Imprecise distribution function associated to a random set

Authors:
Inés Couso;Luciano Sánchez;Pedro Gil
Affiliations:
Department of Statistics and Operational Research, University of Oviedo, C/Calvo Sotelo, s/n 33007 Oviedo, Spain;Department of Computer Science, University of Oviedo, Campus de Viesques, s/n 33271 Gijón 33203, Spain;Department of Statistics and Operational Research, University of Oviedo, C/Calvo Sotelo, s/n 33007 Oviedo, Spain
Venue:
Information Sciences—Informatics and Computer Science: An International Journal
Year:
2004

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Abstract

Some different extensions to random sets of the most common parameters of a random variable share a common rationale: a random set represents the imprecise observation of a random variable, hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable. Following the same principles, in this paper it is proposed a new definition of the distribution function of a random set. This definition is simpler in its formulation and it can be used in more general cases than previous proposals. The properties of the distribution function defined here are discussed: some issues about continuity, convergence of the images of the distribution function, monotonocity and measurability are studied. It is also stated that not all the information conveyed by the random set about the original probability measure (the probability measure induced by the imprecisely observed random variable) is kept by its distribution function.