[Statlist] Next talk: Friday, June 22, 2012 with Richard Nickl, Cambridge University

[Cecilia Rey]

ETH and University of Zurich

Proff. P. Buehlmann - L. Held - H.R. Kuensch -
M. Maathuis -  S. van de Geer - M. Wolf


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We are glad to announce the following talk

Friday, June 22, 2012, 15.15h, HG G 19.1

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  by Richard Nickl, Cambridge University

Title:
Donsker's central limit theorem for Estimating Lévy Measures

Absract:

(This is joint work with M.Reiss)
We consider the problem of statistical inference on the jump  
distribution of a L\'evy process $L_t$ from a sample of equidistant  
'low-frequency' observations of the trajectory of the process. We  
construct a natural estimator $\hat N$ for the cumulative distribution  
function $N$ of the L\'evy measure. Under a polynomial decay  
restriction on the characteristic function $\phi$ a Donsker-type  
theorem is proved, that is, a functional central limit theorem for the  
process $\sqrt n (\hat N_n -N)$ in the space of bounded functions away  
from zero. The limit distribution is a generalised Brownian bridge  
process with bounded and continuous sample paths whose covariance  
structure resembles the 'ill-posedness' of the problem, and which, as  
we show, is efficient, that is, it attains the semiparametric Cramer- 
Rao lower bound in this infinite-dimensional model. The class of L 
\'evy processes covered by our result includes several relevant  
examples such as compound Poisson,
Gamma and self-decomposable processes. The result can be used, as the  
classical Donsker theorem for empirical distributions, for various  
concrete statistical applications, such as the construction of  
confidence bands, Kolmogorov-Smirnov type goodness of fit tests, and  
can serve as an efficient plug-in estimator for various Hadamard  
differentiable functionals.


The abstract is also to be found here:  http://stat.ethz.ch/events/research_seminar



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