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A determination coefficient for a linear regression model with imprecise response
Author(s) -
Ferraro Maria Brigida,
Colubi Ana,
GonzálezRodríguez Gil,
Coppi Renato
Publication year - 2011
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/env.1056
Subject(s) - mathematics , linear regression , proper linear model , estimator , linear model , consistency (knowledge bases) , linear predictor function , statistics , vagueness , econometrics , explanatory power , independence (probability theory) , fuzzy logic , bayesian multivariate linear regression , computer science , artificial intelligence , philosophy , geometry , epistemology
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.