Project acronym MAZEST
Project M- and Z-estimation in semiparametric statistics : applications in various fields
Researcher (PI) Ingrid Van Keilegom
Host Institution (HI) UNIVERSITE CATHOLIQUE DE LOUVAIN
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary The area of semiparametric statistics is, in comparison to the areas of fully parametric or nonparametric statistics, relatively unexplored and still in full development. Semiparametric models offer a valid alternative for purely parametric ones, that are known to be sensitive to incorrect model specification, and completely nonparametric models, which often suffer from lack of precision and power. A drawback of semiparametric models so far is, however, that the development of mathematical properties under these models is often a lot harder than under the other two types of models. The present project tries to solve this difficulty partially, by presenting and applying a general method to prove the asymptotic properties of estimators for a wide spectrum of semiparametric models. The objectives of this project are twofold. On one hand we will apply a general theory developed by Chen, Linton and Van Keilegom (2003) for a class of semiparametric Z-estimation problems, to a number of novel research ideas, coming from a broad range of areas in statistics. On the other hand we will show that some estimation problems are not covered by this theory, we consider a more general class of semiparametric estimators (M-estimators called) and develop a general theory for this class of estimators. This theory will open new horizons for a wide variety of problems in semiparametric statistics. The project requires highly complex mathematical skills and cutting edge results from modern empirical process theory.
Summary
The area of semiparametric statistics is, in comparison to the areas of fully parametric or nonparametric statistics, relatively unexplored and still in full development. Semiparametric models offer a valid alternative for purely parametric ones, that are known to be sensitive to incorrect model specification, and completely nonparametric models, which often suffer from lack of precision and power. A drawback of semiparametric models so far is, however, that the development of mathematical properties under these models is often a lot harder than under the other two types of models. The present project tries to solve this difficulty partially, by presenting and applying a general method to prove the asymptotic properties of estimators for a wide spectrum of semiparametric models. The objectives of this project are twofold. On one hand we will apply a general theory developed by Chen, Linton and Van Keilegom (2003) for a class of semiparametric Z-estimation problems, to a number of novel research ideas, coming from a broad range of areas in statistics. On the other hand we will show that some estimation problems are not covered by this theory, we consider a more general class of semiparametric estimators (M-estimators called) and develop a general theory for this class of estimators. This theory will open new horizons for a wide variety of problems in semiparametric statistics. The project requires highly complex mathematical skills and cutting edge results from modern empirical process theory.
Max ERC Funding
750 000 €
Duration
Start date: 2008-07-01, End date: 2014-06-30
Project acronym VNALG
Project Von Neumann algebras, group actions and discrete quantum groups
Researcher (PI) Stefaan Vaes
Host Institution (HI) KATHOLIEKE UNIVERSITEIT LEUVEN
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary Von Neumann algebras, and more specifically II_1 factors, arise naturally in the study of countable groups and their actions on measure spaces. A central, but extremely hard problem is the classification of these von Neumann algebras in terms of their group/action data. Breakthrough results were recently obtained by Sorin Popa. I presented a combined treatment of these in my Bourbaki lecture notes. In a joint work of Popa and myself, this gave rise to the full classification of certain generalized Bernoulli II_1 factors. In a recent article of mine, it lead for the first time to a family of II_1 factors for which the fusion algebra of finite index bimodules could be entirely computed. Popa's methods open up a wealth of research opportunities. They bring within reach the solution of several long-standing open problems, that constitute the main objectives of the first part of this research proposal: complete descriptions of the finite index subfactor structure of certain II_1 factors, constructions of II_1 factors with a unique group measure space decomposition and computations of orbit equivalence invariants for actions of the free groups. Even approaching these problems would have been completely hopeless just a few years ago. Other constructions of von Neumann algebras arise in the theory of discrete quantum groups. The first rigidity results for quantum group actions on von Neumann algebras constitute the main objective of this second part of the research proposal. Finally, we aim to deal with another connection between quantum groups and operator algebras, through the study of non-commutative random walks and their boundaries. The main originality of this research proposal lies in the interaction between two branches of mathematics: operator algebras and quantum groups. This is clear for the second part of the project and occupies a central place in the first part through subfactor theory.
Summary
Von Neumann algebras, and more specifically II_1 factors, arise naturally in the study of countable groups and their actions on measure spaces. A central, but extremely hard problem is the classification of these von Neumann algebras in terms of their group/action data. Breakthrough results were recently obtained by Sorin Popa. I presented a combined treatment of these in my Bourbaki lecture notes. In a joint work of Popa and myself, this gave rise to the full classification of certain generalized Bernoulli II_1 factors. In a recent article of mine, it lead for the first time to a family of II_1 factors for which the fusion algebra of finite index bimodules could be entirely computed. Popa's methods open up a wealth of research opportunities. They bring within reach the solution of several long-standing open problems, that constitute the main objectives of the first part of this research proposal: complete descriptions of the finite index subfactor structure of certain II_1 factors, constructions of II_1 factors with a unique group measure space decomposition and computations of orbit equivalence invariants for actions of the free groups. Even approaching these problems would have been completely hopeless just a few years ago. Other constructions of von Neumann algebras arise in the theory of discrete quantum groups. The first rigidity results for quantum group actions on von Neumann algebras constitute the main objective of this second part of the research proposal. Finally, we aim to deal with another connection between quantum groups and operator algebras, through the study of non-commutative random walks and their boundaries. The main originality of this research proposal lies in the interaction between two branches of mathematics: operator algebras and quantum groups. This is clear for the second part of the project and occupies a central place in the first part through subfactor theory.
Max ERC Funding
500 000 €
Duration
Start date: 2008-09-01, End date: 2013-08-31
