New Topics in Nonlinear Functional Data Analysis
Sammendrag
Functional data analysis (FDA) comprises statistical methods for data that can be considered as partial or full observations of random curves, surfaces, or related smooth objects. Although much progress has been made in the last decades, a majority of these methods is linear and many important techniques remain absent from this infinite-dimensional branch of statistics. In this dissertation, two key concepts from nonlinear multivariate statistics are introduced to the FDA toolbox: copulas and power variations. Both offer entirely new nonparametric ways to analyse dependence structures of various infinite-dimensional random objects and have immediate applications in fields such as mathematical finance or physics. Due to the intrinsic infinite dimensionality, however, there are significant differences to their corresponding multivariate counterparts which give rise to various mathematical challenges that are addressed in this work.Artikkelliste
Paper I. Benth, F.E., Di Nunno, G. and Schroers, D. “Copula Measures and Sklar’s Theorem in Arbitrary Dimensions”. In: Scandinavian Journal of Statistics, 2021, 49(3), 1144–1183. DOI: 10.1111/sjos.12559. The article is included in the thesis. Also available at: https://doi.org/10.1111/sjos.12559 |
Paper II. Benth, F.E., Di Nunno, G. and Schroers, D. “A Topological Proof of Sklar’s Theorem in Arbitrary Dimensions”. In: Dependence Modeling, 2022, 10(1), 22–28. DOI: 10.1515/demo-2022-0103. The article is included in the thesis. Also available at: https://doi.org/10.1515/demo-2022-0103 |
Paper III. Benth, F.E., Schroers, D., and Veraart, A.E.D. “A Weak Law of Large Numbers for Realised Covariation in a Hilbert Space Setting”. In: Stochastic Processes and their Applications. Vol. 145, (2022), pp. 241–268. DOI: 10.1016/j.spa.2021.12.011. The article is included in the thesis. Also available at: https://doi.org/10.1016/j.spa.2021.12.011 |
Paper IV. Benth, F.E., Schroers, D., and Veraart, A.E.D. “A Feasible Central Limit Theorem for Realised Covariation of SPDEs in the Context of Functional Data”. arXiv:2205.03927. Submitted for publication. To be published. The paper is not available in DUO awaiting publishing. |