A Linear Regression Model for Interval-Valued Response Based on Set Arithmetic

Several linear regression models involving interval-valued variables have been formalized based on the interval arithmetic. In this work, a new linear regression model with interval-valued response and real predictor based on the interval arithmetic is formally described. The least-squares estimation of the model is solved by means of a constrained minimization problem which guarantees the coherency of the estimators with the regression parameters. The practical applicability of the estimation method is checked on a real-life example, and the empirical behaviour of the procedure is shown by means of some simulation studies.

[1]  Dominikus Noll,et al.  Generic Gâteaux-differentiability of convex functions on small sets , 1990 .

[2]  M. Gil,et al.  Least squares fitting of an affine function and strength of association for interval-valued data , 2002 .

[3]  M. Eisen,et al.  Probability and its applications , 1975 .

[4]  Ana Colubi,et al.  Linear Regression Analysis for Interval-valued Data Based on Set Arithmetic: A Review , 2013 .

[5]  Michel Verleysen,et al.  Towards Advanced Data Analysis by Combining Soft Computing and Statistics , 2012, SOCO 2012.

[6]  I. Molchanov.  Theory of Random Sets , 2005 .

[7]  Carlo Bertoluzza,et al.  On a new class of distances between fuzzy numbers , 1995 .

[8]  Ana Colubi,et al.  A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread , 2009, Inf. Sci..

[9]  Ilya Molchanov,et al.  Empirical estimation of distribution quantiles of random closed sets , 1991 .

[10]  Ana Colubi,et al.  A linear regression model for imprecise response , 2010, Int. J. Approx. Reason..

[11]  C. Huber,et al.  Interval censored and truncated data: Rate of convergence of NPMLE of the density , 2009 .

[12]  Ana Colubi,et al.  Least squares estimation of linear regression models for convex compact random sets , 2007, Adv. Data Anal. Classif..

[13]  Phil Diamond,et al.  Least squares fitting of compact set-valued data , 1990 .

[14]  John C. Ham,et al.  Two-stage estimation of structural labor supply parameters using interval data from the 1971 canadian census , 1984 .