Abstract
This thesis consists of five papers (Papers AE) treating problems in nonparametric statistics, especially methods of kernel smoothing applied to density estimation for stochastic processes (Papers AD) 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 continuoustime models.
In Papers A and B we derive expressions for the asymptotic variance of the kernel density estimator of a continuoustime 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 continuoustime version of a leastsquares crossvalidation 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 treats 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 nonparametric regression analysis when data is sampled with a sizebias. Our method covers a wider range of practical situations than previously studied methods and by viewing the problem as a locally weighted leastsquares regression problem, extensions to higher order polynomial estimators are straightforward.
In Papers A and B we derive expressions for the asymptotic variance of the kernel density estimator of a continuoustime 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 continuoustime version of a leastsquares crossvalidation 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 treats 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 nonparametric regression analysis when data is sampled with a sizebias. Our method covers a wider range of practical situations than previously studied methods and by viewing the problem as a locally weighted leastsquares regression problem, extensions to higher order polynomial estimators are straightforward.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  1999 Nov 12 
Publisher  
ISBN (Print)  9162838121 
Publication status  Published  1999 
Bibliographical note
Defence detailsDate: 19991112
Time: 10:15
Place: Centre for Mathematical Sciences MH:B
External reviewer(s)
Name: Bosq, Denis
Title: Prof.
Affiliation: Paris VI, France.

Subject classification (UKÄ)
 Probability Theory and Statistics
Free keywords
 deconvolution
 errorsinvariables
 continuous time
 dependent data
 bandwidth selection
 asymptotic variance
 Density estimation
 kernel smoothing
 size bias.
 Mathematics
 Matematik