A linear regression model for imprecise response

Abstract

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies. © 2010 Elsevier Inc. All rights reserved

On the Generalization Performance of a Regression Model with Imprecise Elements

“…Finally, b m , b l , b r denote the intercepts. The model is well defined also for real (crisp) explanatory variables (see, for more details, Ferraro et al 18,19 ). In that case all the vectors have length equal to p. In this context the dependence relationship is strictly related to the shape of the functions g and h, so we aim at studying the gh-linear dependence between the fuzzy response and the fuzzy explanatory variables.…”

“…In the regression context in the last years the number of publications is grown (see, An et al, 1 Blanco-Fernández et al, 5 Cattaneo and Wiencierz, 8 D'Urso et al, 15 Giordani, 22 Körner and Näther 26 ). In this paper we restrict our attention to a family of regression models with imprecise information previously introduced: Ferraro et al 18,19 and Ferraro and Giordani.…”

Statistical reasoning with set-valued information: Ontic vs. epistemic views

“…Many variants of this method, based on conjunctive fuzzy random sets and various scalar distances exist (see [35] for a recent one) including extensions to fuzzy-valued inputs [38]. These approaches all adopt the ontic view.…”

Similarity based one-sided tests for the expected value and interval data

“…As it is not immediate to handle the asymptotic distribution of T n in practice, bootstrap techniques are applied since they have been shown to be a very useful tool to get a better approximation to the sampling distribution [6,7]. Specifically, we propose the use of a residual bootstrap approach.…”