• LR fuzzy data;
  • regression models;
  • determination coefficient;
  • linear independence test;
  • bootstrap procedure


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.