Mathematical Statistics Seminar

Statistical topological data analysis: Rescaling the persistence diagram

Speaker(s): 
Wolfgang Polonik (UC Davis)
Date: 
Wednesday, July 12, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

A persistence diagram (PD) is one of the basic objects underlying topological data analysis. It is used to analyze topological and geometric features of an underlying space _M_, assuming availability of a random sample from _M_. Existing approaches for such analyses will be reviewed briefly, and their benefits and shortcomings will be discussed. Then we introduce ideas for rescaling PDs, which enables the derivation of novel limit theorems for the total k persistence, and other functionals of PDs.

Geometry of Log-Concave Density Estimation

Speaker(s): 
Bernd Sturmfels (MPI Leipzig)
Date: 
Wednesday, June 28, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We present recent work with Elina Robeva and Caroline Uhler that establishes a new link between geometric combinatorics and nonparametric statistics. It concerns shape-constrained densities on d-space that are log-concave, with focus on the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e.

On cross-validated lasso

Speaker(s): 
Denis Chetverikov (UCLA, USA)
Date: 
Wednesday, June 21, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In this talk, we derive a rate of convergence of the Lasso estimator when the penalty parameter \lambda for the estimator is chosen using K-fold cross-validation; in particular, we show that in the model with the Gaussian noise and under fairly general assumptions on the candidate set of values of \lambda, the prediction norm of the estimation error of the cross-validated Lasso estimator is with high probability bounded from above up to a constant by (s log p/n)^{1/2} (log^{7/8}(pn)), where n is the sample size of available data, p is the number of covariates, and s is the number of non-zer

Kantorovich distance based kernel for Gaussian Processes: estimation and forecast

Speaker(s): 
Jean-Michel Loubes (University Toulouse)
Date: 
Wednesday, June 14, 2017 - 10:00am
Location: 
Hausvogteiplatz 11a, 10117 Berlin, Room 4.13 (4th floor)

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. Here, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes.

Unobserved Heterogeneity and Empirical Bayes Methods

Speaker(s): 
Roger Koenker (Illinois)
Date: 
Wednesday, June 7, 2017 - 10:00am
Location: 
HU Berlin, Heilig-Geist-Kapelle, Spandauerstr. 1, 10178 Berlin

Unobserved heterogeneity is a pervasive feature of modern econometric problems. Recent advances in convex optimization make it possible to efficiently estimate the nonparametric mixture models that underlie such applications and empirical Bayes methods provide a unified decision theoretic framework for interpreting them. This approach will be illustrated with applications to longitudinal models of income dynamics, fraility models in survival analysis and multiple testing.

Quantile-Regression Inference With Adaptive Control of Size

Speaker(s): 
Juan Carlos Escanciano (Indiana University Bloomington)
Date: 
Wednesday, May 31, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Regression quantiles have asymptotic variances that depend on the conditional densities of the response variable given regressors. This talk develops a new estimate of the asymptotic variance of regression quantiles that leads any resulting Wald-type test or confidence region to behave as well in large samples as its infeasible counterpart in which the true conditional response densities are embedded. We give explicit guidance on implementing the new variance estimator to control adaptively the size of any resulting Wald-type test.

Frequency domain likelihood approximations for time series bootstrapping and bayesian nonparametrics

Speaker(s): 
Claudia Kirch (Universität Magdeburg)
Date: 
Wednesday, May 24, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

A large class of time series methods are based on a Fourier analysis, which can be considered as a whitening of the data, giving rise for example to the famous Whittle likelihood. In particular, frequency domain bootstrap methods have been successfully applied in a large range of situations. In this talk, we will first review existing frequency domain bootstrap methodology for stationary time series before generalizing them for locally stationary time series.

Vast network analysis of Limit Order Books

Speaker(s): 
Shi Chen (HU Berlin)
Date: 
Wednesday, May 17, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We propose vast network estimators for limit order books in high dimensional setting, and we argue that limit orders have significant market impacts. Both undirected and directed network estimators are constructed based on consistent estimator for covariance matrix. Furthermore, the undirected estimator is constructed with sparse concentration matrix using graphical lasso, so that the regularized covariance matrix is related to connectedness measure. The directed one is derived from VAR model through penalized variance decomposition.

PCA in an asymmetric norm

Speaker(s): 
Petra Burdejova (HU Berlin)
Date: 
Wednesday, May 17, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of high-dimensional data. However, in many applications such as risk quantification in finance or climatology, one is interested in capturing the tail variations rather than variation around the mean. In this paper, we develop Principal Expectile Analysis (PEC), which generalizes PCA for expectiles. It can be seen as a dimension reduction tool for extreme value theory, where one approximates fluctuations in the \tau-expectile level of the data by a low dimensional subspace.

A review of regularized optimal transport and applications to Wasserstein barycenters

Speaker(s): 
Marco Cuturi (ENSAE / CREST)
Date: 
Wednesday, May 10, 2017 - 10:00am
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

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