A consistent bootstrap linearity test for a regression model with an imprecise response

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.