Research Group "Stochastic Algorithms and Nonparametric Statistics"
Research Seminar "Mathematical Statistics" Winter semester 2009/2010
last reviewed: October, 27, 2009, Christine Schneider
A. Maurer (München)
Similarity and transfer learning
Abstract: Different learning
tasks, such as recognizing the images of capital letters or recognizing
the images of digits 0-9, may share a notion of pattern similarity. Two
images may for example be regarded as similar if one is a differently
scaled version of the other, in which case they are likely to belong to
the same category. If we learn to recognize the similar pairs of a
group of tasks, we may hope to generalize this knowledge to a new task
whose categories obey the same similarity relation. This can greatly
improve learning and classification on the new task, even if only very
little data is available. The talk formalizes the problem of
similarity-learning from examples and gives the derivation of a simple
algorithm which selects a feature-map to minimize a regularized
estimator. Uniform bounds on the estimation error over certain classes
of such feature-maps provide a, theoretical justification. The talk
concludes with a presentation of experimental results.
Erwan Le Pennec (Université Paris Diderot)
Adaptive Dantzig density estimation
Abstract: In the context of
density estimation, we propose a estimator inspired by the Dantzig
estimator of Candes and Tao. The estimated density is the linear
combination of functions of a dictionary which minimizes the l1 norm
of its coefficients under a constraint on the empirical scalar
products. Using sharp concentration inequalities, we propose a data
dependent constraint that leads to oracle inequalities under mild
"local" assumptions on the dictionary. We will show the connection with
the more classical Lasso procedure and explain why our constraint can
be called "minimal". (joint work with K. Bertin and V. Rivoirard)
Alexander McNeil
(Edinburgh)
From Archimedean to Liouville Copulae
Abstract:The Archimedean copula family is used in a number of actuarial and nancial applications
including: the construction of multivariate loss distributions; frailty models for dependent lifetimes;
models for dependent defaults in credit risk. We show how the Archimedean copulas are most usefully
viewed as the survival copulas of so-called simplex distributions, which are scale mixtures of uniform
distributions on simplices. This representation allows us to construct a rich variety of new Archimedean
copulas in dierent dimensions and to solve in principle the problem of generating samples from any
Archimedean copula. It also sheds light on the dependence properties of Archimedean copulas and their
relationship to the mixing or so-called radial distribution of the simplex distribution. Armed with these
insights we generalise the Archimedean copulas to the Liouville copulas, which are the survival copulas
of Liouville distributions, these being scale mixtures of Dirichlet distributions. This generalisation yields
asymmetric, non-exchangeable copulas, whose properties can again be understood in terms of the mixing
or radial distribution of the Liouville distribution.
Abderrahim Taamouti
A Nonparametric Copula based test for Conditional Independence with Applications to Granger Causality
Abstract:This paper proposes a new nonparametric test for conditional independence, which is based on the comparison of Bernstein copula densities using the Hellinger distance. The test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establish local power properties, and motivate the validity of the bootstrap technique that we use in nite sample settings. A simulation study illustrates the good size and power properties of the test. We illustrate the empirical relevance of our test by focusing on Granger causality using nancial time series data to test for nonlinear leverage versus volatility feedback eects and to test for causality between stock returns and trading volume. In a third application, we investigate Granger causality between macroeconomic variables. (joint with Taouk Bouezmarni and Jeroen V.K. Rombouts) tba
Piotr Jaworski
Tail behaviour of copulas + Value at Risk, Conditional Copulas + Market Contagion
Abstract: tba
Paul Doukhan
Weak dependence and some applications
Abstract:Weak dependence as dened in Doukhan and Louhichi is detailed in a lecture notes by
Dedecker et al. 2007; this notion is aimed at providing a reasonnable extension of the standard mixing
conditions such as Rosenblatt 1956's one. Our idea is very elementary and includes more reasonnable
models. After a motivation of our notions we list some examples of times series models of interest. Several
applications may be listed and two of them are detailed. The rst one is quite fundamental for statistical
purpose; we show how to estimate the limit variance in the CLTs. A second application to subsampling
is also considered, with applications to extremes. Quote that our specic conditions need some more
technical diculties due to to the diculty of heredity of our notions.
tba
Jean-Yves Audibert
Risk bounds in linear regression through PAC-Bayesian truncation
Abstract:We consider the problem of predicting as well as the best linear
combination of d given functions in least squares regression, and
variants of this problem including constraints on the parameters of the
linear combination. When the input distribution is known, there already
exists an algorithm having an expected excess risk of order d/n, where n
is the size of the training data. Without this strong assumption,
standard results often contain a multiplicative log n factor, and
require some additional assumptions like uniform boundedness of the
d-dimensional input representation and exponential moments of the
output. This work provides new risk bounds for the ridge estimator and
the ordinary least squares estimator, and their variants. It also
provides shrinkage procedures with convergence rate d/n (i.e., without
the logarithmic factor) in expectation and in deviations, under various
assumptions. The key common surprising factor of these results is the
absence of exponential moment condition on the output distribution while
achieving exponential deviations. All risk bounds are obtained through a
PAC-Bayesian analysis on truncated differences of losses. Finally, we
show that some of these results are not particular to the least squares
loss, but can be generalized to similar strongly convex loss functions.
Jean-Yves Audibert
Risk bounds in linear regression through PAC-Bayesian truncation
Abstract:We consider the problem of predicting as well as the best linear
combination of d given functions in least squares regression, and
variants of this problem including constraints on the parameters of the
linear combination. When the input distribution is known, there already
exists an algorithm having an expected excess risk of order d/n, where n
is the size of the training data. Without this strong assumption,
standard results often contain a multiplicative log n factor, and
require some additional assumptions like uniform boundedness of the
d-dimensional input representation and exponential moments of the
output. This work provides new risk bounds for the ridge estimator and
the ordinary least squares estimator, and their variants. It also
provides shrinkage procedures with convergence rate d/n (i.e., without
the logarithmic factor) in expectation and in deviations, under various
assumptions. The key common surprising factor of these results is the
absence of exponential moment condition on the output distribution while
achieving exponential deviations. All risk bounds are obtained through a
PAC-Bayesian analysis on truncated differences of losses. Finally, we
show that some of these results are not particular to the least squares
loss, but can be generalized to similar strongly convex loss functions.
Arnold Janssen
Anwendungen der Le Cam Theorie in der
Finanzmathematik
Abstract: Le Cam's theory of statistical experiments is a likelihood based universal tool in statistics. In
this talk we will apply this approach to financial models. In particular we are going to talk
about martingale measures, completeness of financial models, pricing of options, and the
approximation of Ito type financial models by discrete models.(joint work with Martin Tietje)
Peter Reinhard Hansen (Stanford University)
Realized GARCH: A Complete Model of Returns and Realized Measures of
Volatility
Abstract: tba
