed by using an appropriate metric between fuzzy data, the notion of random fuzzy set and a bootstrapped central limit theorem for general space-valued random elements. This paper aims to review these ideas and a methodology for the statistical analysis of fuzzy number data which has been developed along the last years.

A set arithmetic-based linear regression model for modelling interval-valued responses through real-valued variables

A new linear regression model for an interval-valued response and a real-valued explanatory variable is presented. The approach is based on the interval arithmetic. Comparisons with previous methods are discussed. The new linear model is theoretically analyzed and the regression parameters are estimated. Some properties of the regression estimators are investigated. Finally, the performance of the procedure is illustrated using both a real-life application and simulation studies.

Linear Regression Analysis for Interval-valued Data Based on Set Arithmetic: A Review

When working with real-valued data regression analysis allows to model and forecast the values of a random variable in terms of the values of either another one or several other random variables defined on the same probability space. When data are not real-valued, regression techniques should be extended and adapted to model simply relationships in an effective way. Different kinds of imprecision may appear in experimental data: uncertainty in the quantification of the data, subjective measurements, perceptions, to name but a few. Compact intervals can be effectively used to represent these imprecise data. Set- and fuzzy-valued elements are also employed for representing different kinds of imprecise data. In this paper several linear regression estimation techniques for interval-valued data are revised. Both the practical applicability and the empirical behaviour of the estimation methods is studied by comparing the performance of the techniques under different population conditions.

On Some Confidence Regions to Estimate a Linear Regression Model for Interval Data

Least-squares estimation of various linear models for interval data has already been considered in the literature. One of these models allows different slopes for mid-points and spreads (or radii) integrated in a unique equation based on interval arithmetic. A preliminary study about the construction of confidence regions for the parameters of that model on the basis of the least-squares estimators is presented. Due to the lack of realistic parametric models for random intervals, bootstrap approaches are proposed. The empirical suitability of the bootstrap confidence sets will be shown by means of some simulation studies.

Inferential studies for a flexible linear regression model for interval-valued variables

Inferential studies for the regression coefficients of a linear model for interval-valued random variables are addressed. Confidence sets and hypothesis tests are investigated and solved through asymptotic and bootstrap techniques. The inferences are based on the least-squares estimators of the model which have been shown to be coherent with the interval arithmetic defining the model and to verify good statistical properties. Theoretical results assure the validity of the procedures. Moreover, some simulation studies and examples are considered to show the empirical behaviour and the practical applicability of the inferences.

A new framework for the statistical analysis of set-valued random elements

Abstract The space of nonempty convex and compact (fuzzy) subsets of R p , K c ( R p ) , has been traditionally used to handle imprecise data. Its elements can be characterized via the support function, which agrees with the usual Minkowski addition, and naturally embeds K c ( R p ) into a cone of a separable Hilbert space. The support function embedding holds interesting properties, but it lacks of an intuitive interpretation for imprecise data. As a consequence, it is not easy to identify the elements of the image space that correspond to sets in K c ( R p ) . Moreover, although the Minkowski addition is very natural when p = 1 , if p > 1 the shapes which are obtained when two sets are aggregated are apparently unrelated to the original sets, because it tends to convexify. An alternative and more intuitive functional representation will be introduced in order to circumvent these difficulties. The imprecise data will be modeled by using star-shaped sets on R p . These sets will be characterized through a center and the corresponding polar coordinates, which have a clear interpretation in terms of location and imprecision, and lead to a natural directionally extension of the Minkowski addition. The structures required for a meaningful statistical analysis from the so-called ontic perspective are introduced, and how to determine the representation in practice is discussed.

A Proposal for Assessing Imprecise Concepts in Spanish Primary and Secondary Schools

Spanish primary and secondary school curricula comprise several contents, learning outcomes and assessment criteria directly related with probability and approximate calculus. Some of them refer to situations modeled by the students, which entail not only uncertainty but also imprecision. For this reason, different techniques including fuzzy logic and fuzzy sets theory could be applied when dealing with this kind of situations in the classroom. Several teaching situations handling imprecise concepts in primary and secondary schools are suggested from a theoretical point of view. These more exible ways of reasoning could be combined with the traditional probability approach, allowing to tackle more general problems and not only those involving exact calculations or specific numerical assignments. Moreover, this type of approaches will provide the students with tools to manage imprecision as a mathematical tool in their personal life.

On Some Concepts Related to Star-Shaped Sets

The convenient theoretical properties of the support function and the Minkowski addition-based arithmetic have been shown to be useful when dealing with compact and convex sets on \(\mathbb {R}^p\). However, both concepts present several drawbacks in certain contexts. The use of the radial function instead of the support function is suggested as an alternative to characterize a wider class of sets—the so-called star-shaped sets—which contains the class of compact and convex sets as a particular case. The concept of random star-shaped set is considered, and some statistics for this kind of variable are shown. Finally, some measures for comparing star-shaped sets are introduced.

Independent k -Sample Equality Distribution Test Based on the Fuzzy Representation

Classical tests for the equality of distributions of real-valued random variables are widely applied in Statistics. When the normality assumption for the variables fails, non-parametric techniques are to be considered; Mann-Whitney, Wilcoxon, Kruskal-Wallis, Friedman tests, among other alternatives. Fuzzy representations of real-valued random variables have been recently shown to describe in an effective way the statistical behaviour of the variables. Indeed, the expected value of certain fuzzy representations fully characterizes the distribution of the variable. The aim of this paper is to use this characterization to test the equality of distribution for two or more real-valued random variables, as an alternative to classical procedures. The inferential problem is solved through a parametric test for the equality of expectations of fuzzy-valued random variables. Theoretical results on inferences for fuzzy random variables support the validity of the test. Besides, simulation studies and practical applications show the empirical goodness of the method.