Research Group "Stochastic Algorithms and Nonparametric Statistics"
Research Seminar "Mathematical Statistics" Summer Semester 2023
|19.04.2023||Dr. Chen Huang (University of Aarhus)|
| Arellano-Bond LASSO estimator for long panel dynamic linear models (hybrid talk)
We consider the estimation and inference for dynamic linear panel models with both large cross-sectional dimension $N$ and long time dimension $T$. The Arellano-Bond (AB) estimator (Arellano and Bond, 1991) is a popular method for dynamic panels. However, it has large bias when $T$ is large. We present a simple least absolute shrinkage and selection operator (LASSO) estimator for the AB method. In particular, we first use LASSO to fit the optimal instrument variables (IV) based on a large group of lags, and then implement the linear IV estimator in the second stage. To further reduce the bias of the estimator, we propose a sample-splitting procedure. Simulations show that the sample-splitting AB-LASSO provides more accuracy in estimation and inference compared to the AB method. Finally, we apply the approach to evaluate the long run effect of democracy on the economic growth.
|26.04.2023||Prof. Dr. Vladimir Spokoiny (WIAS Berlin)|
|Estimation and inference for error-on-operator models (hybrid talk)
Many statistical models can be considered as particular cases of the Error-in-Operator (EiO) model. The list includes high dimensional random design regression, instrumental variables and error-in-variables regression, stochastic diffusion and interacting particle systems models among many others. The talk focuses on theoretical guarantees for the EiO model for finite samples using general theory of semiparametric estimation combined with the recent progress for stochastically linear models.
|03.05.2023||Prof. Dr. Gilles Blanchard (Université Paris Saclay, Frankreich)|
| Stein effect for estimating many vector means: A "blessing of dimensionality" phenomenon (hybrid talk)
Consider the problem of joint estimation of the means for a large number of distributions in R^d using separate, independent data sets from each of them, sometimes also called "multi-task averaging" problem. We propose an improved estimator (compared to the naive empirical means of each data set) to exploit possible similarities between means, without any related information being known in advance. The idea is to shrink each mean towards a convex combination of its neighbors (determined by testing). We prove that this approach provides an improvement in mean squared error that can be quite significant when the (effective) dimensionality of the data is large, and when the unknown means exhibit structure such as clustering or concentration on a low-dimensional set (but which is totally unknown in advance). This is directly linked to the fact that the separation distance for testing is smaller than the estimation error in high dimension and generalizes the well-known James-Stein phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings, which plays an important role in many modern applications. (This is based on joined work with Hannah Marienwald and Jean-Baptiste Fermanian)
|10.05.2023||N.N.||4th Floor, Room 405/406||
|17.05.2023||Prof. Dr. Enno Mammen (Universität Heidelberg)|
|Random planted forest: A directly interpretable tree ensemble (hybrid talk)
We introduce a novel interpretable, tree based algorithm for prediction in a regression setting in which each tree in a classical random forest is replaced by a family of planted trees that grow simultaneously. The motivation for our algorithm is to estimate the unknown regression function from a functional ANOVA decomposition perspective, where each tree corresponds to a function within that decomposition. Therefore, planted trees are limited in the number of interaction terms. The maximal order of approximation in the ANOVA decomposition can be specified or left unlimited. If a first order approximation is chosen, the result is an additive model. In the other extreme case, if the order of approximation is not limited, the resulting model places no restrictions on the form of the regression function. The talk reports on joint work with Munir Hiabu and Joseph T. Meyer.
|24.05.2023||Dr. Jonathan Niles-Weed (New York University)|
|Optimal transport map estimation in general function spaces (online talk)
We consider the problem of estimating the optimal transport map between a (fixed) source distribution P and an unknown target distribution Q, based on samples from Q. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when P and Q have densities bounded above and below and when the transport map lies in a Hölder class), which are often not reflected in practice. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfies a Poincaré inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for bounded densities and Hölder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution and the transport map is given by an infinite-width shallow neural network. Joint work with Divol and Pooladian.
|31.05.2023||Dr. Randolf Altmeyer (University of Cambridge)|
| Statistics for stochastic PDEs: Optimal rates and a nonparametric LAN expansion (hybrid talk)
We consider the problem of estimating the coefficients in second order linear stochastic partial differential equations (SPDE) from spatially discrete or locally averaged observations on a finite time horizon. For a nonparametric diffusivity, we obtain the local asymptotic normality (LAN) property, opening up the road to efficient estimation. The proof reveals interesting aspects of partially observed space-time random fields. It relies on stochastic filtering techniques, on the reproducing kernel Hilbert space of the observation process as well as on the solution of certain Wiener-Hopf integral equations. For lower order terms, we obtain rate-optimal estimators, exhibiting different rates of convergence depending on the differential order. We further establish a general moment-based technique for the joint estimation of diffusivity and volatility parameters, covering both discrete and local measurements.
|21.06.2023||Eddie Aamari (Université Paris Diderot)|
|Minimax manifold estimation: Boundary, noise and computational constraints
In this talk, we will present the optimal rates for the Hausdorff estimation of d-dimensional manifolds M in high ambient dimension. We will focus on the influence of the possible presence of a boundary dM on one hand, and on noise and computational constraints on the other hand. The studied geometric class of target manifolds that we consider reunites and extends the most prevalent C^2-type models: manifolds without boundary, and full-dimensional domains. A Voronoi-based procedure that allows to identify points close to dM will be presented. The noise and computational constraints will be addressed jointly through the statistical query (SQ) framework, which consists in replacing the usual access to samples from a distribution by the access to adversarially perturbed expected values of functions interactively chosen by the learner. This framework provides a natural estimation computational complexity measure, enforces robustness through adversariality, and is closely related to differential privacy. After presenting this framework in details, we will describe a purely geometric algorithm called Manifold Propagation, that reduces the problem to three local geometric routines: projection, tangent space estimation, and point detection. We will then provide constructions of these geometric routines in the SQ framework.
|28.06.2023||First Talk: Alessia Caponera (Universität Milano-Bicocca)|
|10:00 a.m.||Functional estimation of anisotropic covariance and autocovariance operators on the sphere (hybrid talk)
In this talk we present nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. A preliminary exploration of climate data will also be discussed. This is based on joint works with Victor M. Panaretos, Julien Fageot, Matthieu Simeoni and Almond Stöcker.
|28.06.2023||Second Talk: Kartik Waghmare (EPFL Lausanne)|
|11:15 a.m.|| Functional graphical lasso (hybrid talk)
We consider the problem of recovering conditional independence relationships between p jointly distributed Hilbertian random elements given n realizations thereof. We operate in the sparse high-dimensional regime, where n << p and no element is related to more than d << p other elements. In this context, we propose an infinite-dimensional generalization of the graphical lasso. We prove model selection consistency under natural assumptions and extend many classical results to infinite dimensions. In particular, we do not require finite truncation or additional structural restrictions. The plug-in nature of our method makes it applicable to any observational regime, whether sparse or dense, and indifferent to serial dependence. Importantly, our method can be understood as naturally arising from a coherent maximum likelihood philosophy.
|4th Floor, Room 405/406||
|12.07.2023||Mathias Vetter (Universität Kiel)|
|On goodness-of-fit testing for point processes (hybrid talk)
Typical models for point processes like Hawkes processes or inhomogeneous Poisson processes are often of a parametric form where the intensity function or an additional self-exciting component is known up to an unspecified parameter. A lot of research since the seminal paper by Ogata (1978) has been devoted to the estimation of these unknown parameters but even in these rather standard models a consistent goodness-of-fit test has been missing. This talk aims to fill this gap. We will show how to formally set up a bootstrap procedure to allow for goodness-of-fit testing and we will discuss how to prove consistency of the test in the (already quite involved) case of an inhomogenous Poisson processes.
|19.07.2023||Giada Adelfio (Università degli Studi di Palermo)|
|Spatio-temporal point processes: Local second-order statistics for estimation and diagnostics (hybrid talk)
In previous papers, we defined an approach to assesses the goodness-of-fit of spatio-temporal models, weighteing by the inverse of the conditional intensity function. The method accounts for the local weighted second-order statistics, providing a quite general approach for individual diagnostics. Starting from these results, we developed an estimation based on the local second-order characteristics of the weighted process, providing also local estimates. The method does not rely on any particular model assumption on the data, and thus it can be applied for whatever is the generator model of the (even complex) process. In addition, we want to account for the primary role of second-order statistics for improving inference of complex intensity functions, when likelihood could be intractable, and complex interactions may be not negligible for a proper description.
last reviewed: July 13, 2023 by Christine Schneider