Least squares estimation of linear regression models for convex compact random sets

Abstract

Simple and multiple linear regression models are considered between variables whose "values" are convex compact random sets in R p , (that is, hypercubes, spheres, and so on). We analyze such models within a set-arithmetic approach. Contrary to what happens for random variables, the least squares optimal solutions for the basic affine transformation model do not produce suitable estimates for the linear regression model. First, we derive least squares estimators for the simple linear regression model and examine them from a theoretical perspective. Moreover, the multiple linear regression model is dealt with and a stepwise algorithm is developed in order to find the estimates in this case. The particular problem of the linear regression with interval-valued data is also considered and illustrated by means of a real-life example.

Evaluating FOMC forecast ranges: an interval data approach

“…More profound information on the model itself and the estimation process can be found in Blanco-Fernandez et al (2012b), who also present the model's cognate predecessors being the simple, less flexible linear regression models for interval data of Blanco-Fernandez et al (2011) and Gonzalez-Rodriguez et al (2007). The reader may also refer to these papers for some preliminary concepts of the interval data framework including the basics of interval arithmetic.…”

Lasso Estimation of an Interval-Valued Multiple Regression Model

“…The variables to be considered are the ranges of fluctuation of the diastolic blood preasure over the day (y), the pulse rate (x 1 ) and the systolic blood preasure (x 2 ). The dataset can be found in [2] and [5]. In order to make possible the comparison between the estimator proposed in Sect.…”

“…The statistical study of regression models for interval data has been extensively addressed lately in the literature [2][3][4][5]7], deriving into several alternatives to tackle this problem. On one hand, the estimators proposed in [4,7] account the non-negativity constraints satisfied by the spread variables, but do not assure the existence of the residuals.…”

“…Hence, they can lead to ill-defined estimated models. On the other hand, the models proposed in [2,3,5] are formalized according to the natural interval arithmetic and their estimators lead to models that are always well-defined over the sample range.…”

On the Generalization Performance of a Regression Model with Imprecise Elements

“…In the literature, different statistical procedures for imprecise information are proposed (see, for example, Hung, 25 Sun and Wu, 31 Sinova et al 30 ). 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.…”