The Median of a Random Interval

In dealing with real-valued random variables, the median of the distribution is the ‘central tendency’ summary measure associated with its ‘middle position’. When available random elements are interval-valued, the lack of a universal ranking of values makes it impossible to formalize the extension of the concept of median as a middle-position summary measure. Nevertheless, the use of a generalized L 1 Hausdorff-type metric for interval data enables to formalize the median of a random interval as the central-tendency interval(s) minimizing the mean distance with respect to the random set values, by following the alternate equivalent way to introduce the median in the real-valued case. The expression for the median(s) is obtained, and main properties are analyzed. A short discussion is made on the main different features in contrast to the real-valued case.

[1]  Ana Colubi,et al.  Testing 'Two-Sided' Hypothesis about the Mean of an Interval-Valued Random Set , 2008, SMPS.

[2]  M. Chavent,et al.  Trois nouvelles méthodes de classification automatique de données symboliques de type intervalle , 2003 .

[3]  H. Ishibuchi,et al.  Multiobjective programming in optimization of the interval objective function , 1990 .

[4]  Didier Dubois,et al.  Soft Methods for Handling Variability and Imprecision , 2008 .

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

[6]  Miguel López-Díaz,et al.  Interval-valued quantification of the inequality associated with a random set , 2000 .

[7]  Ángela Blanco Fernández.  Análisis estadístico de un nuevo modelo de regresión lineal flexible para intervalos aleatorios , 2009 .

[8]  Ana Colubi,et al.  Testing linear independence in linear models with interval-valued data , 2007, Comput. Stat. Data Anal..

[9]  Antonio Irpino,et al.  Dynamic clustering of interval data using a Wasserstein-based distance , 2008, Pattern Recognit. Lett..

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

[11]  Ana Colubi,et al.  Interval arithmetic-based simple linear regression between interval data: Discussion and sensitivity analysis on the choice of the metric , 2012, Inf. Sci..

[12]  Manuel Montenegro Hermida.  Estadística con datos imprecisos basada en una métrica generalizada , 2003 .

[13]  Rajkumar Roy,et al.  Advances in Soft Computing , 2018, Lecture Notes in Computer Science.

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

[15]  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..