Contributions to the Theory of Nonparametric Regression, with Application to System Identification

The objective in nonparametric regression is to infer a function $m(x)$ on the basis of a finite collection of noisy pairs \(Xᵢ, m(Xᵢ) + Nᵢ)\ⁿᵢ₌₁, where the noise components Nᵢ satisfy certain lenient assumptions and the domain points Xᵢ are selected at random. It is known a priori only that m is a member of a nonparametric class of functions (that is, a class of functions like C 0, 1 which, under customary topologies, does not admit a homeomorphic indexing by a subset of a Euclidean space). The main theoretical contribution of this study is to derive uniform convergence bounds and uniform consistency on bounded intervals for the Nadaraya-Watson kernel estimator and its derivatives. Also, we obtain the corresponding convergence results for the Priestly-Chao estimator in the case that the domain points are nonrandom. With these developments we are able to apply nonparametric regression methodology to the problem of identifying noisy time-varying linear systems.