Nov 20.-22., 2008, G ottingen Estimation of Different Scales in - PowerPoint PPT Presentation

Nov 20.-22., 2008, G¨ ottingen Estimation of Different Scales in Microstructure DFG-SNF Research Group Opening Conference Noise Models ’Statistical Regularisation’ Munk Tony Cai, University of Pennsylvania Introduction Rama Cont, Columbia University Motivating Jos Manuel Corcuera, Universitat de Barcelona example Jean-Pierre Florens, Universit Toulouse I Data transformation Peter Gottschalk, Boston College Janine Illian, University of St Andrews Estimation of Karl Kunisch, Universitt Graz τ 2 Guillaume Lecue, CNRS, LATP, Marseille Jens Perch Nielsen, Cass Business School, London Estimation of Nicolai Meinshausen, University of Oxford σ 2 Ya’acov Ritov, The Hebrew University of Jerusalem Construction of Naftali Tishby, The Hebrew University of Jerusalem sharp estimator for σ 2 Alexandre Tsybakov, Universit Paris VI Numerics Non-constant http: σ and τ //www.stochastik.math.uni-goettingen.de/forschergruppe/ Summary/ Outlook

Estimation of Different The Estimation of Different Scales in Scales in Microstructure Noise Models Microstructure Noise Models from a Munk Nonparametric Regression Perspective Introduction Motivating example Data transformation Axel Munk Estimation of τ 2 Joint work with T. Cai and J. Schmidt-Hieber Estimation of σ 2 Institut f¨ ur Mathematische Stochastik, G¨ ottingen Construction of sharp estimator for σ 2 Numerics www.stochastik.math.uni-goettingen.de/munk Non-constant munk@math.uni-goettingen.de σ and τ Summary/ Outlook

Introduction Estimation of Different Scales in 1 Introduction Microstructure Noise Models Motivating example Munk Data transformation Introduction Motivating 2 Estimation of τ 2 example Data transformation Estimation of 3 Estimation of σ 2 τ 2 Construction of sharp estimator for σ 2 Estimation of σ 2 Numerics Construction of sharp estimator for σ 2 Numerics 4 Non-constant σ and τ Non-constant σ and τ 5 Summary/ Outlook Summary/ Outlook

The model Estimation of Different Suppose we observe (Stein ’87, Gloter and Jacod ’01) Scales in Microstructure Noise Models Y i , n = σ W i / n + τǫ i , n , i = 1 , . . . , n , Munk Introduction where Motivating example (i) W t is a (Standard) Brownian Motion, Data transformation i.i.d Estimation of (ii) ǫ i , n ∼ N (0 , 1), τ 2 (iii) σ, τ > 0 are unknown scale parameters. Estimation of σ 2 Construction of (iv) ǫ i , n and W t are assumed to be independent for all i and all sharp estimator for σ 2 t ∈ [0 , 1]. Numerics Non-constant σ and τ Summary/ Outlook

The model Estimation of Different Suppose we observe (Stein ’87, Gloter and Jacod ’01) Scales in Microstructure Noise Models Y i , n = σ W i / n + τǫ i , n , i = 1 , . . . , n , Munk Introduction where Motivating example (i) W t is a (Standard) Brownian Motion, Data transformation i.i.d Estimation of (ii) ǫ i , n ∼ N (0 , 1), τ 2 (iii) σ, τ > 0 are unknown scale parameters. Estimation of σ 2 Construction of (iv) ǫ i , n and W t are assumed to be independent for all i and all sharp estimator for σ 2 t ∈ [0 , 1]. Numerics Non-constant Statistical Problem σ and τ Estimation of the parameters σ 2 and τ 2 . Summary/ Outlook