Truncated Linear Models for Functional Data

Summary A conventional linear model for functional data involves expressing a response variable Y in terms of the explanatory function X (t), via the model Y=a+∫Ib(t)X(t)dt+error, where a is a scalar, b is an unknown function and I=[0,α] is a compact interval. However, in some problems the support of b or X, I1 say, is a proper and unknown subset of I, and is a quantity of particular practical interest. Motivated by a real data example involving particulate emissions, we develop methods for estimating I1. We give particular emphasis to the case I1=[0,θ], where θ ∈ (0, α], and suggest two methods for estimating a, b and θ jointly; 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.