Research Group "Stochastic Algorithms and Nonparametric Statistics"
Research Seminar "Mathematical Statistics" Winter Semester 2013/14
Place: 
WeierstrassInstitute for Applied Analysis and Stochastics


ErhardSchmidtHörsaal, Mohrenstraße 39, 10117
Berlin 
Time: 
Wednesdays, 10.00 a.m.  12.30 p.m. 
23.10.2013 
Dominique Bontemps (Université Paul Sabatier, Toulouse) 

Bayesian posterior consistency and contraction rates in the shape invariant model 
30.10.2013 
Prof. Denis Belomestny (Universität Essen) 
Hausvoglteiplatz 11a, Raum 4.13 
On one inverse problem of nancial mathematics with
error in the operator 
06.11.2013 
Stéphane Gaïffas (École Polytechnique Palaiseau, France) 

Link prediction in graphs with timeevolving features 
13.11.2013 
Mathieu Sart (Université Sophia Antipolis, Nizza) 
Hausvoglteiplatz 11a, Raum 4.13 
Estimation of the transition density of a Markov chain 
20.11.2013 
VictorEmmanuel Brunel (ENSAE/ParisTech, France) 

Adaptive estimation of convex polytopes 
27.11.2013 
Marc Hoffmann (Université Paris Dauphine) 

Statistical estimation of a growthfragmentation model
observed on a genealogical tree 
04.12.2013 
Jia Li (Duke University, Durham, USA) 

Inference on volatility functional dependence 
11.12.2013 
Professor Matt Wand (University of Technology, Sydney, Australia) 
Hausvoglteiplatz 11a, Raum 4.13 
Variational Approximations in Statistics I 
18.12.2013 
Professor Matt Wand (University of Technology, Sydney, Australia) 

Variational Approximations in Statistics II

25.12.2013 
Xmas! 


01.01.2013 
Happy New Year! 


08.01.2014 
Prof. Angelika Rohde (RuhrUniversität Bochum) 

Adaptation to lowest density regions with application to support recovery 
15.01.2014 
Nicolai Meinshausen (ETH Zürich) 

Challenges for highdimensional inference 
22.01.2014 
Prof. Thomas Kneib (GeorgAugustUniversität Göttingen) 

Structured additive distributional regression 
29.01.2014 
Dr. Axel Bücher (RuhrUniversität Bochum) 

When uniform weak convergence fails: empirical processes
for dependence functions via epi and hypographs 
05.02.2014 



12.02.2014 
Bo Markussen (University of Copenhagen) 

Functional data, operator calculus, and statistical
computation 
last reviewed: January 21, 2014, Christine Schneider
Dominique Bontemps (Université Paul Sabatier, Toulouse)
Bayesian posterior consistency and contraction rates in the shape invariant model
Abstract:In this work, we consider the socalled Shape Invariant Model which stands for the estimation of a function f0 submitted to a random translation of law g0 in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process P(f0,g0) as well as f0 and g0.
In this perspective, we adopt a Bayesian point of view and find prior on f and g such that the posterior distribution concentrates around P(f0,g0) at a polynomial rate when n goes to infinity. We obtain a logarithmic posterior contraction rate for the shape f0 and the distribution g0. We also derive logarithmic lower bounds for the estimation of f0 and g0 in a frequentist paradigm.
Prof. Denis Belomestny (Universität Essen)
On one inverse problem of nancial mathematics with
error in the operator
Abstract: In this talk we consider a calibration problem for the so called Markov Functional Models
(MFM) from the statistical point of view. It is shown that at each step of the calibration procedure one
has to solve a nonlinear inverse problem with the operator estimated in the previous step. We propose a
regularisation method and derive the optimal convergence rates.
Stéphane Gaïffas (École Polytechnique Palaiseau, France)
Link prediction in graphs with timeevolving features
Abstract:
Estimation of the transition density of a Markov chain
Abstract:
VictorEmmanuel Brunel (ENSAE/ParisTech, France)
Adaptive estimation of convex polytopes
Abstract: We consider a sample of $n$ i.i.d. random variables, uniformly distributed in some unknown polytope $P$ in $\R^d$. We propose a maximum likelihood estimator whose accuracy, defined as the expectation of the Lebesgue measure of its symmetric difference with the true polytope, is at most of the order $r(\ln n)/n$, when $r$, the number of vertices of $P$, is known. Using a concentration inequality for this estimator, we develop a procedure in order to estimate $P$ adaptively with respect to $r$, based on a model selection method. The adaptive estimator achieves the same rate as for the case of known $r$.
Marc Hoffmann (Université Paris Dauphine)
Statistical estimation of a growthfragmentation model
observed on a genealogical tree
Abstract: We model the growth of a cell population by a piecewise deterministic Markov branching
tree. Each cell splits into two osprings at a division rate B(x) that depends on its size x. The size of
each cell grows exponentially in time, at a rate that varies for each individual. We show that the mean
empirical measure of the model satises a growthfragmentation type equation if structured in both size
and growth rate as state variables. We construct a nonparametric estimator of the division rate B(x) based
on the observation of the population over dierent sampling schemes of size n on the genealogical tree.
Our estimator nearly achieves the rate nôs=(2s+1) in squaredloss error asymptotically. When the growth
rate is assumed to be identical for every cell, we retrieve the classical growthfragmentation model and our
estimator improves on the rate nôs=(2s+3) obtained in a related framework through indirect observation
schemes. Our method is consistently tested numerically and implemented on Escherichia coli data.
Jia Li (Duke University, Durham, USA)
Inference on volatility functional dependence
Abstract: We develop inference theory for models involving possibly nonlinear transforms of the elements of the spot covariance matrix of a multivariate continuoustime process observed at high frequency. The framework can be used to study the relationship among the elements of the latent spot covariance matrix and processes defined on the basis of it such as systematic and idiosyncratic variances, factor betas and correlations. The estimation is based on matching modelimplied moment conditions under the occupation measure induced by the spot covariance process. We prove consistency and asymptotic mixed normality of our estimator of the (random) coefficients in the volatility model and further develop model specification tests. We apply our inference methods to study variance and correlation risks in nine sector portfolios comprising the S\&P 500 index. We document sectorspecific variance risks in addition to that of the market and timevarying heterogeneous correlation risk among the marketneutral components of the sector portfolio returns.
(Joint with V. Todorov and G. Tauchen)
Professor Matt Wand (University of Technology, Sydney, Australia)
Variational Approximations in Statistics I
Abstract:Variational approximations facilitate approximate inference
for the parameters in complex statistical models and provide
fast, deterministic alternatives to Monte Carlo methods. However,
much of the contemporary literature on variational approximations
is in Computer Science rather than Statistics, and
uses terminology, notation, and examples from the former field.
In this series of lectures we explain variational approximation in statistical
terms. In particular, we illustrate the ideas of variational approximation
using examples that are familiar to statisticians.
Professor Matt Wand (University of Technology, Sydney, Australia)
Variational Approximations in Statistics II
Abstract: Variational approximations facilitate approximate inference
for the parameters in complex statistical models and provide
fast, deterministic alternatives to Monte Carlo methods. However,
much of the contemporary literature on variational approximations
is in Computer Science rather than Statistics, and
uses terminology, notation, and examples from the former field.
In this series of lectures we explain variational approximation in statistical
terms. In particular, we illustrate the ideas of variational approximation
using examples that are familiar to statisticians.
Nicolai Meinshausen (ETH Zürich)
Challenges for highdimensional inference
Abstract: Highdimensional inference has progressed rapidly in the last years.
I will give a brief introduction and show examples from biology,
neuroscience and physics,
while also mentioning theoretical underpinnings.
Some challenges of inference for large data sets will be discussed,
including computational feasibility and inhomogeneity.
I will highlight some more recent developments about the feasibility of
constructing confidence intervals
for regression coefficients when the number of variables exceeds the
number of observations.
Prof. Thomas Kneib
Structured additive distributional regression
Abstract:In this talk, we present a generic Bayesian framework for inference in distributional regression models in which each parameter of a potentially complex response distribution and not only the mean is related to a structured additive predictor. The latter is composed additively of a variety of different functional effect types such as nonlinear effects, spatial effects, random coefficients, interaction surfaces or other (possibly nonstandard) basis function representations. To enforce specific properties of the functional effects such as smoothness, informative multivariate Gaussian priors are assigned to the basis function coefficients. Inference is then based on efficient Markov chain Monte Carlo simulation techniques where a generic procedure makes use of distributionspecific iteratively weighted least squares approximations to the full conditionals. We provide detailed guidance on practical aspects of model choice including selecting an appropriate response distribution and predictor specification. The importance and flexibility of Bayesian structured additive distributional regression to estimate all parameters as functions of explanatory variables and therefore to obtain more realistic models, is exemplified in several applications with complex response distributions. We will also introduce extensions to multivariate distributional regression
Dr. Axel Bücher (RuhrUniversität Bochum)
When uniform weak convergence fails: empirical processes
for dependence functions via epi and hypographs
Abstract:For copulas whose partial derivatives are not continuous everywhere on the interior of the
unit cube, the empirical copula process does not converge weakly with respect to the supremum distance.
This makes it hard to verify asymptotic properties of inference procedures for such copulas. To resolve the
issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence
theory is developed. Convergence with respect to the new metric is related to epi and hypoconvergence
and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform
convergence, whereas under mild side conditions, it implies Lp convergence. Even in cases where uniform
convergence fails, weak convergence with respect to the new metric is established for empirical copula and
tail dependence processes. No additional assumptions are needed for tail dependence functions, and for
copulas, the assumptions reduce to existence and continuity of the partial derivatives almost everywhere
on the unit cube. The results are applied to obtain asymptotic properties of minimum distance estimators,
goodnessoft tests and resampling procedures.
Nicolai Meinshausen (ETH Zürich)
Challenges for highdimensional inference
Abstract: Highdimensional inference has progressed rapidly in the last years.
I will give a brief introduction and show examples from biology,
neuroscience and physics,
while also mentioning theoretical underpinnings.
Some challenges of inference for large data sets will be discussed,
including computational feasibility and inhomogeneity.
I will highlight some more recent developments about the feasibility of
constructing confidence intervals
for regression coefficients when the number of variables exceeds the
number of observations.
Bo Markussen (University of Copenhagen)
Functional data, operator calculus, and statistical
computation
Abstract: Functional data consists of observations of continuous curves, e.g. human growth curves or
chemical spectrograms measured by gas chromatography, say. In practice the underlying functions are
observed at nitely many sample points giving a multivariate observation. Suppose that we have observed
n curves at the same p sample points. Functional data analysis, however, is dierent from classical multi
variate statistics since the pdimensional observations of each of the n samples originate from underlying
continuous curves. Special problems related to functional data includes data representation, registration,
and regularization. See the monograph by Ramsay and Silverman (2005) for an introduction. Data repre
sentation is often done by a functional basis for the underlying curves, e.g. using splines or wavelets, where
the coecients are estimated from the multivariate observations. This method retains the data in a nitely
dimensional framework, and the needed statistical computations are often done using sparse matrices. In
this talk we advocate a dierent computational approach. The idea is to embed the discrete observation
of each curve into the originating function space, and then to do the computations on genuine functions
and not on vectors of basis coecients. This approach implies that the sparse matrix computations are
replaced by operator calculus. In Markussen (Bernoulli, 2013) this approach was implemented for constant
coecient dierential operators, where there exists semiexplicit formulae for the associated Greens func
tions, and in Rakt and Markussen (CSDA, 2014) the methodology was generalized to image data. In this
talk I will discuss recent developments using a much more versatile class of integral operators, including
a discussion of the associated numerical computations. The methods will be exemplied in the context of
functional regression, where the discretely observed curves are used as covariates for a univariate (or low
dimensional) response.
Bharath Sriperumbudur (University of Cambridge)
RKHSinduced innite dimensional exponential families:
Density estimation and beyond
Abstract: In the rst part of the talk, we consider the problem of estimating densities in an innite
dimensional exponential family indexed by functions in a reproducing kernel Hilbert space. Since standard
techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves)
do not yield practically useful estimators, we propose an estimator based on the score matching method
introduced by Hyvarinen, which involves solving a simple linear system. We show that the proposed
estimator is consistent, and provide convergence rates under smoothness assumptions. We also empirically
demonstrate that the proposed method outperforms the standard nonparametric kernel density estimator.
The second part of the talk deals with certain mean element and covariance operator associated with
the above considered exponential family. For an appropriate choice of kernel, we show that these objects
uniquely characterize the probability measure and then provide an ecient approach for nonparametric
twosample testing and independence testing.
Based on various joint works with Kenji Fukumizu (Institute of Statistical Mathematics, Tokyo) and Arthur
Gretton (Gatsby Unit, University College London).