Research Group "Stochastic Algorithms and Nonparametric Statistics"

Abstract:

Paul von Bünau and Franz Kiraly (TU Berlin)

Stationary subspace analysis and ideal regression

Abstract:In the rst part of this talk, we introduce Stationary Subspace Analysis (SSA), a sourceseparation method which linearly factorizes observed data into stationary and non-stationary components. We present an SSA algorithm for the mean and covariance matrix, discuss conditions for identiability and our simulation results. In an application to EEG analysis, we demonstrate how SSA reveals meaningful stationary and non-stationary brain sources. In the second part of this talk, we present ideal regression, an algebraic-geometric approach to estimating polynomial relations. Ideal regression is a generalization of classical regression setups. We demonstrate how to obtain consistent estimators and how to prove properties for these estimators, using techniques from algebraic geometry. We present a set of possible applications, and show that SSA can be viewed as a special case of ideal regression, exhibiting an algorithmic estimator which we evaluate in simulations. (This is joint work with F. Meinecke, D. Bly

Alois Kneip (Universität Bonn)

Factor models and variable selection in high dimensional regression analysis

Abstract:The paper considers linear regression problems where the number of predictor variables is possibly larger than the sample size. The basic motivation of the study is to combine the points of view of model selection and functional regression by using a factor approach: it is assumed that the predictor vector can be decomposed into a sum of two uncorrelated random components re ecting common factors and specic variabilities of the explanatory variables. It is shown that the traditional assumption of a sparse vector of parameters is restrictive in this context. Common factors may possess a signicant in uence on the response variable which cannot be captured by the specic eects of a small number of individual variables. We therefore propose to include principal components as additional explanatory variables in an augmented regression model. We give nite sample inequalities for estimates of these components. It is then shown that model selection procedures can be used to estimate the parameters of the augmented model, and we derive theoretical properties of the estimators. Finite sample performance is illustrated by a simulation study.

Johannes Schmidt-Hieber (Vrije Universiteit Amsterdam)

Confidence statements for qualitative features in deconvolution

Abstract:Suppose that we observe data from a deconvolution model, that is, we observe an i.i.d. sample from an unknown distribution under additive noise. In many practical problems the main interest lies not in pointwise reconstruction of the true density but rather in answering qualitative questions, for instance about the number of modes or the regions of increase and decrease. In this talk, we derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. Important examples covered within this framework are to test for local monotonicity or local convexity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. Theoretically we derive convergence rates for the distance between the multiscale statistic and a distribution-free approximation and study the power of the constructed tests. In a second part, we illustrate our method by numerical simulations. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for point- wise estimation are very slow. This is joint work with Axel Munk (Goettingen) and Lutz Duembgen (Bern).

Céline Duval (CREST)

Adaptive wavelet estimation of a renewal reward process

Abstract:We study the nonparametric estimation of the jump density of a renewal reward process from the discrete observation of one trajectory over [0,T]. We consider the microscopic regime when the sampling rate tends to 0 as T goes to infinity. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates that vanish with T at arbitrary polynomial rates. First we consider the particular case of a compound Poisson process, where computations are explicit, and then expand the result to renewal reward processes.

Jan Johannes (Université catholique de Louvain, Belgium)

Adaptive estimation in function linear Models

Abstract:We consider the nonparametric estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. The theory in this presentation covers both the estimation of the slope function or its derivatives (global case) as well as the estimation of a linear functional of the slope function (local case). We propose an estimator of the slope function which is based on dimension reduction and additional thresholding. Moreover, replacing the unknown slope function by this estimator we obtain in the local case a plug-in estimator of the value of linear functional evaluated at the slope. It is shown that in both the global and the local case these estimators can attain minimax optimal rates of convergence up to a constant. However, the estimator of the slope function requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski (2011). It is shown that the adaptive estimator with data-driven choice of the dimension parameter can attain the lower minimax risk bound in the global case up to a constant and in the local case up to a logarithmic factor, and this over a variety of classes of slope functions and covariance operators.

Vladimir Spokoiny (WIAS Berlin)

Bernstein - von Mises theorem for quasi-posterior

Abstract: Bernstein - von Mises Theorem is one of the most remarkable result in Bayesian inference. It claims that under rather weak conditions on the model and on the prior, the posterior distribution is asymptotically close to a normal distribution with the mean at the MLE and the covariance matrix which is inverse of the total Fisher information matrix.This talk extends this result to the situation when the likelihood function is possibly misspecified, the sample size is fixed and does not tend to infinity, and the parameter dimension is large relative to sample size. A further extension to hyperpriors and Bayesian model selection is discussed as well.

Prof. Ingo Steinwart (Universität Stuttgart)

Statistical Learning: Classification, Density Level Estimation, and Clustering

Abstract:The last decade has witnessed an explosion of data collected from various sources. Since in many cases these sources do not obey the assumptions of classical statistical approaches, new automated tools for interpreting such data have been developed in the machine learning community. Statistical learning theory tries to understand the statistical principles and mechanisms these tools are based on. The talk presents some developments in this direction and illustrates how new statistical insights led in turn to the design of new data analysis tools. In the first part I will talk about support vector machines for binary classification and density level set estimation, where the focus lies on fully automated, adaptive learning methods and the mathematical theory behind them. The second part then focuses on (semi-)adaptive methods for clustering, where clusters are defined to be the connected components of some density level sets. Besides others we discuss adaptive modifications of the popular DBSCAN clustering algorithm.

Abstract:

Nicolas Verzelen (INRA, Montpellier)

Minimax risks for sparse regression: a review

Abstract:Consider the standard Gaussian linear regression model. Numerous works have been devoted to building efficient estimators of the unknown parameter when it dimension p is much larger than the number of observations n. In such a situation, a classical approach amounts to assume the parameter is approximately sparse. In this talk, we will review recent minimax bounds over classes of k-sparse parameter vectors. Such bounds shed light on the limitations due to high-dimensionality. Interestingly, an elbow effect occurs when the number k log(p/k) becomes large compared to n. Practical implications of this phenomenon will also be discussed.

Shujie Ma (University of California, Riverside)

Simultaneous variable selection and estimation in semiparametric modeling of longitudinal/clustered data

Abstract: We consider the problem of simultaneous variable selection and estimation in additive partially linear models for longitudinal/clustered data. We propose an estimation procedure via polynomial splines to estimate the nonparametric components and apply proper penalty functions to achieve sparsity in the linear part. Under reasonable conditions, we obtain the asymptotic normality of the estimators for the linear components and the consistency of the estimators for the nonparametric components. We further demonstrate that, with proper choice of the regularization parameter, the penalized estimators of the nonzero coecients achieve the asymptotic oracle property. The nite sample behavior of the penali- zed estimators is evaluated with simulation studies and illustrated by a longitudinal CD4 cell count dataset.

Thomas Royen (FH Bingen)

Some exact distributions derived from gamma random variables including Greenwood's statistic

Abstract: In this lecture the exact distribution of Greenwood's statistic - the square sum of spacings - is obtained as a special case among several distributions derived from independent gamma random variables. The exact distributions of the "basic-statistics", i.e. sum of squares, sample variance and sample coecient of variation are represented by dierent methods as power series, convex combinations of cumulative gamma distribution functions, several orthogonal series and univariate integrals over well known transcendental functions. Some results are extended to not identically distributed random variables. These methods have been used to compute a table with nine 8-digit-percentage points of Greenwood's statistic for sample sizes up to n=100.

Cecilia Mancini (University of Florence)

Spot Volatility Estimation Using Delta Sequences

Abstract:We introduce a class of nonparametric spot volatility estimators based on delta sequences and conceived to include many of the existing estimators as special cases. The full limit theory with unevenly sampled observations under inll asymptotics and nite time-horizon is rst derived in the pure diusive settings. We then extend our class of estimators to include Poisson jumps or nancial microstructure noise in the observed price process. As a development of our results, we perform an accurate analysis of the Fourier method by studying the distribution theory of the spot volatility estimator and by improving the computational aspects of the algorithm. Empirical evidence from the S & P500 stock index futures market is also provided. (This is joint work with Vanessa Mattiussi and Roberto Ren)

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