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Continuous-Time Models in Kernel Smoothing

Sköld, Martin LU (1999) In Doctoral Theses in Mathematical Sciences 1999:5.
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 a 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... (More)
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 a 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 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 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. (Less)
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author
opponent
  • Prof. Bosq, Denis, Paris VI, France.
organization
publishing date
type
Thesis
publication status
published
subject
keywords
deconvolution, errors-in-variables, continuous time, dependent data, bandwidth selection, asymptotic variance, Density estimation, kernel smoothing, size bias., Mathematics, Matematik
in
Doctoral Theses in Mathematical Sciences
volume
1999:5
pages
100 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Centre for Mathematical Sciences MH:B
defense date
1999-11-12 10:15
external identifiers
  • other:LUNFMS-1009-1999
ISSN
1404-0034
ISBN
91-628-3812-1
language
English
LU publication?
yes
id
7f330c7a-9980-4d09-9e92-32f4b625b6b1 (old id 19290)
date added to LUP
2007-05-25 15:47:20
date last changed
2016-09-19 08:44:52
@phdthesis{7f330c7a-9980-4d09-9e92-32f4b625b6b1,
  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.<br/><br>
<br/><br>
In Papers A and B we derive expressions for the asymptotic variance of the kernel density estimator of a 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 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 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.},
  author       = {Sköld, Martin},
  isbn         = {91-628-3812-1},
  issn         = {1404-0034},
  keyword      = {deconvolution,errors-in-variables,continuous time,dependent data,bandwidth selection,asymptotic variance,Density estimation,kernel smoothing,size bias.,Mathematics,Matematik},
  language     = {eng},
  pages        = {100},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  series       = {Doctoral Theses in Mathematical Sciences},
  title        = {Continuous-Time Models in Kernel Smoothing},
  volume       = {1999:5},
  year         = {1999},
}