A specification test for the Single Functional Index Model

The problem of specifying the link function that models the dependence structure between two random elements is a very important task in regression analysis. In the framework of scalar on function regression, the link function is described through a real valued operator acting on a functional space and it is difficult to visualize and hence select a coherent specification. A test for specification in such a framework that exploits semi-parametric principles is illustrated, in particular by exploiting the Single Functional Index Model. The test statistic is a special form of U-statistic and once it is defined, its asymptotic null distribution is derived under suitable conditions. The finite sample performances of the test are analyzed through a simulation study by using both the asymptotic p-value and some bootstrap approaches. To demonstrate the potentialities of the method, an application to a spectrometric real dataset is performed.