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 &amp; 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

#### 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.
##### Scheda breve Scheda completa
2011
bootstrap procedure; determination coefficient; lr fuzzy data; linear independence test; regression models
01 Pubblicazione su rivista::01a Articolo in rivista
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]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/340120
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