Forschungsgruppe "Stochastische Algorithmen und Nichtparametrische Statistik"
Research Seminar "Mathematical Statistics" Winter Semester 2016/2017
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| 19.10.16 | Dr. Itai Dattner (University of Haifa) |
| Statistical learning of dynamic systems Dynamic systems are ubiquitous in nature and are used to model many processes in biolo- gy, chemistry, physics, medicine, and engineering. In particular, systems of (deterministic or stochastic) di erential equations are commonly used for the mathematical modeling of the rate of change of dynamic processes. These systems describe the interrelationships between the variables involved, and depend in a complicated way on unknown quantities (e.g., initial values, constants or time dependent parameters). Learning dynamic systems involves the "standard" statistical problems such as studying the identi ability of a model, estimating model parameters, predicting future states of the system, testing hypotheses, and choosing the "best" model. However, modern dynamic systems are typically very complex: nonlinear, high dimensional and only partly measured. Moreover, data may be sparse and noisy. Thus, statistical learning (inference, prediction) of dynamical systems is not a trivial task in practice. In this talk we will present some recent theoretical results and methodologies concerning identi ability and estimation of dynamic systems. We will also discuss real data examples coming from diverse areas such as infectious diseases and biology. |
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| 26.10.16 | Dr. Alexandra Carpentier (Universität Potsdam) |
| Uncertainty quantifcation through adaptive and honest confidence
sets Empirical uncertainty quanti cation of estimation procedures can be simple in parametric, low dimensional situations. However, it becomes challenging and often problematic in high and in nite dimensional models. Indeed, adaptivity to the unknown model complexity becomes key in this case, and uncertainty quantifiation becomes akin to model estimation. - Such model-adaptive uncertainty quantifation can be formalised through the concept of adaptive and honest confience sets. Recent results related to this concept will be presented. - Model estimation, or model testing, is a specific types of composite-composite testing problems. General theory and tools for this kind of problems will be presented, in particular for quantifying the impact of the null hypothesis on the testing rate. |
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| 02.11.16 | Olga Klopp (Universtité Paris-Ouest Nanterre) |
| Network models and sparse graphon estimation Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of statistical es- timation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network and derive optimal rates of convergence for this problem. Our results cover the important setting of sparse networks. We also establish upper bounds on the mini- max risk for graphon estimation when the probability matrix is sampled according to a graphon model. The problem of estimation of the matrix of connection probabilities of a network can be viewed as a particular case of a general matrix sequence model. In this model, we observe the noisy entries of a matrix and assume that the signal matrix is ßtructured", that is, it can be factorized using sparse factors. This model includes a number of interesting problems such as the mixture of Gaussian, sparse dictionary learning, stochastic block models and mixture membership models. In the second part of the talk, I will consider the problem of statistical estimation of the signal matrix for this model and derive optimal rates of estimation. Keywords. Inhomogeneous random graph; Networks; Oracle inequality; Sparse graphon. |
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| 09.11.16 | Dr. Johannes Schmidt-Hieber (Leiden University) |
| Asymptotic equivalence between density estimation and the
Gaussian white noise model revisited Asymptotic equivalence means that two statistical models have the same asymptotic pro- perties with respect to all decision problems with bounded loss. In nonparametric statistics, asymptotic equivalence has been found useful as it allows in some situations to switch to simpler models. One of the most famous results is Nussbaums theorem which states that nonparametric density estimation is asymptotically equivalent to a Gaussian shift model provided that the densities satisfy some smoothness assumptions and are bounded away from zero. In this talk we review the literature and study to which extent one can relax the assumption that the densities must be uniformly bounded away from zero. We derive moreover the optimal rates of the Le Cam de ciencies. A part of the talk will be devoted to explaining the lower and upper bounds in more detail. As application, we mention Poisson intensity estimation with low count data. This is joint work with Kolyan Ray. |
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| 16.11.16 | Prof. Thorsten Dickhaus (Universität Bremen) |
| HVP 11a, R.4.13 | Statistical properties of Bernstein copulae with applications in
multiple testing (joint work with André Neumann, Taras Bodnar and Dietmar Pfeifer) A general way to estimate continuous functions consists of approximations by means of Bernstein polynomials. Sancetta and Satchell (2004) proposed to apply this technique to the problem of approximating copula functions. The resulting so-called Bernstein copulae are nonparametric copula estimates with some desirable mathematical features like smoothness.We extend previous statistical results regarding bivariate Bernstein copulae to the multivariate case and study their impact on multiple tests. In particular, we utilize them to derive asymptotic confidence regions for the family-wise error rate (FWER) of simultaneous test procedures which are empirically calibrated by making use of Bernstein copulae approximations of the dependency structure among the test statistics. This extends a similar approach by Stange, Bodnar and Dickhaus (2015) in the parametric case. A simulation study quantifies the gain in FWER level exhaustion and, consequently, power which can be achieved by exploiting the dependencies, in comparison with common threshold calibrations like the Bonferroni or the Sidak correction. Finally, we demonstrate an application of the proposed methodology to real-life data from insurance. |
| 23.11.16 | Shi Chen (Humboldt Universität zu Berlin, IRTG 1792) |
| Network dynamics of high-frequency trading data: Evidence from
NASDAQ market We propose a robust connectedness estimator for limit order books in high dimensional setting, and we argue that limit orders have significant market impacts. The estimator is constructed based on sparse precision matrix using graphical lasso, so that the regularized covariance matrix is related to connectedness measure. The microstructure noise embedded in high frequency data is removed by pre- averaging estimation. Furthermore, we provide a jump-robust estimator for connectedness of NASDAQ firms from different industrial sectors. Based on these insights, our results successfully track the network dynamics. |
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| 30.11.16 | Dr. Gwennaélle Mabon und Prof. Markus Reiß (Humboldt-Universität Berlin) |
| Estimation of linear and nonlinear functionals in nonparametric
boundary models For nonparametric regression with one-sided errors and a boundary curve model for Pois- son pR oint processes we consider rst the problem of ecient estimation for linear functionals of the form f(x)w(x)dx with unknown f and known w. We propose a simple blockwise estimator and then build up a nonparametric maximum-likelihood approach. Both methods allow for estimation with optimal rate n( +1=2)=( +1) under -Hlder smoothness or monotonicity constraints (analogue of = 1). Surprisin- gly, the nonparametric MLE approach enjoys additionally non-asymptotic eciency properties (UMVU: uniformly minimum variance among all unbiased estimators). Given uniform observations supported on a convex set, the theory extends to unbiased estimation of the volume of this convex set. In a second step, we consider estimation of nonlinear functionals of the form R (f(x))dx for known weakly di erentiable . In view of Lp -norms a primary example is (x) = jxjp. Even in that case unbiased estimation is feasible and optimal convergence rates can be derived. As an application we discuss Lp-separation rates in nonparametric testing. The proofs rely essentially on martingale stopping arguments for counting processes and the underlying point process geometry. The estimators are easily computable and some simulation results are presen- ted. Surprising di erences with standard models like Gaussian white noise are discussed. Diculties with establishing a Bayesian Bernstein-von Mises result are pointed out. (based on joint work with Leonie Selk, Nikolay Baldin and Martin Wahl) |
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| 07.12.16 | Prof. Vladimir Spokoiny (WIAS Berlin) |
| Adaptive weights clustering The paper discusses a new method of unsupervised learning for high dimensional data based on the idea of adaptive weights from Polzehl and Spokoiny (2000). The procedure recovers the unknown clustering structure without any prior information about the number of clusters, their size, distance between clusters, etc. The approach extends the popular k-mean and density based clustering procedures by using dynamically updated local weights. Theoretical results describe two major features of the method: propagation within a homogeneous region and separation between two different regions. Numerical results show state-of-art performance of the new procedure. |
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| 14.12.16 | No Seminar |
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| 21.12.16 | No Seminar |
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| 04.01.17 | No Seminar |
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| 11.01.17 | Markus Pelger (Stanford University) |
| Estimating latent asset-pricing factors We develop an estimator for latent factors in a large-dimensional panel of financial data that can explain expected excess returns. Statistical factor analysis based on Principal Component Analysis (PCA) has problems identifying factors with a small variance that are important for asset pricing. Our estimator searches for factors with a high Sharpe-ratio that can explain both the expected return and covariance structure. We derive the statistical properties of the new estimator based on new results from random matrix theory and show that our estimator can find asset-pricing factors, which cannot be detected with PCA, even if a large amount of data is available. Applying the approach to portfolio and stock data we find factors with Sharpe-ratios more than twice as large as those based on conventional PCA. Our factors accommodate a large set of anomalies better than notable four- and five-factor alternative models. |
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| 18.01.17 | Alexey Naumov (Skoltech, Moscow) |
| Bootstrap confidence sets for spectral projectors of sample covariance Let X_1, ... ,X_n be i.i.d. sample in R^p with zero mean and the covariance matrix S. The problem of recovering the projector onto the eigenspace of S from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector P_r on the subspace of the r th eigenvalue and its empirical counterpart \hat{P}_r in terms of the effective trace of S. This paper offers a bootstrap procedure for building sharp condence sets for the true projector P_r from the given data. This procedure does not rely on the asymptotic distribution of || P_r - \hat{P}_r ||_2 and its moments, it applies for small or moderate sample size n and large dimension p . The main result states the validity of the proposed procedure for nite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples. |
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| 25.01.17 | Dr. Magalie Fromont-Renoir (Université Rennes) |
| Family-wise separation rates for multiple testing This joint work with Matthieu Lerasle (Univ. Paris-Saclay, France) and Patricia Reynaud- Bouret (Univ. Cote d'Azur, France) is devoted to the question of the theoretical evaluation of multiple testing procedures.Where as many rst kind error-related evaluation criteria have been de ned, as generali- zations or relaxations of the historical Family-Wise Error Rate (FWER), very few second kind error-related criteria have been proposed in the multiple testing literature. Starting from a parallel between some tests of multiple hypotheses and some tests of a single hypothesis, based on aggregation approaches known to lead to minimax adaptivity properties, we extend the notion of Separation Rate, at the core of the minimax theory for nonparametric single hypothesis tests, to the multiple testing eld.We thus introduce the notion of weak Family-Wise Separation Rate (wFWSR) and its stronger counterpart, the Family-Wise Separation Rate (FWSR), leading to an emergent minimax theory for multiple tests whose FWER is controlled. We present some illustrations in various classical Gaussian frameworks, that corroborate several expected results under particular conditions on the tested hypotheses, but also give more surprising results. |
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| 01.02.17 | Fabienne Comte (Université Paris Descartes) |
| Laguerre basis for inverse problems related to nonnegative
random variables I will present, through two main examples, the speci c properties of the Laguerre basis and show that it is a very convenient tool to solve estimation problems on R+. The rst example is the regression-convolution model: an estimator of the unknown underlying function is built in two steps (deconvolution step, regression step) which are explained and discussed. Then, a risk study is conducted, that shows as usual that a bias-variance tradeo must be performed. A model selection device is shown to solve this question. The second example concerns a simpler multiplicative model, for which a projection estimators of the density of the hidden variables are built and discussed. The speci c properties of the Laguerre basis with respect to these solutions are enhanced. Rates of convergence in relation with Sobolev-Laguerre spaces are presented. To conclude, several other problems solved with the Laguerre bases are listed. |
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| 08.02.17 | Katharina Proksch (Universität Göttingen) |
| Multiscale scanning in inverse problems - With applications to nanobiophotonics We propose a multiscale scanning method to determine active components of a quantity f w.r.t. a dictionary U from observations Y in an inverse regression model Y = Tf + with operator T and general random error . To this end, we provide uniform con dence statements for the coecients ('; f); ' 2 U, under the assumption that (T?)1(U) is of wavelet-type. Based on this we obtain a decision rule that allows to identify the active components of U, i.e. (f; ') 6= 0; ' 2 U, at controlled, family-wise error rate. Our results rely on a Gaussian approximation of the underlying multiscale statistic with a novel scale penalty adapted to the ill- posedness of the problem. The important special case of deconvolution is discussed in detail. Further, the pure regression case, when T = id and the dictionary consists of moving windows of various sizes (scales), is included, generalizing previous results for this setting. Simulations support our theory and we illustrate the potential of the method as an inferential tool for imaging. As a particular application we discuss super-resolution microscopy and analyze experimental STED data to locate single DNA origami. |
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| 15.02.17 | Lásló Györfi (Budapest University of Technology and Economics) |
| The role of machine learning in the nonparametric prediction of time
series
The main purpose of this paper is to consider the prediction of stationary time series for various losses: squared loss (regression problem), 0; 1 loss (pattern recognition) and log utility (growth optimal portfolio selection). We are interested in universal prediction rules, which are consistent for all possible stationary and ergodic processes. Such rules can be constructed using aggregation techniques of machine learning by combining elementary rules (experts) in data dependent way. |
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last reviewed: February 7, 2016, by Christine Schneider