ABSTRACT

In the previous chapter, we studied how to reconstruct the individual functions from observed discrete functional data so that the reconstructed individual functions of a functional data set can be approximately modeled as i.i.d realizations of an underlying stochastic process. In this chapter, we introduce two important stochastic processes, namely, Gaussian process and Wishart process. These two processes play a central role in this book. Some of their important properties are investigated in Section 4.2 where we show that the squared L2-norm of a Gaussian process is a χ2-type mixture, and the trace of a Wishart process is also a χ2-type mixture. In Section 4.3, methods for approximating the distribution of a χ2-type mixture are introduced. The ratio of two independent χ2-type mixtures is called an F -type mixture. Some methods for approximating the distribution of an F -type mixture are given in Section 4.4. These methods are useful for statistical inferences about functional data. As applications of these methods, a one-sample problem for functional data is introduced in Section 4.5 where some basic inference techniques for functional data are introduced. Some remarks and bibliographical notes are given in Section 4.7. The chapter concludes with technical proofs of some main results in Section 4.6 and some exercises in Section 4.8.