Variance function estimation via model selection

Volume 37 / 2010

Teresa Ledwina, Jan Mielniczuk Applicationes Mathematicae 37 (2010), 387-411 MSC: 62J99, 62F12, 62G20. DOI: 10.4064/am37-4-1


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.


  • Teresa LedwinaInstitute of Mathematics
    Polish Academy of Sciences
    Kopernika 18
    51-617 Wrocław, Poland
  • Jan MielniczukInstitute of Computer Science
    Ordona 21
    01-237 Warszawa, Poland
    Warsaw University of Technology
    Plac Politechniki 1
    00-661 Warszawa, Poland

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