(NMST611) Advanced Statistical Seminar (2023/2024)
Wednesday: 15:40 - 17:20 | Prezenčne v Praktiku KPMS
The advanced statistical seminar consists of presentations delivered (typically in person) by invited foreign speakers or departmental guests. Assorted topics from modern statistics -- theory and applications -- are usually covered by the talks.
Seminar schedule (Winter term 2023/2024)
- 04.10.2023 | 15:40 | Andrew McCormack
Technical University of Munich, Germany
Title: Information Geometry and Asymptotics for Kronecker Covariances
The Kronecker covariance structure for array data posits that the covariances along comparable modes, such as rows and columns, of an array are similar. Over and above being a plausible model for many types of data, the Kronecker covariance assumption is especially useful in high-dimensional settings, where unconstrained covariance matrix estimates are typically unstable. In this talk we explore asymptotics and information geometric aspects of estimators of Kronecker covariance matrices. The asymptotic properties of two estimators, the maximum likelihood estimator and an estimator based on partial traces are contrasted. It is shown that the partial trace estimator is inefficient, where the relative performance of this estimator can be quantified in terms of a principle angle between tangent spaces. We also discuss a consistency property of the partial trace estimator and demonstrate that for higher-order tensors the covariance matrix can be consistently estimated in a fixed-n and large-p regime.
------------------------------------------------------------------------------------------- - 18.10.2023 | 15:40 | Jan Beran
University of Konstanz, Germany
Title: On seasonal empirical processes and the change of seasons
Abstract: Seasonal empirical processes play an essential role when analyzing seasonal time series.
In this talk we consider limit theorems under long memory assumptions. A uniform reduction principle, a functional limit theorem and a test for changes in exceedences are derived. The test is used to shed some light on the question whether climate change has led to a shift of seasons.
------------------------------------------------------------------------------------------- - 01.11.2023 | 15:40 | Vali Asimit
University of London, United Kingdom
Title: Constructing Optimal Portfolios under Risk Budgeting
Abstract: We provide a mathematical characterization where general risk preferences are considered for risk parity/budgeting portfolio construction problems. Statistical inferences are determined for those portfolios when risk preferences are ordered by variance and by Conditional Value-at-Risk. We demonstrate that when the returns are jointly elliptically distributed the risk budgeting problems will have the same solution for any risk measures that are homogeneous of the same order. Any risk measure that is a function of centred moments would lead to the same risk budgeting portfolios, for elliptical returns. For the general problem when distribution of returns is not known, we demonstrate the existence of a solution to the risk budgeting problem for any homogeneous risk preferences. A novel Conditional Value-at-Risk estimator is proposed, which is shown to perform very well on non i.i.d observations, based on simulated and real-life data, especially during periods of bull market and irrational exuberance. Our numerical results show superior performance of risk parity portfolios in terms of various measure of performance such as Sharpe ratio and diversification when comparing with other benchmark portfolios including the equally weighted portfolio.
------------------------------------------------------------------------------------------- - 15.11.2023 | 15:40 | Ulrike Schneider
Vienna University of Technology, Austria
Title: A Unified Framework for Pattern Recovery in Penalized Estimation and its Geometry
Abstract: We consider the framework of penalized estimation where the penalty term is given by a polyhedral norm, or more generally, a polyhedral gauge, which encompasses methods such as LASSO (and many variants including the generalized LASSO), SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a general notion of patterns based on subdifferentials and formalize an approach to measure pattern complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected by the procedure with positive probability, the so-called accessibility condition. We also introduce the stronger noiseless recovery condition which can be shown to play exactly the same role as the well-known irrepresentability condition for the LASSO in that the probability of pattern recovery in our general framework is bounded by 1/2 if the condition is not satisfied. Finally, we prove that the noiseless recovery condition can indeed be relaxed when turning to so-called thresholded penalized estimation: in this setting, the accessibility condition is already sufficient (and necessary) for sure pattern recovery provided that the signal of the pattern is large enough. We demonstrate how our findings can be interpreted through a geometrical lens throughout the talk and illustrate our results for LASSO and SLOPE in particular.
------------------------------------------------------------------------------------------- - 29.11.2023 | 15:40 | Thomas Verdebout
Université libre de Bruxelles, Belgium
Title: Asymptotic power of Sobolev tests for uniformity on hyperspheres
Abstract: One of the most classical problems in multivariate statistics is considered, namely, the problem of testing isotropy, or equivalently, the problem of testing uniformity on the unit hypersphere. Rather than restricting to tests that can detect specific types of alternatives only, we consider the broad class of Sobolev tests. While these tests are known to allow for omnibus testing of uniformity, their non-null behavior and consistency rates, unexpectedly, remain largely unexplored. To improve on this, we thoroughly study the local asymptotic powers of Sobolev tests under the most classical alternatives to uniformity, namely, under rotationally symmetric alternatives. We show in particular that the consistency rate of Sobolev tests does not only depend on the coefficients defining these tests but also on the derivatives of the underlying angular function at zero.
------------------------------------------------------------------------------------------- - 13.12.2023 | 15:40 | Veronika Rockova
University of Chicago Booth School of Business, USA
Title: Adaptive Bayesian Predictive Inference
Abstract: Bayesian predictive inference provides a coherent description of entire predictive uncertainty through predictive distributions. We examine several widely used sparsity priors from the predictive (as opposed to estimation) inference viewpoint. Our context is estimating a predictive distribution of a high-dimensional Gaussian observation with a known variance but an unknown sparse mean under the Kullback-Leibler loss. First, we show that LASSO (Laplace) priors are incapable of achieving rate-optimal performance. This new result contributes to the literature on negative findings about Bayesian LASSO posteriors. However, deploying the Laplace prior inside the Spike-and-Slab framework (for example with the Spike-and-Slab LASSO prior), rate-minimax performance can be attained with properly tuned parameters (depending on the sparsity level sn). We highlight the discrepancy between prior calibration for the purpose of prediction and estimation. Going further, we investigate popular hierarchical priors which are known to attain adaptive rate-minimax performance for estimation. Whether or not they are rate-minimax also for predictive inference has, until now, been unclear. We answer affirmatively by showing that hierarchical Spike-and-Slab priors are adaptive and attain the minimax rate without the knowledge of sn. This is the first rate-adaptive result in the literature on predictive density estimation in sparse setups. This finding celebrates benefits of a fully Bayesian inference.
------------------------------------------------------------------------------------------- - 03.01.2024 | 15:40 | Werner Müller
Johannes Kepler University Linz, Austria
Title: News on the virtual noise method in the optimal design of experiments
Abstract: The optimal design of experiments for correlated processes is an increasingly relevant and active research topic. Present methods have restricted possibilities to judge their quality. To fill this gap, the virtual noise approach is complemented by a convex formulation leading to an equivalence theorem comparable to the uncorrelated case and to an algorithm giving an upper performance bound against which alternative design methods can be judged. Moreover, a method for generating exact designs follows naturally. Estimation problems in a finite design space are exclusively considered with a fixed number of elements. A comparison of some classical examples from the literature as well as a real application is provided.
