Truncated linear models for functional data

Abstract

A conventional linear model for functional data involves expressing a response variableYin terms of the explanatory functionX(t), via the model, whereais a scalar,bis an unknown function andis a compact interval. However, in some problems the support ofborX,say, is a proper and unknown subset of, and is a quantity of particular practical interest. Motivated by a real data example involving particulate emissions, we develop methods for estimating. We give particular emphasis to the case, whereθ∈ (0,α], and suggest two methods for estimatinga,bandθjointly; we introduce techniques for selecting tuning parameters; and we explore properties of our methodology by using both simulation and the real data example mentioned above. Additionally, we derive theoretical properties of the methodology and discuss implications of the theory. Our theoretical arguments give particular emphasis to the problem of identifiability.