# Research Group "Stochastic Algorithms and Nonparametric Statistics"

## Research Seminar "Mathematical Statistics" Summer Semester 2013

last reviewed: July 4, 2013, Christine Schneider

Andrija Mihoci (C.A.S.E., HU Berlin)

Local adaptive multiplicative error models for high-frequency forecasts

Abstract: We propose a local adaptive multiplicative error model (MEM) accommodating time-varying parameters. MEM parameters are adaptively estimated based on a sequential testing procedure. A data- driven optimal length of local windows is selected, yielding adaptive forecasts at each point in time. Analyzing one-minute cumulative trading volumes of ve large NASDAQ stocks in 2008, we show that local windows of approximately 3 or 4 hours are reasonable to capture parameter variations while balancing modeling bias and estimation (in)eciency. In forecasting, the proposed adaptive approach signicantly outperforms a MEM where local estimation windows are xed on an ad hoc basis.

Dr. Micha Pesta(Charles University, Prag/Czech Republic)

Asymptotic consistency and inconsistency of the chain ladder

Abstract: The distribution-free chain ladder reserving method belongs to the most frequently used approaches in the general insurance. It is well known, see [1], that the estimators $widehatf_j$ of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In [2], we deal with some asymptotic properties of $widehatf_j$. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors $f_j$ are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well. References: [1] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. em ASTIN Bulletin, bf 23, 2, 213-225. [2] Pešta, M. and Hudecová, Š. (2012). Asymptotic consistency and inconsistency of the chain ladder. em Insurance: Mathematics and Economics, bf 51, 2, 472-479.

Alexey Kulik (Taras Shevchenko University, Kiev)

Limit theorems and statistical inference in Markov models

Abstract:The ``martingale problem approach'' for proving averaging principle and diffusion approximation type theorems for functionals of an ergodic Markov process will be discussed. Applications to statistical inference will be illustrated by two examples concerning asymptotic properties of MLE studied (a) for discretely observed solutions to SDE's with jumps; (b) for autoregressive models with hidden Markov input.

Matthias Scherer (TU München)

On the construction and use of factor copula models

Abstract: Modeling the dependence structure of high-dimensional random vectors is not an easy task. Nevertheless, it is required in many applications in the financial industry. Often, one faces a tradeoff between models that are rather simple but computationally efficient on the one hand, and very flexible dependence structures that become unhandy as the dimension of the problem increases on the other hand. Several popular families of copulas, especially when based on a factor-model construction, are extendible. Even though such structures are very convenient in large dimensions (due to the factor model / conditional i.i.d. structure), the assumption of conditional i.i.d. may be over-simplistic for real situations. One possibility to overcome extendibility without giving up the general structure is to consider hierarchical (or nested) extensions of the dependence structure in concern. Heuristically speaking, the dependence structure of hierarchical copulas is induced by some global stochastic factor affecting i.i.d. components and by additional group-specific factors that only affect certain sub-vectors. We present a survey of recent developments on hierarchical models, such as hierarchical Archimedean and Marshall-Olkin type dependence structures, and unify the literature by introducing the notion of h-extendibility. This definition generalizes extendible models in a natural way to hierarchical structures. Finally, we sketch applications to credit risk and insurance portfolios.

Ismael Castillo (Université Pierre et Marie Curie, Paris)

Some results on frequentist analysis of Bayesian posterior distributions

Abstract: In this talk I will discuss recent work on Bayesian analysis of procedures in non- and semi-parametric
settings. First, I will talk about conditions guaranteeing the asymptotic normality of the marginal posterior distribution - the so-called Bernstein-von Mises theorem - in semiparametric settings and give some examples. Second, a notion of nonparametric Bernstein-von Mises theorem will be introduced and some applications discussed (this part is joint work with Richard Nickl).

Lajos Horvath (University of Utah)

Change point detection: models and approaches

Abstract: I'll discuss some of the rst change point results in the literature and the possible mathe-
matical approaches to these problems. One of the most important results is the change point detection
in regression when the likelihood and related methods are used. In the last part the regression method is
extended to functional data. Several applications will illustrate the applicability of the theoretical results.

Jörg Breitung (Universität Bonn)

Challenges for the analysis of macroeconomic panel data

Abstract: Macroeconomic panel data typically involve aggregate variables from various countries such
as output (GDP), employment, in
ation rates, wages etc. In contrast to the large N, small Tframework,
the two dimensions of a typical macroeconomic dataset are more balanced, often providing a comparable
number of time periods and countries (regions). Although this is inconsequential for the analysis based on
the linear static panel data framework, it becomes crucial when estimating a dynamic model. A second
important feature of macroeconomic data is cross-section dependence among countries. In many cases this
dependence cannot be accommodated by a simple function of the geographical distance but also depends
on trade relations and the level of economic development. Furthermore, cross-country data often exhibit a
much richer pattern of heterogeneity that cannot be represented just by letting the intercept vary across
countries. While it is infeasible to allow for individual specic regression coecients in a large N, small
Tpanel framework, this may be a suitable option when analyzing macroeconomic data. Finally, potential
problems arising from nonstationary variables become more relevant if the number of time periods is
comparable to the number of countries. My talk focuses on an asymptotic framework for (stationary)
dynamic panel data models, where N and T tends to innity at the same rate, which seems to be more
appropriate for typical macroeconomic applications.

Andreas Andresen (WIAS Berlin)

Introduction to Spokoiny's finite sample analysis of maximum likelihood estimators

Abstract: In the paper "parametric estimation, finite sample theory" (2012) V. Spokoiny introduces a finite sample approach to describe the behaviour of quasi maximum likelihood estimators. The resulting statements can be viewed as extensions of known asymptotic properties to the finite sample setting. The talk presents and explains in some detail the ideas and techniques of this approach. A convergence result for a sequential alternating maximization scheme to calculate the profile maximum likelihood estimator shall illustrate the benefits of this technique.

Markus Pauly (Universität Düsseldorf)

Weighted logrank permutation tests for randomly censored
survival data

Abstract: In biomedical research weighted logrank tests are frequently applied to compare two samples
of randomly right censored survival times. We address the question how to combine a couple of weighted
logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone
of alternatives which are given by hazard rates. This leads to a new class of semiparametric projection-
type tests. We show that these tests can be carried out as permutation tests and discuss their asymptotic
properties. A simulation study together with the analysis of a classical dataset illustrate the advantages.

Anne Leucht (Universität Mannheim)

Degenerate U-statistics under weak dependence: Asymptotics, bootstrap and applications in statistics

Abstract: Numerous test statistics can be approximated by statistics of U- or V-type. In the case of i.i.d. random variables the limit distributions can be derived by a spectral decomposition of the kernel if the latter is square integrable. This method has been adopted for mixing and associated random variables, respectively. However, in the dependent case this approach requires some care. Most of the results in the literature have been derived under restrictive assumptions on the associated eigenvalues and eigenfunctions whose validity is quite difficult or even impossible to verify for many concrete examples in statistical hypothesis testing. In this talk, we devise new approaches to the asymptotic distributions of degenerate U- and V-statistics for weakly dependent random variables. We avoid any high-level assumption that can hardly be checked in applications. Instead only some moment constraints and smoothness assumptions concerning the kernel are required.
The limit distributions of U- and V-statistics for both independent and weakly dependent observations cannot be used directly since they depend on certain parameters which in turn depend on the underlying situation in a complicated way. Therefore, problems arise as soon as critical values for test statistics of U- and V-type have to be determined. The bootstrap offers a convenient way to circumvent these problems. There are already various papers on the validity of different bootstrap methods for degenerate U-statistics of i.i.d. data. Here, we derive consistency of parametric as well as model-free bootstrap methods for those statistics in the case of weakly dependent observations.
Finally, we apply our results to construct various hypothesis test of L2-type for time series.

Bharath Sriperumbudur (University of Cambridge)

RKHS-induced innite dimensional exponential families:
Density estimation and beyond

Abstract: In the rst part of the talk, we consider the problem of estimating densities in an innite
dimensional exponential family indexed by functions in a reproducing kernel Hilbert space. Since standard
techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves)
do not yield practically useful estimators, we propose an estimator based on the score matching method
introduced by Hyvarinen, which involves solving a simple linear system. We show that the proposed
estimator is consistent, and provide convergence rates under smoothness assumptions. We also empirically
demonstrate that the proposed method outperforms the standard non-parametric kernel density estimator.
The second part of the talk deals with certain mean element and covariance operator associated with
the above considered exponential family. For an appropriate choice of kernel, we show that these objects
uniquely characterize the probability measure and then provide an ecient approach for non-parametric
two-sample testing and independence testing.
Based on various joint works with Kenji Fukumizu (Institute of Statistical Mathematics, Tokyo) and Arthur
Gretton (Gatsby Unit, University College London).