Simultaneous Confidence Regions in a Flexible Linear Regression for Random Intervals: A Bootstrap Approach
Angela Blanco-Fernandez, Ana Colubi, Gil Gonzalez-Rodriguez
Statistics and Operational Research, University of Oviedo, Oviedo, Asturias, Spain

When real experimental data take interval values (representing, for instance, fluctuations or variations of a magnitude along a given period of time), they are usually modeled by means of the so-called interval-valued random variables, or random intervals. The linear relationship between two random intervals can be formalized through an interval arithmetic-based linear model with flexible and versatile properties. The LS estimation of this model has been addressed in Blanco-Fernandez et al., 2011. The closed form of the LS estimators is found as the solution of an optimization problem with constraints. In this way, it is guaranteed that they keep the coherency with the considered regression model. In the interval scenario exact parametric methods are not feasible yet for inferential studies, since no realistic parametric models to describe the distribution of the random intervals have been shown to be widely applicable in practice. Inferential studies for the linear model can be developed by means of asymptotic techniques, based on the study of the limit distributions of the regression estimators (see, for instance, Gil et al., 2007). To improve the results for finite sample sizes, bootstrap methods are widely considered (see, for instance, Colubi, 2009, Blanco-Fernandez et al., 2010). In this work, a bootstrap algorithm for the construction of simultaneus confidence regions for the regression function is proposed. The procedure is based on the classical method of paired bootstrap, since both intervals in the linear model are considered as random elements (see Efron & Tibshirani, 1993). The technique is reinforced by means of its application over a real-life example, and some simulation studies.

Blanco-Fernandez, A., Corral, N., Gonzalez-Rodriguez, G., 2011. Estimation of a flexible simple linear model for interval data based on the set arithmetic. Submitted.

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Gil, M.A., Gonzalez-Rodriguez, G., Colubi, A. Montenegro M., 2007. Testing linear independence in linear models with interval-valued data. Computational Statistics & Data Analysis 51, 3002-3015.

Blanco-Fernandez, A., Corral, N., Gonzalez-Rodriguez, G., Palacio, A., 2010. On some confidence regions to estimate a linear regression model for interval data. In: Borgelt, C. et al. (Eds.) Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing 77, 33-40.