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 (C) 2010 John Wiley & Sons, Ltd.
A determination coefficient for a linear regression model with imprecise response / Ferraro, MARIA BRIGIDA; A., Colubi; G., Gonzalez Rodriguez; Coppi, Renato. - In: ENVIRONMETRICS. - ISSN 1180-4009. - 22:4(2011), pp. 516-529. [10.1002/env.1056]
A determination coefficient for a linear regression model with imprecise response
FERRARO, MARIA BRIGIDA;COPPI, Renato
2011
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 (C) 2010 John Wiley & Sons, Ltd.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.