ABSTRACT
In Chapter 6 we have studied the functional linear model (FLM) (6.4) under the assumption that the time-independent design matrix X is a full-rank matrix. In that chapter, under some mild regularity conditions, various pointwise, L2-norm-based, F -type, and bootstrap tests for the FLM (6.4) were discussed. As in classical linear models, in many situations, however, the design matrixX in an FLM may be ill-conditioned, that is, it is non-full-rank. The associated FLM is called an ill-conditioned or non-full-rank FLM, which can be defined as follows:
y(t) = Xβ(t) + v(t), v(t) ∼ SPn(0, γIn), t ∈ T , (7.1)
where y(t) : n × 1 is a vector of response functions, X : n × (p + 1) is an ill-conditioned time-independent design matrix with rank k < (p + 1) < n, v(t) : n × 1 is a vector of subject-effect functions, and β(t) : (p + 1) × 1 is a vector of coefficient functions. When the first column of X is 1n, an ndimensional vector of ones, the first component of β(t) models the intercept function of the non-full-rank FLM (7.1). In this chapter, we aim to extend the methodologies developed for the full-rank FLM (6.4) in Chapter 6 for the above non-full-rank FLM (7.1).
