Variance function estimation via model selection

Teresa Ledwina¹ ; Jan Mielniczuk²

¹ Institute of Mathematics Polish Academy of Sciences Kopernika 18 51-617 Wrocław, Poland
² Institute of Computer Science Ordona 21 01-237 Warszawa, Poland and Warsaw University of Technology Plac Politechniki 1 00-661 Warszawa, Poland

Applicationes Mathematicae, Tome 37 (2010) no. 4, pp. 387-411

Résumé

The problem of estimating an unknown variance function in a random design Gaussian heteroscedastic regression model is considered. Both the regression function and the logarithm of the variance function are modelled by piecewise polynomials. A finite collection of such parametric models based on a family of partitions of support of an explanatory variable is studied. Penalized model selection criteria as well as post-model-selection estimates are introduced based on Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) methods of estimation of the parameters of the models. The estimators are defined as ML or REML estimators in the models with dimensions determined by respective selection rules. Some encouraging simulation results are presented and consistency results on the solution pertaining to ML estimation for this approach are proved.

Détail

DOI : 10.4064/am37-4-1

Keywords: problem estimating unknown variance function random design gaussian heteroscedastic regression model considered regression function logarithm variance function modelled piecewise polynomials finite collection parametric models based family partitions support explanatory variable studied penalized model selection criteria post model selection estimates introduced based maximum likelihood restricted maximum likelihood reml methods estimation parameters models estimators defined reml estimators models dimensions determined respective selection rules encouraging simulation results presented consistency results solution pertaining estimation approach proved