Research Group "Stochastic Algorithms and Nonparametric Statistics"

Abstract:

Christian Bayer (WIAS Berlin)

A forward-reverse representation for conditional diffusions

Abstract: We construct an estimator for the conditional expectation of a functional of the path of a diffusion process given its initial and terminal values. The estimator is based on a representation of the conditional expectation by an (unconditional) expectation of a functional of the diffusion and its associated reverse process, as introduced by Milstein, Schoenmakers and Spokoiny (2004), which can be computed by Monte Carlo simulation. We analyze the speed of convergence and present a numerical example. (Joint work with John Schoenmakers.)

Holger Dette (Ruhr-Universität Bochum)

Of Copulas, Quantiles, Ranks, and Spectra an L1- approach to spectral analysis

Abstract: We present an alternative method for the spectral analysis of a strictly stationary time series {Yt}tÎZ. We define a ''new'' spectrum as the Fourier transform of the differences between copulas of the pairs (Yt, Yt-k) and the independence copula. This object is called copula spectral density kernel and allows separating marginal and serial aspects of a time series. We show that it is intrinsically related to the concept of quantile regression. Like in quantile regression, which provides more information about the conditional distribution than the classical location-scale model, the copula spectral density kernel is more informative than the spectral density obtained from the autocovariances. In particular the approach provides a complete description of the distributions of all pairs (Yt, Yt-k). Moreover, it inherits the robustness properties of classical quantile regression, because it does not require any distributional assumptions such as the existence of finite moments. In order to estimate the copula spectral density kernel we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted L1-projections of the ranks of the observed time series onto a harmonic regression model. We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions. The finite-sample properties of the new methodology and its potential for applications are briefly investigated by simulations and a short empirical example.

M. Höhle ( LMU & RKI)

Bayesian nowcasting during the large EHEC/HUS outbreak in Germany, 2011

Abstract: During May July 2011, Germany was confronted with a large outbreak of gastrointestinal disease caused by STEC O104:H4. A total of 2,987 cases of diarrhoea without the haemolyticuremic syndrome (HUS) complication and 855 cases of HUS were attributable to the outbreak, making this one of the largest STEC outbreaks ever. During the outbreak it was vital to have daily information on current epidemic trends, in order to judge if the outbreak was ongoing, assess the impact of control measures and perform capacity planning. However, such real-time tracking is complicated by inherent delay in public health reporting systems between the occurrence of the event, e.g. time of symptom onset or hospitalization, and the time the report becomes available in the surveillance database. Following Donker et al. (2011) such delay-adjusting tracking procedures are in the public health setting called nowcasts. We address the nowcasting task in the statistical framework of the occurred-but-not-reported-events problem (Lawless, 1994). Here, estimation of the delay distribution takes the inherent right-truncation of the data generating process into account. We propose a novel Bayesian inference framework for the problem, which better reects uncertainty due to both estimation of the delay distribution and count data sampling. Speciacally, we use the proposed nowcast procedure to predict the daily number of HUS hospitalizations during the STEC O104:H4 outbreak. Retrospectively, on basis of the full epidemic curve, we perform a quantitative evaluation of different nowcast schemes based on proper scoring rules for count data (Czado et al, 2009). An implementation of the algorithm is made available in the R package surveillan- ce (Hhle, 2007). The presented material is joint work with Matthias an der Heiden, Robert Koch Institute.)

Jan Vecer (Frankfurt School of Finance and Management)

Reference asset invariant price processes

Abstract: A large class of models of the price is reference asset (numeraire) invariant, for instance the geometric Brownian motion model or jump models with symmetric jump sizes. This property means that the returns of the price process X_Y(t) under measure associated with the reference asset Y have the same distribution as the returns of the price process Y_X(t) under the measure associated with the reference asset X. This is of a practical importance when the price and the inverse price are both used such as in the case of the foreign exchange markets (for instance euro-dollar or dollar-euro exchange rates)

E. De Vito (University of Genova)

Kernel methods for support estimation

Abstract: I will present some results about set estimation, where the goal is to estimate the support of a probability distribution from a sample of points identically and independently distributed according to it. A new class of algorithms is proposed based on a regularized kernel regression estimation. The consistency of the estimator with respect to the Hausdorff distance is provided under the assumption that the kernel is of positive type and satisfies a suitable separating property. The convergence rate depends on the decay of the eigenvalues of the integral operator associated to the kernel. In the second part of the talk I will show some experiments on synthetic and real data and I will discuss the choice of the parameters, which is a delicate issue.

Christoph Breunig (Universität Mannheim)

Specification testing in nonparametric instrumental quantile regression

Abstract: In econometrics, there are many environments which require nonseperable modeling of a structural disturbance. In this setting, the case of possibly endogenous regressors has been studied recently. Regarding this literature, key assumptions to obtain identification and estimation results are: Valid instruments must be at hand and the model must be monotonic in the nonseparable, continuously distributed scalar disturbance. If one of these assumptions is violated a function solving the model equation need not to exist. In this talk we consider a test whether such a function exists. Our test statistic is asymptotically normally distributed under correct specification and consistent against any alternative model. Under a sequence of local alternatives the asymptotic distribution of our test is derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases.

Dag Tjoestheim (University of Bergen)

Using local Gaussian correlation to test for independence, copula structure and financial contagion

Abstract: Local Gaussian correlation is a local dependence concept that is based on approximating a bivariate density function by a bivariate Gaussian distribution locally. The correlation of the approximating Gaussian at a given location is taken as a measure of local dependence at that location. The background, theory and properties of the local Gaussian correlation will be summarized, and a number of applications will be presented: Characterizing positive and negative nonlinear dependence and independence testing. Describing copulas in terms of local Gaussian correlation. Detecting contagion for financial crises.

W. K. Härdle

Quantile regression in high dimensions with single index models

Abstract: In this paper, we use a projection based model specification. A balance between statistical precision and variable selection may be achieved by combining semiparametric ideas with variable selection techniques. More precisely, we pursue a single index strategy in combination with some variable selection methods that are applicable to a variety of calibration situations including labor market discrimination analysis and conditional value at risk estimation. Moreover, the weight composite loss function we consider is a flexible estimation framework to address the robustness and efficiency questions. In the setting with the dimensionality much larger than the sample size, we derive theoretical results on the asymptotic behavior of our estimation. The methodology is evaluated further in simulations and real data examples.

Abstract:

Emmanuel Boissard (WIAS)

Convergence of empirical measures in Wasserstein distance

Abstract: We discuss non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein metrics. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.

Abstract:

Evgeny Spodarev (Ulm University)

Limit theorems for excursion sets of stationary random fields

Abstract: In this talk, we give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we present a number of limit theorems of central type for the volume of excursions of stationary (quasi--, positively or negatively) associated random fields with stochastically continuous realisations for a fixed excursion level. This class includes in particular Gaussian, Poisson shot noise, certain infinitely divisible, $\alpha$--stable and max--stable random fields satisfying some extra dependence conditions. Functional limit theorems (with the excursion level being an argument of the limiting Gaussian process) are proved. For stationary isotropic $C^1$--smooth Gaussian random fields similar results are available also for the surface area of the excursion set. The case of the excursion level increasing to infinity and the asymptotics of the mean volume, surface area and the Euler characteristic of excursions of non--stationary smooth Gaussian random fields with a unique point of maximum variance are considered as well.

Richard Samworth (Cambridge)

Log-concavity: New theory and methodology

Abstract:Log-concave densities on $\mathbb{R}^d$ form an attractive infinite-dimensional class that includes many standard parametric families and has several useful properties. I will discuss ideas of log-concave projection and its relevance for density estimation, regression and testing problems. In the second half of the talk, I will show how related ideas can be used to introduce a new estimation framework in Independent Component Analysis models.

Abstract:

Siegfried Hörmann (ULB, Brussels)

PCA for functional time series: basics, applications and extensions

Abstract:Data in many elds of science are sampled from processes that can most naturally be described as functional. Examples include growth curves, tem- perature curves, curves of nancial transaction data and patterns of pollu- tion data. Functional data analysis (FDA) is concerned with the statistical analysis of such data. Clearly, functions are intrinsically innite dimensional objects and it seems evident that tools for reducing dimensionality are even more impor- tant than in the multivariate context. A key technique for dimension re- duction and analyzing functional data is the functional principal component analysis (FPCA). This talk consists of two parts. In the rst part we will give a basic introduction to FDA and FPCA. In particular we focus on the case when functional data are observed sequentially, i.e. they form a functional time series. As an application we present the problem of curve prediction based on FPCs. Though quite useful, FPCs have also some serious drawbacks when ap- plied in the sequential setting. Indeed, they are not providing optimal dimension reduction as they do in the i.i.d. case. This is because (stan- dard) FPCA acts in a static way and ignores possible serial dependence of the functional observation. Moreover, though cross-sectionally uncor- related for a xed observation, the FPC-score vectors have non-diagonal cross-correlations. This means that we cannot analyze them componentwise (like in the i.i.d. case) but need to consider them as vector time series which are less easy to handle and interpret. This motivates the second part of our talk on dynamic FPCA. Dynamic FPCA is a recently introduced method which provides the adequate (in a certain way optimal) dimension reduction for functional time series. In a real data example and some simulations we show the drastic improvement it brings compared to the static approach. The talk is a patchwork of some recent papers co-authored with A. Aue, D. Dubart-Norinho, M. Hallin, L. Kidzinski and P. Kokoszka.

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