A determination coefficient for a linear regression model with imprecise response

Authors

  • Maria Brigida Ferraro,

    Corresponding author
    1. Dipartimento di Statistica, Probabilitá e Statistiche Applicate, Sapienza Università di Roma, Piazzale Aldo Moro 5 00185, Rome, Italy
    • Dipartimento di Statistica, Probabilità e Statistiche Applicate, Sapienza Università di Roma, Piazzale Aldo Moro 5 00185, Rome, Italy.
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  • Ana Colubi,

    1. Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, Calle Calvo Sotelo S/N 33007, Oviedo, Asturias, Spain
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  • Gil González-Rodríguez,

    1. European Centre for Soft Computing, Edificio Científico-Tecnológico, Calle Gonzalo Gutiérrez Quirós S/N 33600 Mieres, Asturias, Spain
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  • Renato Coppi

    1. Dipartimento di Statistica, Probabilitá e Statistiche Applicate, Sapienza Università di Roma, Piazzale Aldo Moro 5 00185, Rome, Italy
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Abstract

Fuzzy sets are often used to handle the imprecision/vagueness that affects some characteristics in environmental sciences. A determination coefficient is introduced in order to quantify the degree of relationship between an imprecise response variable and a scalar explanatory predictor in a linear regression problem. An estimator of such coefficient useful to measure the goodness of fit of the model is proposed and its strong consistency is proved. Moreover, a specific linear independence testing procedure is established and both the asymptotic significance level and the power under local alternatives are established. Since the asymptotic results require large samples, a consistent bootstrap approach is developed. The empirical behavior of the suggested methods is illustrated by means of some simulations and real-life examples. Copyright © 2010 John Wiley & Sons, Ltd.

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