Some applications of stochastic process techniques to statistics
Appears in the following Collection
- Matematikk 
AbstractThe thesis studies certain mathematical aspects of model selection, statistical estimation theory and probability using stochastic process tools.
The paper “The copula information criterion” is the paper with most direct statistical applications and concerns the so-called model selection problem: Suppose you have several seemingly equally good statistical models of a phenomena – which one should you use? This problem can be approached in many ways, and in classical statistical models, the problem is considered to a large extent solved by the so-called AIC formula. This formula is used also in more non-standard statistical models, and our paper shows that this may lead to misleading results. By generalizing the AIC, we provide a correct model selection formula for multivariate statistical models called copula models, which are commonly used in financial applications. However, the story is somewhat more complex: If we follow the original motivation of the AIC formula, we show that no such formula can exist for the most commonly used copula models. We then generalize another motivation of the AIC formula to our setting, and find a generally applicable formula. The moral of the story is that the AIC formula must be re-derived for non-classical models, and the result may be drastically different: it may even not exist!
The paper “On the errors committed by sequences of estimator functionals” is a paper on probability, with applications to confidence intervals and computer simulations. Suppose you have a computer program that approximates the behavior of a statistical system to a higher and higher precision as the computation time increases. When should you stop the computation? We provide an approximate answer to the point when even if you would continue indefinitely further, the resulting gain in precision would not improve significantly.
The paper “Estimation and Inference for Jump Regression Models” concerns the estimation of discontinuous phenomena. It exemplifies that in non-standard statistical models (for example those describing discontinuities), the standard tools of classical statistics are sub-optimal, and even from a classical perspective so-called Bayesian estimators are superior. We quantify these statements precisely, and provide an improvement to the computation process for estimating such models.
List of papers. Papers III and IV are removed from the thesis due to copyright restrictions.
Paper I Steffen Grønneberg and Nils Lid Hjort. On the errors committed by sequences of estimator functionals. Mathematical Methods of Statistics, 2011, Vol. 20, No. 4, pp. 327–346. (c) Allerton Press, Inc., 2011. The original publication is available at www.springerlink.com doi:10.3103/S106653071104003X
Paper II Steffen Grønneberg. The copula information criterion and its implications for the maximum pseudo likelihood estimator. Pages 113-138 in: Dependence Modeling, Vine Copula Handbook, Dorota Kurowicka and Harry Joe (Ed.), Copyright 2011 World Scientific Pub. With permission from World Scientific Publishing Co. Pte. Ltd.
Paper III Steffen Grønneberg and Nils Lid Hjort. The copula information criteria. Manuscript, not yet published.
Paper IV Steffen Grønneberg, Gudmund Hermansen, and Nils Lid Hjort. Estimation and inference for jump regression models. Manuscript, not yet published.