A Consistent Bootstrap Linearity Test for a Regression Model with an Imprecise Response
Maria Brigida Feraro1, Gil Gonzalez Rodriguez2, Ana Colubi2
1Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Rome, Italy; 2Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Oviedo, Asturias, Spain

A new linearity test for a regression model with an imprecise response is investigated. The imprecise variable is modeled by means of a particular fuzzy random variable determined by three values: the center, the left and the right spread. In details, we jointly consider three models whose responses are the center and two transforms of the left and the right spreads. In this context it is important to check if the relationship is indeed linear. The proposed linearity test is based on empirical processes of the regressors marked by the residuals.

Taking into account some properties of the linear combinations, it can be used a linear combination of the test statistics referred to each model or a test statistic of a model in which the response is a linear combination of the three responses. The problem is faced with a bootstrap approach. The asympotic distribution of the test statistic is checked and the consistency of the bootstrap test is discussed.


Efron, B., Tibshirani, R.J. 1993. An introduction to the bootstrap, Chapman & Hall, New York.

Ferraro, M.B., Coppi, R., González-Rodríguez, G., Colubi, A., 2010a. A linear regression model for imprecise response. International Journal of Approximate Reasoning 51, 759–770.

Ferraro, M.B., Colubi, A., González-Rodríguez, G., Coppi, R., 2010b. A determination coefficient for a linear regression model with imprecise response. Environmetrics (in press, doi:10.1002/env.1056).

Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. Journal of Mathematical Analysis and Applications 114, 409–422.

Stute W (1997) Nonparametric model checks for regression. Ann Statist 25:613–641

Stute W, Gonzalez Manteiga W, Presedo Quindimil M (1998) Bootstrap approximations in model checks for regression. J Amer Stat Assoc 93:141–149

Zadeh, L.A., 1965. Fuzzy sets. Information and Control 8, 338–353.

Keywords: fuzzy random variables; regression model; linearity test; bootstrap approach

Biography: Ph.D. in Statistical Methodology - Sapienza University of Rome.

Postdoc Position, Department of Statistical Sciences, Sapienza University of Rome - sector: SECS-S/01 Topic: Statistical Methodology concerning techniques of explorative and inferential analysis.