Continuous-Time Models in Kernel Smoothing

Martin Sköld

Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
1999

ISBN 91-628-3812-1
LUNFMS--1009--1999

Abstract:

This thesis consists of five papers (Papers A-E) treating problems in non-parametric statistics, especially methods of kernel smoothing applied to density estimation for stochastic processes (Papers A-D) and regression analysis (Paper E). A recurrent theme is to, instead of treating highly positively correlated data as "asymptotically independent", take advantage of local dependence structures by using continuous-time models.

In Papers A and B we derive expressions for the asymptotic variance of the kernel density estimator of continuous-time multivariate stationary process and relate convergence rates to the local character of the sample paths. This is in Paper B applied to automatic selection of smoothing parameter of the estimators. In Paper C we study a continuous-time version of a least-squares cross-validation approach to selecting smoothing parameter, and the impact the dependence structure of data has on the algorithm.  A correction factor is introduced to improve the methods performance for dependent data. Papers D and E treat two statistical inverse problems where the interesting data are not directly observable. In Paper D we consider the problem of estimating the density of a stochastic process from noisy observations. We introduce a method of smoothing the errors and show that by a suitably chosen sampling scheme the convergence rate of independent data methods can be improved upon. Finally in Paper E we treat a problem of non-parametric regression analysis when data is sampled with a size-bias. Our method covers a wider range of practical situations than previously studied methods and by viewing the problem as a locally weighted least-squares regression problem, extensions to higher order polynomial estimators are straightforward.

Key words:

Density estimation, kernel smoothing, asymptotic variance, bandwidth selection, dependent data, continuous time, errors-in-variables, deconvolution, size bias.