abstracts of hamburger stochastik-tage 2000...6 plenary lecture functional limit laws in statistics...

Hamburger Stochastik-Tage 2000Universität Hamburg

21.–24. März 2000Zusammenfassungen

German Open Conference on Probabilityand Statistics

University of HamburgMarch 21 to 24, 2000

Abstracts

Scientific Programme Committee

Claudia Klüppelberg, TU München (Chair)Holger Dette, U BochumHans-Jürgen Engelbert, U JenaEnno Mammen, U HeidelbergGeorg Neuhaus, U HamburgMichael Nussbaum, WIAS BerlinVolker Schmidt, U UlmMartin Schweizer, TU BerlinAnton Wakolbinger, U Frankfurt

Local Organizing Committee

Georg Neuhaus (Chair)Dietmar PfeiferHans DadunaGerhard HübnerChristian HennigDirk NitschkeBero RoosKristin BetancourtJohana Nešlehová

1

Contents

Sections and Chairpersons 2

Plenary Lectures 3Opening Lecture 5Plenary Lecture 6Closing Lecture 7

Invited Lectures 9Sec. 1: Medical Statistics and Biometry 11Sec. 2: Stochastic Methods in Optimization and OperationsResearch 11Sec. 3: Asymptotic Statistics, Nonparametrics and Resampling 12Sec. 4: Statistics of Stochastic Processes and Time Series 12Sec. 5: Monte-Carlo-Methods, Simulation and Image Processing 13Sec. 6: Data Analysis and Design of Experiments 13Sec. 7: Model Choice in Statistics 13Sec. 8: Limit Theorems and Large Deviations 14Sec. 9: Statistics of Extremes and Subexponential Distributions 14Sec. 10: Insurance and Finance 15Sec. 11: Stochastic Geometry and Spatial Statistics 15Sec. 12: Stochastic Networks and Random Graphs 15Sec. 13: Stochastic Models in Biology und Physics 16Sec. 14: Stochastic Analysis 16Sec. 15: Dynamical Systems and Ergodic Theory 17Sec. 16: Quality Control and Reliability Theory 17

Contributed Lectures 19

Index of Authors 121

2

Sections and Chairpersons1 Medical Statistics and Biometry

Niels Keiding (Kopenhagen)

2 Stochastic Methods in Optimization and Operations ResearchKarl-Heinz Waldmann (Karlsruhe)

3 Asymptotic Statistics, Nonparametrics and ResamplingLutz Dümbgen (Lübeck)

4 Statistics of Stochastic Processes and Time SeriesJens-Peter Kreiß (Braunschweig)

5 Monte-Carlo-Methods, Simulation and Image ProcessingClaudia Czado (München)

6 Data Analysis and Design of ExperimentsUrsula Gather (Dortmund)

7 Model Choice in StatisticsFriedrich Liese (Rostock)

8 Limit Theorems and Large DeviationsGerd Christoph (Magdeburg)

9 Statistics of Extremes and Subexponential DistributionsThomas Mikosch (Groningen)

10 Insurance and FinanceChristian Hipp (Karlsruhe)

11 Stochastic Geometry and Spatial StatisticsMartina Zähle (Jena)

12 Stochastic Networks and Random GraphsRudolf Mathar (Aachen)

13 Stochastic Models in Biology und PhysicsAnton Bovier (Berlin)

14 Stochastic AnalysisMichael Röckner (Bielefeld)

15 Dynamical Systems and Ergodic TheoryMichael Scheutzow (Berlin)

16 Quality Control and Reliability TheoryWaltraud Kahle (Magdeburg)

17 Open SectionGerhard Hübner (Hamburg)

3

Plenary Lectures

5

Opening Lecture

Wavelets in statistics: beyond the standard assumptions

Bernard W. Silverman (Dept. of Mathematics, University of Bristol)

The original application of wavelets in statistics was to the estimation of a curvegiven observations of the curve plus white noise at 2J regularly spaced points. Therationale for the use of wavelet methods in this context is reviewed briefly. Variousextensions of the standard statistical methodology are discussed. These will includesome or all of curve estimation in the presence of correlated and nonstationarynoise, the estimation of (0−1) functions, the handling of irregularly spaced dataand data with heavy-tailed noise, and deformable templates in image and shapeanalysis. A Bayesian approach can be used to encapsulate the notion that mostof the wavelet coefficients are zero, and properties of this will be considered. Thenon-decimated, or translation-invariant, wavelet transform provides a substantialstatistical improvement over the standard discrete wavelet transform. There isa fast algorithm for finding all the within-level covariances in the discrete wavelettransform of a sequence with arbitrary band-limited covariance structure. Practicalapplications drawn from neurophysiology, meteorology and palaeopathology arepresented. Finally, some directions for possible future research are outlined.

6

Plenary Lecture

Functional Limit Laws in Statistics

Paul Deheuvels (Université Paris VI)

To motivate our approach, we first consider a classical density estimation setup.Let X1, X2, . . . denote i. i. d. random variables with a uniformly continuous den-sity f on R. The usual fixed bandwidth kernel Akaike-Parzen-Rosenblatt densityestimator is defined by fn(x) = (nhn)−1

∑ni=1K((x−Xi)/hn), where K is a func-

tion of bounded variation on R fulfilling∫RK(u) du = 1, and hn, n = 1, 2, . . . is

a sequence of positive constants such that hn → 0 and nhn/ log n → ∞. We firstobserve that these conditions are enough to ensure that( nhn

2 log(1/hn)

)1/2supx∈R±{fn(x)− Efn(x)} →P

(supx∈R

f(x)∫RK2(u) du

)1/2.(1)

Variants of (1) have been established by several authors (Silverman (1978), Stute(1982), Hall (1990), Deheuvels and Mason (1992), Deheuvels (1992), among others),with restrictions on either hn or the kernel K, the latter being usually assumed tobe boundedly supported, condition which may be shown to be superfluous in thissetting.

A more general weighted version of (1) has been investigated by Deheuvels andEinmahl (2000) in the framework of randomly censored observations. In this case,it may be shown, under appropriate assumptions on hn, that

( nhn2 log(1/hn)

)1/2supx∈I

ψn(x){fn(x)− Efn(x)} →P(

supx∈I

f(x)ψ2(x)∫I

K2(u) du)1/2

,

(2)

where I is a subinterval of R on which is defined a suitably regular function ψ withsupx∈I f(x)ψ2(x) < ∞, and where ψn denotes a uniformly consistent estimator ofψ.

Results of this type turn out to follow directly from functional limit laws forlocal empirical processes. Further examples of the kind are due to Mason (1988),Deheuvels and Mason 1990,1992,1994), Deheuvels (1992), and Einmahl and Mason(1998), among others.

The previous applications serve to illustrate the fact that functional limit lawsconstitute a powerful tool to describe the limiting behavior of a large class of non-parametric statistics. Our main purpose is to give an overview of this methodology.

7

Closing Lecture

Modeling Credit Derivatives

Wolfgang M. Schmidt (Deutsche Bank AG Frankfurt)

Credit derivatives are a relatively new class of financial derivative products. Theyallow for the customized transfer of the risk of default or deterioration of a certaindebtor or even a whole portfolio of credits. The valuation of credit derivativesis based upon the no-arbitrage pricing theory. The talk gives an overview of thevarious models used for pricing and hedging financial derivatives related to creditevents. We review existing theoretical approaches as well as their use in practice.Finally, some new developments and open modeling problems are outlined.

9

Invited Lectures

Invited Lectures 11

Sec. 1: Medical Statistics and Biometry

Regression Analysis of sampled Event History Cohort Datain Epidemiology

Per Kragh Andersen (Department of Biostatistics, University of Copenhagen)

Epidemiological studies are often based on event history data data obtained byfollowing large population-based cohorts over time. The rates of occurrence of thevarious events may then be analysed using, e. g. Cox-type regression models, thestatistical properties of which are well-known (see, e. g. “Statictical Models Based onCounting Processes”, Andersen, Borgan, Gill and Keiding, Springer-Verlag, 1993,Section VII.2).

Retrieving covariate information for everyone in the cohort may, however, byvery costly and various ways of the sampling from the cohort to reduce costs havebeen suggested. Such sampling designs include the nested case-control design andthe case-cohort design. These designs, their rationales, and their statistical analysiswill be reviewed (see, e. g. Borgan, Goldstein and Langholz, Ann. Statist, 1995 andSørensen and Andersen, Biometrika, 2000). Examples of their use in epidemiologicalinvestigations will be given.

Sec. 2: Stochastic Methods in Optimization andOperations Research

Comparison of queues with different discrete-time arrivalprocesses

Arie Hordijk (Mathematical Institute, University Leiden)

An open stochastic queueing network with one input node is considered. The net-work dynamics are assumed to satisfy a linear recursion in the so-called (max,+)-algebra. The epoch of the beginning of the n-th firing time of a FIFO-stochasticevent graph satisfies such a linear recursion for each transition. A special case is atandem network of FIFO-single-server queues. Suppose that a stationary sequenceof potential arrival epochs is given. The actual arrivals to the network are con-trolled by an admission sequence, which determines whether a potential arrival willbe admitted to the network. In optimal routing control the objective is to findthe optimal admission sequence. Here, we compare different admission sequencesthrough the performance of the stochastic network. Using the theory on multimod-ular functions, we derive a comparison lemma. Applications of this lemma will bediscussed, e. g. the fluid scaling decreases the travelling time (waiting time for onequeue) in the stochastic increasing convex ordering. Using a regularization proce-dure, which has been used in the theory on balanced sequences, we derive a secondcomparison lemma and the most regular arrival process for a fixed arrival intensity.We show that it provides a stochastic lower bound for any Markov arrival processwith the same stationary arrival intensity. We give a counterexample showing thatthe intuitive extension of the Ross conjecture is not true in this context, and wepresent the appropriate extension which is true.

12 Invited Lectures

Sec. 3: Asymptotic Statistics, Nonparametrics andResampling

Wavelets in nonparametric curve estimation

Michael H. Neumann (Wirtschaftswissentschaftliche Fakultät,Humboldt-Universität zu Berlin)

During the last years wavelets have become one of the standard tools in nonparamet-ric curve estimation. After their introduction into statistics by French statisticiansthey have attracted increasing attention because of their potential in connectionwith nonlinear regularisation techniques introduced by Donoho and Johnstone.

In the first part of the talk a brief introduction to the basic ideas in wavelet-based curve estimation is given. Reasons for the superiority of nonlinear waveletestimators over traditional nonparametric estimators (kernel, spline) are explainedin the context of the Gaussian white noise model. In the second part a generalizationto more complex estimation problems (non-Gaussian noise, multivariate curves) isdescribed.

References

Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Waveletshrinkage: asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57, 301–369.Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methodsfor stationary non-Gaussian time series. J. Time Ser. Anal. 17, 601–633.Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropicfunction classes and application to adaptive estimation of evolutionary spectra.Ann. Statist. 25, 38–76.Neumann, M. H. (2000). Multivariate wabelet thresholding. Statistica Sinica, Toappear.Dahlhaus, R. and Neumann, M. H. (2000). Locally adaptive fitting of semiparamet-ric models to nonstationary time series. Stoch. Proc. Appl., accepted for publication.

Sec. 4: Statistics of Stochastic Processes and Time Series

Recursive Monte Carlo Filters for State Space Models

Hans R. Künsch (Seminar für Statistik, ETH Zentrum Zürich)

A state space (or hidden Markov) model consists of an unobservable state process(Xt) assumed to be Markovian and a sequence of observations (Yt) which are con-ditionally independent given (xt) and such that the conditional distribution of Ytdepends on xt only. This model class has found a wide variety of applications,and I will present some of these in a first part. The main part of the talk isconcerned with filtering, that is computation of the conditional distribution of Xtgiven (y1, . . . , yt). This filter distributions obey a certain recursion that consists ofalternating applications of Markov transitions and Bayes formula. However, exactcomputations are usually not feasible, and recursive Monte Carlo approximationshave received much attention recently. I will discuss different variants of the basicsimulation algorithm with respect to speed and accuracy. Finally, I will addressthe issue of error propagation for the basic algorithm. Because Bayes formula istypically not contractive with respect to the prior, the straightforward analysis failsand more subtle arguments are needed.

Invited Lectures 13

Sec. 5: Monte-Carlo-Methods, Simulation and ImageProcessing

The Propp-Wilson algorithm with read-once randomness

Olle Häggström (Mathematical Statistics, Chalmers University of Technology,Göteborg)

In the last few years, there has been a lot of interest in simulation algorithms thatare “perfect”, in the sense that the initialization bias of the usual Markov chainMonte Carlo methods is completely removed. The most famous (and so far mostimportant) example is the Propp-Wilson algorithm. I will give a gentle introductionto this algorithm, and then discuss a recent improvement (due to Wilson) whichallows the use of a read-once source of random bits.

Sec. 6: Data Analysis and Design of Experiments

Tests based on Regression Depth

Peter J. Rousseeuw , Stefan Van Aelst , Mia Hubert (Dept of Mathematics andComputer Science, University of Antwerp)

The deepest regression is a method for linear regression introduced by Rousseeuwand Hubert (1999). It is the fit with maximal regression depth. We propose anapproximate algorithm for fast computation of the deepest regression in higherdimensions. From the distribution of the regression depth function we derive testsfor the true unknown parameters in the linear regression model. We also constructconfidence regions by applying depth to bootstrapped regression estimates. Forbivariate datasets we use the maximal regression depth to construct a test forlinearity versus convexity/concavity. Finally, the deepest regression is applied topolynomial regression and to the Michaelis-Menten model.

References

Rousseeuw, P.J., and Hubert, M. (1999), “Regression Depth,” Journal of the Amer-ican Statistical Association, 94, 388-402.

Sec. 7: Model Choice in Statistics

Tests of fit based on the integrated empirical process andtheir efficiencies

Norbert Henze (Institut für Mathematische Stochastik, Universität Karlsruhe),Yakov Yu. Nikitin (Dept. of Mathematics and Mechanics, St. Petersburg

University)

We consider tests of fit based on functionals of the suitably integrated empiri-cal process. A key limiting process is the integrated Brownian bridge. The newtest statistics are compared with their classical “non-integrated” counterparts withrespect to local Bahadur efficiency in case of shift alternatives. Whereas the in-tegrated Kolmogorov-Smirnov test is locally optimal for the logistic distribution,an integrated ω1n-statistic turns out to be locally Bahadur optimal for the “root-logistic” distribution.

14 Invited Lectures

Sec. 8: Limit Theorems and Large Deviations

Asymptotic Distribution of Quadratic Forms

Friedrich Götze (Fakultät für Mathematik, Universität Bielefeld)

We shall study approximations for the distribution of nondegenerate quadraticforms of sums of indepenedent random vectors by χ2 and Gaussian distributions.In this problem optimal rates of convergence in the number of summands n areshown for higher dimensional vectors.

The methods used in the proofs allow as well the investigation of asymptoticapproximations of quadratic forms of n-dimensional vectors with independent com-ponents with application e. g. to quadratic forms of AR-processes. This is jointwork with A. Tikhomirov.

Furthermore, the methods developed in the latter cases suffice in higher dimen-sions to answer well known questions about the uniform distribution of values ofquadratic forms on lattices in connection with conjectures by Davenport and Op-penheim. Moreover, optimal bounds for the error of the volume approximation forthe number of lattice points in large ellipsoids are obtained.

Sec. 9: Statistics of Extremes and SubexponentialDistributions

Long strange segments and long range dependence

Gennady Samorodnitsky (School of OR and IE, Cornell University), PeterMansfield (School of Accounting and Finance, University of Tasmania, Hobart),

Svetlozar T. Rachev (Institute of Statistics and Mathematical Economics,University of Karlsruhe)

Long range dependence is supposed to be the kind of dependence that not onlydissipates slowly with time; it is qualitatively different from ordinary, short rangedependence. The notion of long range dependence has been traditionally associatedwith slow decay of correlations. Looking solely at correlations, however, does notallow one even to talk about long range dependence in a stochastic process withinfinite variance; and infinite variance models have acquired prominence in the last10 years. Furthermore, even if the variance is finite, the information carried bycorrelations is fairly limited if the actual process is far from being a Gaussian one.

A possible alternative approach to the phenomenon of long range dependence isto look at implications of the latter. That is, one looks at a particular importantfunctional of a stochastic process. One looks then for a kind of a “phase transition”in the behavior of this functional.

We study long strange intervals in a linear stationary stochastic process withregularly varying tails. It turns out that the length of the longest strange intervalgrows, as a function of the sample size, at different rates in different parts of the pa-rameter space. We argue that this phenomenon may be viewed in a fruitful way asa phase transition between short and long range dependence. We prove a limit the-orem that may form a basis for statistical detection of long range dependence. Oursuggested statistic is then applied to a number of financial and telecommunicationdata sets.

Invited Lectures 15

Sec. 10: Insurance and Finance

Business risk vs. financial market risk. Which comes first? Amodel for a corporate strategy

Michael Taksar (Department of Applied Mathematics, State University of NewYork at Stony Brook)

We consider a problem in which the liquid assets or reserves of a company aremodeled by a diffusion process. At each moment of time the management of thecompany makes a decision of the amount of dividends paid-out to the shareholders.There is also a possibility to reduce risk exposure by conducting a less aggressivebusiness activity, which also results in a smaller potential profit. Mathematicallythis corresponds to decreasing simultaneously drift and diffusion coefficients of thecontrolled process. In addition the reserve of the company can be invested in astock market in which asset prices follows Black-Scholes model.

Of a particular interest is the example of an insurance company, in which therisk control gets of a more tangible form, such as reinsurance. We formulate thecorresponding stochastic control models without and with controllable investments.The later means that at each moment the company decides what fraction of itsreserve to put in risky and risk-free assets. For each of the cases the optimal policyis found and described in a close form.

Sec. 11: Stochastic Geometry and Spatial Statistics

Random Fractals

Kenneth Falconer (Math. Institute, University of St. Andrews)

This talk will survey some random fractal constructions and their properties. Thesewill include statistically self-similar fractals, fractal percolation and fractional Brow-nian surfaces. The talk will be illustrated by many examples and refer to recentwork and unsolved problems.

Sec. 12: Stochastic Networks and Random Graphs

Infinite tandem queueing networks

Jean Mairesse (CNRS, Université Paris)

In an infinite tandem network, queues are connected in such a way that the inter-departures from a queue form the inter-arrivals into the next one. The basic model isthat of single-server queues with infinite buffers and a FCFS discipline, the servicesat different queues being independent. One can consider several variants of themodel, depending on the assumptions on the services (exponential (M), i. i. d. (GI)or ergodic (G)), and on the boundary conditions (one-sided or two-sided infinitestream of customers).

In such tandem networks, the total time spent by a customer in a sequence ofqueues can be represented as a longest path in an infinite periodic oriented graphwith a specific structure.

It is of interest to extend the model by considering longest paths in more generalinfinite periodic oriented graphs. By doing this, one can represent sojourn times ininfinite tandem of queues with limited buffers, or in infinite tandem of event graphs

16 Invited Lectures

(a sub-class of Petri nets). We present an overview of the results known on thesemodels.

Sec. 13: Stochastic Models in Biology und Physics

Droplet Growth in Fluids

Frank den Hollander (Katholieke Universiteit Nijmegen)

For a supersaturated gas to be able to condense to a liquid, the gas must createa “critical droplet” of the liquid phase inside the gas phase. This critical droplettriggers the phase transition. In certain experimental situations, however, the for-mation of this droplet takes a very long time. During this time the gas is ina “metastable state”, characterized by many unsuccesful attempts to create thedroplet.

We describe a model of a two-dimensional lattice gas where particles performso-called Kawasaki dynamics, i. e., particles hop around, are not allowed to occupythe same site, and when two particles sit next to each other they feel a bindingenergy that slows down their dissociation. We look at this model in the limit of lowtemperature and low density, and present two theorems. The first theorem identifiesthe size and the shape of the critical droplet and the time of its creation. The secondtheorem describes the behavior of the lattice gas just prior to the creation of thecritical droplet.

Because particles are conserved under Kawasaki dynamics, the study of metasta-bility and nucleation is much more difficult than under non-conserved dynamics,such as Glauber spin-flip dynamics in the Ising model. The conservation law in-troduces non-local phenomena, which need to be understood and controlled. As aresult, the metastable behavior of the lattice gas exhibits interesting new aspects.

Sec. 14: Stochastic Analysis

On a space of BV functions over the abstract Wiener space

Masatoshi Fukushima (Department of Mathematics, Kansai University)

Let (E,H, µ) be an abstract Wiener space and we denote by ∇u the H-derivativeof u ∈ FC1b . Let H be the class of functions ρ ∈ L1+(E;µ) satisfying the ray Hamzacondition in every direction ` ∈ E∗. For ρ ∈ H,

Eρ(u, v) =∫E

〈∇u(z),∇v(z)〉H ρ(z) µ(dz), u, v ∈ FC1b .

is a well defined closable symmetric form on L2(E; ρ · µ) and its closure denotedby (Eρ,Fρ) is a quasi-regular local, conservative Dirichlet form on L2(F, ρ dµ),where F is the support of the measure ρdµ. Accordingly there exists an associatedconservative diffusion process Mρ = (Xρt , Pz) on F , which is called a distortedOrnstein-Uhlenbeck process. A function ρ ∈ L1(E;µ) is said to be a BV function(ρ ∈ BV (E) in notation) if

V (ρ) = supG∈(FC1b )E∗ ,‖G‖H(z)≤1

∫E

∇∗G(z)ρ(z)µ(dz)

Invited Lectures 17

is finite. If ρ ∈ H ∩ BV (E) ∩ Lp(E;µ), p > 1, then there exist a positive finitemeasure ‖Dρ‖ on E charging no Eρ-exceptional set and a ‖Dρ‖-measurable functionσρ : E −→ H such that ‖σρ(z)‖H = 1 ‖Dρ‖-a. e. and the next equation holds:∫

E

∇∗G(z)ρ(z)µ(dz) =∫E

〈G(z), σρ(z)〉H‖Dρ‖(dz) , ∀G ∈ (FC1b )E∗ .

Further, the diffusion Mρ satisfies the stochastic equation :

Xρt −Xρ0 = Bt −

12

∫ t0

Xρs ds+12

∫ t0

σρ(Xρs )dLρs

for an E-valued Brownian motion Bt and a PCAF Lρt of Mρ with Revuz measure

‖Dρ‖. A measurable set Λ ⊂ E is called Caccioppoli if IΛ ∈ BV (E). In this case,the support of the measure ‖DIΛ‖ is contained in ∂Λ and the above equationsreduce to the Gauss formula and the Skorohod equation respectively.

Sec. 15: Dynamical Systems and Ergodic Theory

News from the theory of random dynamical systems

Ludwig Arnold (Institut für Dynamische Systeme, Universität Bremen)

We first recall the definition of a random dynamical system, covering in particularsolutions of random and stochastic differential and difference equations. For acomprehensive exposition see the monograph [1].

We then report on progress made during the last couple of years in the followingareas:

(1) Stochastic bifurcation theory (classification of 1-dimensional case, new insightinto stochastic Hopf bifurcation, further results of Baxendale)

(2) Random attractors (existence via transforming a stochastic differential equa-tion into a random differential equation)

(3) Lyapunov’s second method for random dynamical systems (what are theproper concepts of stability, attraction and Lyapunov function?)

(4) Order-preserving random dynamical systems (limit set trichotomy)(5) Jordan normal form for products of random matricesAll results will be discussed by way of simple examples and numerical case stud-

ies, including the Brusselator, the Lorenz system and the Duffing-van der Pol os-cillator with parametric noise, and the Duffing oscillator with additive noise.

References

[1] Arnold, L.: Random dynamical systems. Springer, Berlin, 1998.

Sec. 16: Quality Control and Reliability Theory

Reliability Assessment and Predictive Inference: SomeThoughts on the Foundations

Elja Arjas (Rolf Nevanlinna Institute, University of Helsinki)

18 Invited Lectures

In this review I will discuss some foundational ideas which arise naturally in thecontext of reliability assessment. I will focus on situations in which such assess-ments are updated continuously on the basis of monitored failure data. This makesthe assessments a dynamic process, with discontinuities arising from unpredictableobservations. In the considered approach, probabilistic modeling of system struc-tures and their reliabilities and statistical inference from observed data will beconsidered under a single predictive (Bayesian) framework. The role and the inter-pretation of the notion of infinite exchangeability are discussed, highlighting boththe strong normative implications of this assumption and its practical limitationsin the context of reliability. The central themes of the talk are illustrated by simpleexamples.

19

Contributed Lectures


Convergence of Conditional Gibbsmeasures and theEquivalence of Gibbsensembles


Stefan Adams (Mathematisches Institut, Universität München)

We first show for the Ising-model on the lattice Z in dimension 1 that the Gibb-smeasures conditioned on a microcanonical constraint for the energy converge tothe Gibbsmeasures which maximize the entropy relative to the constraint. Thisresult answers the equivalence of Gibbsensembles on the level of measures in aformulation, which would mean that the microcanonical and the grand canonicalGibbsdistributions in finite boxes have the same infinite-volume limits. We willintroduce three different types of equivalence of the Gibbsensembles in order toshow that still there is a lack of proof on the level of measures with non-periodicboundary conditions. In our result we do not need periodic boundary conditionslike in [Csi87] and [Geo93] and we have the result for the whole lattice Z insteadof Z+ in [Men93]. If we have not periodic boundary conditions the limit measureis not necessary translation invariant, and therefore we need a different approach.And we have the result for the Ising-model in d = 1 that the accumulation points ofthe sequence of the microcanonical Gibbsdistributions Mηn,�,ρ for the mean energy� and mean magnetization ρ belong to the set GΘ(βΦ) of Gibbsmeasures for aninteraction βΦ.

In the second part we will generalize these results to an arbitrary finite statespace E and microcanonical conditions like in [Geo93] in the way that we use themethods of types like in [Csi87], Large Deviation Results for Markov Chains andGibbsmeasures and our results.

References

[Csi87] Csiszar, I., Cover, T.M., Choi, B.S., Conditional limit theorems underMarkov conditioning, IEEE Trans. Inform. Theory 33, 788-801, 1987.

[Geo93] Georgii, H.O., Large Deviations and Maximum Entropy Principle ForInteracting Random Fields an Zd, The Annals of Probability 21, No. 4,1845-1875, 1993.

[Men93] Menzel, P., A limit theorem for one-dimensinal Gibbsmeasures underconditions on the empirical field, Stochastic Processes and their Appli-cations 44, 347-359, 1993.

Optimal Decision Rules in the Stochastic Model ofInvestment

Sec. 2: Stochastic Methods in Optimization and Operations Research

Vadim I. Arkin , Alexander D. Slastnikov (Central Economics and MathematicsInstitute Moscow)

As an object of investment it will be considered a project of a creation of a newenterprise. Investments I, necessary for such a project are considered to be instan-taneous and irreversible (sunk costs).

At any moment the investor can either accept the project and start with theinvestment, or delay the decision before obtaining new information on the environ-ment (prices on the production, demand, etc.).

We consider that the profit from the project (including all taxes and paymentsexcept income tax) is described by stochastic process Π = (πt, 0 ≤ t

22 Contributed Lectures

Tax system significantly influences the investor behavior and can be representedas a triplet (γf , γr, ν), where γf and γr are income tax rates into federal and regionalbudgets, and ν is a duration of tax holidays.

Present value of the investor return from the project started at the moment t Vτdepends on tax system and process πt.

The problem of the investor is:

E (Vτ − I) e−ρτ → maxτ,

where maximum is considered over all Markovian moments τ (“investment rules”).The process of the project profits is described by stochastic integral equation

πt = π0 +

t∫0

πs(αds+ σ dws)−Nt(τ)∑j=0

ξjπθj−0 ,

where wt, t ≥ 0 is a Wiener process, Nt(τ) =

{Nt−τ , if t ≥ τ ,0, if t < τ ,

(Ns, s ≥ 0) is

Poisson process, (ξi, i ≥ 0) are independent (on πt) random variables.For this model we obtain the optimal investment rule in explicit form.

Measurement Error Correction in Lifetime DataSec. 16: Quality Control and Reliability Theory

Thomas Augustin (Department of Statistics, University of Munich)

A typical problem in regression analysis is the presence of measurement error.Often there are variables of particular interest which cannot be directly observedor measured correctly. However, if one ignores the measurement error by justplugging in substitutes or incorrect measurements (“naive estimation”), then all theparameter estimates must be suspected to be severely biased. Error-in-variablesmodelling provides a methodology, which is serious about that fact and developsprocedures to adjust for the measurement error. For a summary of the state of theart in linear and nonlinear models see: Cheng and van Ness (1999) and Caroll,Ruppert, and Stefanski (1995).

After a brief introduction into the topic of error-in-variables in general, the talkturns to the problem of measurement error in parametric duration models. In thecontext of a flexible structural model based on mixtures of normals general resultsare derived which allow to construct measurement error corrected quasi-likelihoodestimates in any accelerated failure time model, i. e. in most of the commonlyused parametric duration models (e. g. in the Weibull model, the Gamma modeland the log-logistic model). The procedure proposed is also general enough to dealsimultaneously with covariate measurement error as well as with measurement errorin the duration variable itself. A straightforward extension to censored observations,however, seems to be not possible. Even under non-informative random censorship,the quasi-score function depends on the typically unknown censoring distributionin quite a complex way. Several ideas are brought up for discussion how to makethe approach also suitable for censored data.

References

Augustin, T. (1999): Correcting for measurement error in parametric durationmodels by quasi-likelihood. SFB Discussion Paper 157, University of Munich. Ac-cepted for publication in: Biometrical Journal.Caroll, R. and Ruppert, D. and Stefanski, L.A. (1995): Measurement Errorin Nonlinear Models. Chapman and Hall, London.


Cheng, C.-L. and Van Ness, J.W. (1999): Statistical Regression with Measure-ment Error. Arnold, London.

Conductivity Parameters of a High-Contrast RandomMedium

Sec. 5: Monte-Carlo-Methods, Simulation and Image Processing

Antonia D. Avdeenko (Novosibirsk)

The paper is closely related with the paper by [A.G.Kolpakov Numerical Experi-ments for Conductivity of a High-Contrast Random Medim] presented to the Con-ference.

The mentioned above paper demonstrates that the dependence “effective dielec-tric constant-volume fraction V of the disks” is a percolation type function for amedium randomly filled with randomly distributed disks. In the present paper theresults concerning the determination of the parameters of the function are pre-sented.

The experimental (obtained from numerical experiments) graphics “effective di-electric constant – volume fraction V of the disks” look like a graphics of the func-tion C(V − V0)p + C0. The aim of the work is check this hypothesis and calculate(evaluate) the numerical values of the parameters V0, p and C0.

The estimates for V0, p and C0 will be presented.

Distance measures and substitution processes in molecularphylogeny

Sec. 13: Stochastic Models in Biology und Physics

Ellen Baake (Zoologisches Institut München)

Molecular phylogeny, the art of reconstructing the history of a sample of sequencesfrom the leaves of a tree, is often performed within the framework of additive metrictrees [1]. If distances between leaves are tree-additive, the tree topology as well asthe branch lengths may be reconstructed. A very general distance concept is the so-called “paralinear distance” [2, 3], which is a function of pairwise joint probabilitiesof the letters at the leaves. It is tree-additive if the underlying substitution processis a Markov process on a tree, where the transition matrices are allowed to varyfrom branch to branch.

We address two questions:1) How are the branch lengths belonging to the paralinear distance related to

the substitution process on the tree?2) Can information about the substitution process (as opposed to just the tree

geometry) be inferred from pairwise joint distributions of letters at the leaves?The answers are:1) The paralinear branch length may be interpreted as the sum of relative proba-

bility fluxes, integrated over the branch, and averaged over the forward and reversedirections of the Markov process, where the averaging reflects our ignorance of theposition of the root [4].

2) The Markov transition matrices across branches may not be inferred sepa-rately, but return-trip matrices (forward and back again across a branch) may beinferred, up to conjugacy in the case of internal branches [5].


References

[1] M. S. Waterman et al., J. Theor. Biol. 64 (1977), 199–217.[2] M. A. Steel, Appl. Math. Lett. 7 (1994), 19–25.[3] J. A. Lake, PNAS 91 (1994), 1455–1459.[4] E. Baake, A. v. Haeseler, Theor. Pop. Biol. 55 (1999), 166–175.[5] E. Baake, Math. Biosci. 154 (1998), 1–21.

Diffraction theory of stochastic point sets ISec. 13: Stochastic Models in Biology und Physics

Michael Baake (Institut für Theoretische Physik, Universität Tübingen)

The diffraction measure (also called diffraction spectrum) is the Fourier transformof the autocorrelation measure of a translation bounded complex measure in n-dimensional space. It is a positive measure with unique decomposition into pp,sc, and ac components, relative to Lebesgue measure. In this joint work withMoritz Hoeffe, the mathematical frame is introduced and various general resultsare derived, in particular on the diffraction of Dirac combs confined to a subset of alattice or a model set. This provides the basis for the investigation of lattice gasesand their generalizations.

On the topological structure of random fractalsSec. 11: Stochastic Geometry and Spatial Statistics

Christoph Bandt (Institut für Mathematik und Informatik, Arndt-UniversitätGreifswald)

Random fractals are obtained in a natural way from many spatial-temporal randomprocesses, for example percolation clusters or limit distributions of certain branch-ing processes and interacting particle systems. On the other hand there is a moreconstructive method to produce random fractals by iteration of randomly chosenvectors of similarity maps. While this construction is well understood, it is notclear what kind of distributions of mappings will guarantee connectivity properties,or will lead to the natural examples mentioned before.

In the lecture we shall apply the method of neighbour maps for deterministicfractals to the random case. In particular we study separation and connectivityproperties of the construction set or construction measure.

GUEs and QueuesSec. 13: Stochastic Models in Biology und Physics

Yuliy Baryshnikov (EURANDOM Eindhoven)

Consider the process Dk, k = 1, 2, . . . , given by

Dk = sup0=t0


Heuristic policies for queueing networksSec. 2: Stochastic Methods in Optimization and Operations Research

Nicole Bäuerle (Abteilung Mathematik VII, Universität Ulm)

Control problems in stochastic queueing networks are hard to solve. Therefore,it is important to derive good heuristic policies. In this talk we will show howto construct such policies for general networks. The construction relies on theknowledge of the optimal control for the associate fluid problem. It can be shownthat this class of policies is asymptotically optimal under fluid scaling. We willidentify situations in which a one-to-one translation of the fluid optimal controlgives an optimal policy for the stochastic network problem. We will also show somenumerical results.

The value of initial investment informationSec. 10: Insurance and Finance

Dirk Becherer (Fachbereich Mathematik, Technische Universität Berlin),Jürgen Amendinger (HypoVereinsbank AG München)

We consider an investor maximizing his expected utility from terminal wealth withportfolio decisions based on the available information flow. This investor faces theopportunity to acquire some additional information I. The subjective fair value ofthis information for the investor is equal to the amount of money that he can payfor I, such that this cost is balanced by the informational advantage in terms ofmaximal expected utility. We calculate this value for common utility functions inthe situation of a complete market.

Effects of outliers on the analysis of multivariate dataSec. 6: Data Analysis and Design of Experiments

Claudia Becker (Fachbereich Statistik, Universität Dortmund)

Outliers in datasets can affect the statistical analysis in various ways. Especially formultivariate data, the effects of outliers are not even all known. This begins withthe simple standardization of observations, which is used in many statistical proce-dures. Thus, different methods like classical principal component analysis (Crouxand Haesbroeck, 1999), Sliced Inverse Regression (Li, 1991, Gather, Hilker andBecker, 1999), cluster analysis (Cuesta-Albertos, Gordaliza and Matran, 1997),or classical outlier identification procedures (Becker and Gather, 1999) may bestrongly influenced by the occurrence of outliers. The paper shows in which waysthis can happen. Consequently, there is a need for robustified versions of classi-cal multivariate methods. The talk therefore discusses possible robustifications ofvarious multivariate procedures.

References

Becker, C., Gather, U. (1999), The Masking Breakdown Point of Multivariate Out-lier Identification Rules, J. Amer. Statist. Assoc., 94, 947–955.Croux, C., Haesbroeck, G. (1999), Principal Component Analysis Based on Ro-bust Estimators of the Covariance or Correlation Matrix: Influence Functions andEfficiencies, Preprint IS-P 5, Institut de Statistique et de Recherche Opérationelle,Université Libre de Bruxelles.Cuesta-Albertos, J.A., Gordaliza, A., Matran, C. (1997), Trimmed k-Means: AnAttempt to Robustify Quantizers, Ann. Statist., 25, 553–576.


Gather, U., Hilker, T., Becker, C. (1999), A Robustified Version of Sliced InverseRegression, to appear in: Proceedings of the Workshop on Statistical Methodologyfor the Sciences: Environmetrics and Genetics, Ascona, 23.05.-28.05.1999.Li, K.-C. (1991), Sliced Inverse Regression for Dimension Reduction (with discus-sion), J. Amer. Statist. Assoc., 86, 316–342.

Bayes Tests of Power One and Sequential Detection inExponential Families


Martin Beibel (Institut für Mathematische Stochastik, Universität Freiburg)

Let Ω denote an interval on the real line R. Let (Pθ, θ ∈ Ω) denote a one-parameterexponential family of distributions on R. Let θ0 denote a point in the interior ofΩ. Let X1, X2, . . . be a sequence of independent random variables distributedaccording to Pθ for some unknown θ ∈ Ω. We consider the problem of testingthe null hypothesis θ = θ0 versus the alternative θ 6= θ0. We suppose that thestrategies at hand are stopping times T of the observed sequence X1, X2, . . . andthat stopping always means deciding in favour of the alternative. This impliesthat it is desirable not to stop at all if the null hypothesis holds true and to stopas soon as possible if the alternative holds true. Let Likn(θ) denote the likeli-hood Likn(θ) = dPθ/dPθ0 |σ(X1, . . . , Xn) and I(θ) the Kullback-Leibler informa-tion I(θ) = Eθ log(Lik1(θ)). Let p denote a prior density on Ω. Let for c > 0 andany stopping time T of X1, X2, . . . denote L(c, T ) the Bayes risk

L(c, T ) = P0(T c−1}, where

Z(n) =∫

Ω

ν(θ)−1 Likn(θ)p(θ) dθ ,

with

ν(θ) =1I(θ)

exp

[−∞∑n=1

1nEθ0

{min

(Likn(θ), 1

)}].

We further extend this approach to sequential detection problems. A careful analy-sis of the overshoot logZ(Nc)− log(1/c) plays a central role in our arguments. Weuse general results from nonlinear renewal theory and nonlinear parking problems.

Firm Pension: Case Study NovartisSec. 10: Insurance and Finance

Thomas Benesch (Institut für Wirtschafts- und Betriebswissenschaften Graz)

In Austria the system for old age provision, is dominated by the public pensionsystem. Most of the pensions (about 93 %), drawn by retired persons in Austria,derives from the public pension system. The average net income replacement rate,for social security pensioners in the private sector, amounts to about 72 percent ofthe income earned before retirement. This is, in international comparison, rathergenerous. At present, the financial gap, particularly affects whoever earns over theupper limit income threshold, has few premium years, and who retires before theofficial retirement age. The present financing problems, have their cause mainly inthe unfavourable situation of the labour market. Weak employment growth, and


the implementation of generous early retirement insurance schemes, have seriouslystrained the pension system. The ageing of the population, will create additional,long term, financing problems, over the next decades. Important kinds of the firmpension, are the pension funds and the direct provision promise. The spread ofpension funds, has increased quickly, because of recent tax incentives. In Austriaabout 11 % of the employees have a claim on a firm pension.

With the help of the Novartis direct pension promise, it is possible to receive anoptimum of 70 % of the last income level, even for incomes far above upper limitincome threshold. Novartis is a world wide leader in the fields of Life Sciences, andcontributes through different products and services to the health and well beingof human beings. The firm arose in 1996 through the fusion merger of Ciba andSandoz. The interests of the concern are focused on health, agriculture and nutri-tion. The corporate headquarters of Novartis, one of the largest European businessconcerns, is in Basel, Switzerland. Novartis shares are quoted in the Swiss stockexchange, and are traded from there in London and in the USA. World-wide, No-vartis engages 87,000 employees from different cultures, with different qualificationsand duties in over 100 countries. Turnover in 1997 was over 265 Billion AustrianShillings. The Austrian Novartis Group engage a total of over 2,400 employs andgarnered a turnover of 10.5 Million Austrian shillings in 1997.

With the direct pension promise, the employer promises the employee irrevoca-bly, to give a provision when he retires. The employer bears the risk. The financingof the provisions, are paid from the current budget. If the provisions are givenwith right tax deductible pension postponements, can be accumulated. 50 % of thisobligation has to be held in securities.

Identity Change with Newsgroup Questionings: CaseExample European Parliamentary Elections

Sec. 6: Data Analysis and Design of Experiments

Thomas Benesch (Institut für Wirtschafts- und Betriebswissenschaften,Technische Universität Graz)

In the real life there is the possibility to change the identity, e. g. because of discon-tent with the own identity, vocationally caused changes of the residence, change ofname by social obligations or by a witness protection program.

In the EDP the identity is shaped of so called roles, which has rights and obli-gations. E. g. a person at a computer can be at the same time administrator anduser.

Within the area of the Internet identities are added by newsgroups, nicknamesin chatrooms, login with various email addresses such as Hotmail. The identity inthe Internet reduces to a IP (Internet Protocol) or to the sender of a email.

A questionnaire, i. e. an email was sent to newsgroups, which refers to a WWWsite, on which the questionnaire is about the European parliament elections inparty-oriented newsgroups were executed. Since 26 May 1999 up to the Europeanparliamentary election we have received 49 responses.

It was clearly shown that with topics, which are delicate or emotional occupied,like politics topic, the respondent wants to weight their opinion superproportionallyand therefore using different identities while filling out the questionnaires. Thusthe sex or the age was changed. Further it became clear that answers of respondentcame predominantly, who has an Internet access free of charge.


Anomalous Behaviour for the Random Corrections to theCumulants of Random Walks in Fluctuating Random Media


Maria Simonetta Bernabei (Institut für Angewandte Mathematik, RheinischeFriedrich-Wilhelms Universität Bonn)

The Central Limit Theorem (CLT) for a model of discrete-time random walkson the lattice Zν in a fluctuating random environment was proved for almost-allrealizations of the space- time environment, for all ν ≥ 1 and it was proved thatthe random correction to the average of the random walk for ν ≥ 3 is finite. In thepresent paper we consider the cases ν = 1, 2 and prove the CLT for the randomcorrection to the first two cumulants. The rescaling factor for the average is T

14 for

ν = 1 and (lnT )12 , for ν = 2, for the covariance is T 1−

ν4 , ν = 1, 2.

Gibbs measures with respect to Brownian motionSec. 13: Stochastic Models in Biology und Physics

Volker Betz , Jozsef Lörinczi (Zentrum Mathematik, Technische UniversitätMünchen), Fumio Hiroshima (Department of Mathematics, Hokkaido University)

We perturb Brownian motion by a single site and a pair potential. For finite volumethis leads to the probability measure µa,bT on B(C([−T, T ],Rd)) given by

dµa,bT (ω) =1

Za,bTexp

(∫ T−T

V (ωs) ds−∫ T−T

∫ T−T

W (ωs − ωt, s− t) ds dt

)dWTa,b ,

where T ∈ R, a, b ∈ Rd, WTa,b is pinned Brownian motion starting in a at time −Tand ending in b at time T , V : Rd → R and W : Rd×R→ R are suitable potentialsand Za,bT is the normalizing constant. We prove existence of a limiting probabilitymeasure on B(C(R),Rd)) for the µa,bT as T → ∞ and consider the problem ofuniqueness, dependence on boundary conditions, and support properties of thismeasure.

For a particular choice of W the measure is Nelsons model of quantum fieldtheory. Using this connection, we will prove the exponential decay of the particledensity in the ground state.

An asymptotic model for regression modelsSec. 3: Asymptotic Statistics, Nonparametrics and Resampling

Wolfgang Bischoff (Insitut für Mathematische Stochastik, UniversitätKarlsruhe)

We consider a general regression model with experimental region [a, b]. For eachdesign the observations can be embedded into C[0, 1]; hence the observations can beconsidered as stochastic process in C[0, 1]. If the sequence of designs converges suit-ably, then the corresponding sequence of stochastic process converges to a Gaussianprocess. This Gaussian process can be considered as an asymptotic model of the re-gression model and the corresponding design. If the regression model is linear, thenoptimal procedures can be determined. The consequence of the above considera-tions for a nonlinear model is the contents of the talk “Efficiency of the Kolmogorofftest in an asymptotic model of a nonlinear regression model” by Frank Miller.


References

W. Bischoff (1998) A functional central limit theorem for regression models, Ann.Stat. 26, 1398 - 1410.W. Bischoff, F. Miller (1998) Asymptotically optimal tests and designs for testingthe mean in regression models with applications to change-point problems, Ann.Inst. Statist. Math., to appear.W. Bischoff, F. Miller (1999) The F-test is asymptotically optimal in linear regres-sion models with unknown distribution of the error, Preprint.

Asymptotic results and a Markovian approximation for theM(n)/M(n)/s+GI system


Andreas Brandt (Wirtschaftswissenschaftliche Fakultät, Humboldt-Universitätzu Berlin), Manfred Brandt (Konrad-Zuse-Zentrum für Informationstechnik Berlin

(ZIB))

In this paper for the M(n)/M(n)/s+GI system, i. e. for a s-server queueing systemwhere the calls in the queue may leave the system due to impatience, we presentnew asymptotic results for the intensities of calls leaving the system due to im-patience and a Markovian system approximation where these results are applied.Furthermore, we present a new proof for the formulae of the conditional density ofthe virtual waiting time distributions, recently given by Movaghar for the slightlyless general M(n)/M/s + GI system. Also we obtain new explicit expressions forrefined virtual waiting time characteristics as a byproduct.

Nonparametric Analysis of Multicenter Clinical TrialsSec. 1: Medical Statistics and Biometry

Edgar Brunner (Abteilung Medizinische Statistik, Universität Göttingen)

Nonparametric methods for the analysis of data from multi-center trials with twotreatments are considered where the sample sizes may be unequal and the distribu-tion functions are not assumed to be continuous. Some recently published methods(Akritas, Arnold and Brunner, 1997; Akritas and Brunner, 1997; Brunner, Munzeland Puri, 1999) for the analysis of such data in unbalanced designs are used toderive statistics for fixed as well as for mixed models where the hypotheses are for-mulated by means of the distribution functions (Akritas and Arnold, 1994). Unlikein the literature, unequal weights are used for the centers according to the samplesizes. The effect of the weights on the pre-assigned level as well as on the powerof the tests is investigated for the case of independent observations as well as fordesigns with repeated measures. The procedures are applied to a multi-center trialwith ordered categorical data.

Risk Investment ModelsSec. 10: Insurance and Finance

F. Thomas Bruss (Département de Mathématique, ISRO, Université Libre deBruxelles)


How should we invest capital into a sequence of investment opportunities, if, by thecharacteristics of the problem, our interest must focus on trying to invest in thevery best opportunity?

We motivate and present new “weak-information” models to tackle such ques-tions, in particular in view of venture capital investment strategies. In general weassume that an investment which turns out best within a target group yields alucrative, possibly time-dependent, rate of return, that non-invested capital keepsits risk-free value, whereas “wrong” investments lose their value.

Several models are examined mainly for the so-called no-information case, in-cluding models for an unknown number of opportunities. Optimal strategies andvalues are found, also for different utility functions, and a few examples are explic-itly solved. We also include results for the so-called full-information case, where,in addition, the quality distribution of investment opportunities is supposed to beknown.

Some results presented in this talk are joint work with Thomas S. Ferguson,(UCLA).

Keywords: Competitive performance, the Kelly betting system, general utility,hedging, secretary problems, no-information, full-information, random number ofopportunities.

AMS 1991 Subject Classification: primary 60G40, secondary 90A80.

Decompounding: an estimation problem for compoundPoisson distributions

Sec. 3: Asymptotic Statistics, Nonparametrics and Resampling

Boris Buchmann (Institut für Mathematische Stochastik, UniversitätHannover)

The compound Poisson distribution associated with a distribution Q on the realline and some λ > 0 is defined to be the distribution of S = X1 +X2 + . . .+XN ,with N , X1, X2, . . . independent, N Poisson-distributed with parameter λ and Qthe distribution of the X-variables. We consider the problem of non-parametricestimation of Q from a sample of values from S.

Higher Order Asymptotic Behavior of AdaptationProcedures in Nonparametric Regression


Olaf Bunke , P. Illouga (Institute of Mathematics, Humboldt-University Berlin)

We consider the estimation of smooth regression functions under replicated inde-penent observations with homogeneous variance at fixed design points. Especiallyinteresting cases for applications are factorial designs with near means at someneighbouring cells. We consider linear estimators of means or of regression func-tions which depend on a smoothing parameter h, as e. g. kernel estimators, ridgeor penalized least squares. We define adaptive estimators substituting h by a datadependent h and we calculate for increasing replication sizes a 2nd order approx-imation of their risk. The increment of this approximation over the MSE or riskfor an estimator with optimal smoothing parameter h is positive and is shown tobe the same for the different adaptation procedures, as e. g. “classical plug-in”,ordinary and full cross-validation or minimization of an unbiased risk estimator.Under regularity conditions we calculate a 3rd order risk approximation for theadaptive estimators. In the case of ridge or penalized least squares and of kernel


estimators with Gaussian kernel it turns out, that the classical plug-in is in generalbetter than minimizing an unbiased risk estimator in the sense of a smaller 3rdorder risk approximation. The last adaptation procedure is in general better thana cross-validation procedure.

An approach for the separation between stochastic andchaotic time series


Anja M. Busse (LS Computergestützte Statistik, Universität Dortmund)

For the description and the analysis of time series it is useful to initially introduce acoarse classification in order to be able to choose the most appropriate tools for themore detailed analysis, which could e. g. be concerned with a sensible choice of theforecasting window. One possible classification is to discriminate between stochas-tic processes with stationary distributions and chaotic processes, the asymptoticdistributions of wich are called unstable. In the context of forcasting this discrim-ination can e. g. provide information about the predictability of the process. Inthe case of a chaotic time series the prediction accuracy can decrease considerablyalready after only a few time-steps. Such a discrimination between deterministicchaotic and stochastic linear time series can be achieved by analyzing the Lya-punov spectrum or the largest Lyapunov exponent of the time series (this is oftenjust referred to as the Lyapunov exponent). Originally, the Lyapunov exponent wasdefined for non-stochastic, deterministic systems, however it is possible to analyzeit in a stochastic framework. This is the aim of this work. Since the Lyapunovexponent is a discriminatory parameter of theasymptotic distribution, it has to beestimated from agiven time series. In the literature various approaches for suchanestimation have been suggested and will be reviewed. However all these methodsare susceptible to interference. Noisy time series, missing numerical stability orlimited number of data often yield a bad estimator. Consequently the classificationinto a chaotic or non-chaotic system is possibly incorrect. In order to solve thisproblem knowledge about the distribution of the estimator for the Lyapunov expo-nent would be beneficial. It is shown that such an interpretation of the Lyapunovexponent in the context of statistics is only possible if ergodicity is guaranteed andthat in this case the distribution of the Lyapunov exponent estimator can in prin-ciple be calculated. Additionally, examples are provided for misclassifications oftime series due to bad estimators for the Lyapunov exponent.

Regression depth and overlapSec. 6: Data Analysis and Design of Experiments

Andreas Christmann (Universität Dortmund), Peter J. Rousseeuw(Department of Mathematics and Computer Science, Universitaire Instelling

Antwerp)

It is shown that the recent notion of regression depth (Rousseeuw and Struyf (1998),Rousseeuw and Hubert (1999)) can be used as a data-analytic tool to measurethe amount of separation between successes and failures in the binary responseframework. Extending this algorithm allows us to compute the overlap in data setswhich are commonly fitted by logistic regression models. The overlap is the numberof observations that would need to be removed to obtain complete or quasicompleteseparation, i. e. the situation where the logistic regression parameters are no longeridentifiable and the maximum likelihood estimate does not exist. It turns out that


the overlap is often quite small. The method is also useful in other regressionmodels with binary response variables, e. g. in the probit model. Programs and aninterface to S-PLUS are available to compute the regression depth and to measurethe overlap.

References

Albert, A. and Anderson, J.A. (1984). On the existence of maximum likelihoodestimates in logistic regression models. Biometrika, 71, 1-10.Christmann, A. and Rousseeuw, P.J. (1999). Measuring overlap in logistic regres-sion. University of Dortmund, SFB 475, Technical report.Künsch, H.R., Stefanski, L.A. and Carroll, R.J. (1989). Conditionally unbiasedbounded-influence estimation in general regression models, with applications togeneralized linear models. J. Amer. Statist. Assoc., 84, 460-466.Rousseeuw, P.J. and Hubert, M. (1999). Regression Depth. With discussion.J. Amer. Statist. Assoc., 94, 388-433.Rousseeuw, P.J. and Struyf, A. (1998). Computing location depth and regressiondepth in higher dimensions. Statistics and Computing, 8, 193-203.Santner, T.J. and Duffy, D.E. (1986). A note on A. Albert and J.A. Anderson’sconditions for the existence of maximum likelihood estimates in logistic regressionmodels. Biometrika, 73, 755-758.

Slow Convergence to Discrete Stable LawSec. 8: Limit Theorems and Large Deviations

Gerd Christoph (Otto-von-Guericke-Universität Magdeburg)

Discrete stable distributions and their domains of attractions were considered inSteutel and van Harn (1979). If a non-negative integer-valued random variablebelongs to the non-normal domain of attraction of a certain discrete stable lawthen convergence rates are slow in limit theorems.

Limit theorems for sequential order statisticsSec. 8: Limit Theorems and Large Deviations

Erhard Cramer (Fachbereich Mathematik, Universität Oldenburg)

Sequential order statistics can be seen from two different points of view. On theone hand, they serve as a model for so-called sequential k-out-of-n systems. Thetechnical structure of these systems is that of ordinary k-out-of-n systems, but thedistributional assumptions are relaxed: After the breakdown of some componentthe underlying life-time distribution of the components at work may be changed.Effects of a component failure on the remaining elements of the system are modelledby adjusting the life-time distribution after each failure. On the other hand, manywell-known models of ordered random variables, e. g., ordinary order statistics,record values, progressive type II censored order statistics and Pfeifer’s records,can be regarded as sequential order statistics (in the distribution theoretical sense).Hence, results obtained for sequential order statistics in distribution theory and instatistical inference can be interpreted within these particular models.

In this talk, limit theorems for extreme sequential order statistics are established.The limit distributions are classified with respect to the underlying parameters ofthe sequential order statistics. Choosing particular parameter values, well-knownlimit theorems for the particular models result.


Analysis of Multivariate Categorical Response DataSec. 5: Monte-Carlo-Methods, Simulation and Image Processing

Claudia Czado (SCA Zentrum Mathematik, Technische Universität München)

The development of adequate models for multivariate categorical response datawith covariate adjustment has been an active research area in the last years. In thecase, where interest is focused on marginal and association parameters, generalizedestimating equations (GEE) (see for example Lipsitz, Laird and Harrington (1991)and Liang, Zeger and Qaqish (1992)) and likelihood (see for example Fitzmauriceand Laird (1993) and Molenberghs and Lesaffre (1994)) based methods have beenproposed. The number of parameters required for the full specification of thesemodels grows exponentially with the length of the response vector. Therefore,the analysis is often focused on marginal and first order parameters. In this case,the multivariate probit model (Ashford and Sowden (1970)) becomes an attractivealternative to the above models. The application of the multivariate probit modelhas been hampered by the intractability of the maximum likelihood estimator,when the length of the response vector is large. We show that this difficulty canbe overcome by the use of Markov Chain Monte Carlo methods and illustrate themethod on unemployment data arising from panel studies (Czado (1999)).

References

Ashford, J.R. and Sowdon, R.R. (1970) Multivariate probit analysis, Biometrics,26, 535-546.Czado, C. (1999) Multivariate Regression Analysis of Panel Data with Binary Out-comes applied to Unemployment Data, to appear in Statistical Papers.Fitzmaurice, G.M. and Laird, N.M. (1993). A likelihood-based method for analysinglongitudinal binary responses, Biometrika, 80, 1, 141-151.Liang, K.-Y, Zeger,S.L. and Qaqish, B. (1992). Multivariate regression analyses forcategorical data (with discussion). J.R. Statist. Soc. B, 54, 3-40.Lipsitz, S.R., Laird, N.M. and Harrington, D.P. (1991). Generalized estimatingequations for correlated binary data: using the odds ratio as a measure of associa-tion, Biometrika, 78, 153-160.Molenberghs, G. and Lesaffre, E. (1994). Marginal Modeling of Correlated OrdinalData Using a Multivariate Plackett Distribution. J. Amer. Statist. Soc. , 89, No.426, 633-644.

Monotonicity and dependence properties of sojourn andcycle times in closed networks of queues


Hans Daduna (Fachbereich Mathematik, Universität Hamburg), Ryszard Szekli(Mathematical Institute, Wroclaw University)

We consider closed Gordon-Newell networks of exponential queues with cyclic struc-ture (tandem systems). We first prove that the conditional cycle time is stochas-tically increasing if the initial joint queue length vectors of two cycles are orderedincreasingly with respect to the partial sum ordering. It follows that for tandemsin equilibrium the cycle times are stochastically increasing in the population sizeof the network.

These observations are the starting point for the investigation of the correlationstructure of sojourn times and of successive cycle times, e. g.:

The vector of successive sojourn times during a cycle is negatively associated.


The vector of two successive cycle times is negatively associated.The correlation between the n-th and m-th cycle decreases to 0 with |m−n| → ∞.Feedback-sojourn times are positively associated.We discuss generalizations to networks with general topology.

Laplace operators and diffusions in tensor bundles overPoisson spaces

Sec. 14: Stochastic Analysis

Alexei Daletskii (Institut für Angewandte Mathematik, Stochastik, UniversitätBonn)

The talk is based on the joint work of S. Albeverio, A.Daletskii and E. Lytvynov[ADL1], [ADL2]. There is a growing interest in geometry and analysis on Poissonspaces, i. e. on the spaces of locally finite configurations in non-compact manifolds,equipped with Poisson measures. In [AKR1], an approach to these spaces as toinfinite dimensional manifolds was initiated.We develop this point of view. Wedefine spaces of differential forms over Poisson spaces. Further, we define andstudy Laplace operators acting in these spaces. We show in particular that thecorresponding de Rham Laplacian can be expressed in terms of the Dirichlet op-erator on functions on the Poisson space and the Witten Laplacian on the initialmanifold associated with the intensity of the corresponding Poisson measure. Wegive a probabilistic interpretation and investigate some properties of the associatedsemigroups.

References

[ADL1] S. Albeverio, A. Daletskii, E. Lytvynov: Laplace operators and diffusionsin tangent bundles over Poisson spaces, Preprint SFB 256, UniversitätBonn, 1999.

[ADL2] S. Albeverio, A. Daletskii, E. Lytvynov: Laplace operators and diffu-sions in tensor bundles over Poisson spaces, Preprint SFB 256, UniversitätBonn, 1999.

[AKR1] S. Albeverio, Yu. Kondratiev, M. Röckner: Analysis and geometry onconfiguration spaces, J. Func. Anal. 154 (1998), 444–500.

On a generalized Pickands’s constantsSec. 9: Statistics of Extremes and Subexponential Distributions

Krzysztof Dȩbicki (Mathematical Institute, University of Wroc law)

Pickands’s constants play important role in the exact asymptotic of extreme valuesfor Gaussian stochastic processes. By the generalized Pickands’s constant Hη wemean the following limit

Hη = limT→∞

Hη(T )T

,

where Hη(T ) = exp exp(maxt∈[0,T ]

(√2 η(t)− σ2η(t)

)), η(t) is a centered Gaussian

process with stationary increments and variance function σ2η(t).Under some mild conditions on σ2η(t) we prove thatHη is well defined and we give

a comparison criterion for generalized Pickands’s constants. Moreover we prove atheorem where Hη(T ) appear. This theorem extends known in the literature resultof Pickands for some stationary Gaussian processes.


Local maximin property of tests in infinite dimensonalGaussian shift experimentsSec. 7: Model Choice in Statistics

Peter Dencker (Fachbereich Mathematik, Universität Rostock)

Infinite dimensional Gaussian shift experiments appear as limit experiments in non-parametric problems in statistics, e. g. in goodness of fit tests. Optimal tests forthe limit experiment are used to construct tests for finite sample sizes. These testsare by construction asymptotically optimal.

In the case of a finite codimension of the null hypothese H0, the χ2-statisticprovides a Maximin-α-Test for the test of H0 versus the alternative parameter setwith a fixed distance δ > 0 from the null hypotheses (see e. g. Strasser (1985)).

In goodness of fit tests with and without estimated parameters the null hypothe-ses in the limit experiment has infinite codimension and a χ2-type statistic doesnot exist. A way out are weighted quadratic statistics. We choose nonnegativeweights ηl, such that

∑∞l=1 ηl is finite. Our new test statistic is a wheighted sum of

squares T =∑∞l=1 ηlX

2l , where Xl are i. i. d. according to N(0, 1). Such statistics

assign different weights to different directions and are often applied. The naturalquestion ist the possible optimality. To answer this question we use the concept ofthe local maximin property of tests introduced and studied in Giri, Kiefer (1964) forstatistics appearing in multivariate analysis. Using this approach we prove a localmaximin property of the α-test based on T . An essential step is to characterize afamily of sets on which the α-test based on T have in some sense “locally constant”power.

This optimality criteria will be translated into a locally asymtotic optimalityconcept of the orignal problem (as in the classical LAN theory, see e. g. Witting,Müller-Funk (1995)).

This optimality concept motivates the use of quadratic test statistics in nonpara-metric problems, e. g. in goodness of fit problems.

References

Giri, N., Kiefer, J. (1964). Local and asymptotic minimax properties of multi-variate tests. Ann. Math. Stat. 35, 21-35.Strasser, H. (1985). Mathematical Theory of Statistics. de Gruyter, Berlin.Witting,, H. Müller-Funk, U. (1995). Mathematische Statistik II. Teubner,Stuttgart.

The effect of different regulation schemes for simultaneousevents on the performance behaviour of discrete time

queueing systemsSec. 2: Stochastic Methods in Optimization and Operations Research

Bernadette Desert (Hamburg), Hans Daduna (Fachbereich Mathematik,Universität Hamburg)

We consider tandem networks of discrete time Bernoulli servers with state depen-dent service rates and a state dependent Bernoulli arrival stream at the first nodeof the tandem. As the problems are formulated within a discrete time framework,arrivals and departures may occur simultaneously. We present different regulationschemes for these simultaneous events and investigate their effects on customeroriented performance measures. It turns out that the stationary distribution ofa system depends on the arrival regime imposed upon the model. The order in


which the events take place also influences the perspective of a customer arrivingat the first node of the tandem or jumping between two arbitrary nodes inside thenetwork. We therefore focus our attention on the arrival theorems and prove thatunder certain conditions a customer always observes the time stationary behaviourof the system during his jumps; in other words, we obtain an analogue of the well-known PASTA property. In this respect, we have to distinguish between local andglobal control for the regulation of simultaneous jumps. Moreover, the arrival theo-rems enable us to compute performance measures such as the individual customers’waiting and passage time distributions. It turns out that for a particular regulationscheme which is known in the literature, Little’s formula fails to hold.

Optimal designs for rational modelsSec. 6: Data Analysis and Design of Experiments

Holger Dette (Institut für Mathematik, Ruhr-Universität Bochum)

In this talk D-optimal for inverse quadratic polynomial models are presented. Thedesign problem is equivalent to an eigenvalue problem for a Schrödinger equa-tion which is solved analytically and, specifically, the D-optimal designs put equalmasses at the points ±1 and the zeros of a sum of ultraspherical polynomials. Aconjecture extends these analytical results to more general cases.

References

H. Dette, L.Haines, L. Imhof (1999). Optimal designs for rational models andweighted polynomial regression. Annals of Statistics, to appear.

Asymptotic Expansions of the Robbins-Monro ProcessSec. 8: Limit Theorems and Large Deviations

Jürgen Dippon (Mathematisches Institut A, Universität Stuttgart)

To estimate the root x0 of an unknown regression function f : R → R, whosefunction values f(x) at points x can be observed with some random error V only,Robbins and Monro (1951) suggested to run the recursion Xn+1 = Xn − anYn withYn = f(Xn) + Vn. Under regularity assumptions, the normalized Robbins-Monroprocess (Zn), given by Zn := (Xn − x0)/Var(Xn), is asymptotically normal.

In this talk we present Edgeworth expansions which provide approximations ofthe distribution function of Zn up to an error of order o(1/

√n) and o(1/n), for

instance

P

(Xn − x0√Var (Xn)

≤ x

)= Φ(x) +

1√np1(x)φ(x) +

1np2(x)φ(x) + o

(1n

),

where Φ and φ denote the distribution function and the density of N(0, 1)-distri-bution, respectively, p1 and p2 are polynomials.

As corollaries we obtain asymptotic confidence intervals for the unknown pa-rameter x0 whose coverage probability errors are of order O(1/n). Further resultsconcern Cornish-Fisher expansions of the quantile function of Zn, an Edgeworthcorrection of the distribution function and a stochastic expansion in terms of powersof both 1/

√n and a normal random variable.

The proofs use ideas of Helmers, Callaert, Janssen, Veraverbeke, Bickel, Goetzeand van Zwet who investigated Edgeworth expansions for L- and U -statistics.


A Statistical Approach To Case Based Reasoning, WithApplication To Breast Cancer Data


Jürgen Dippon , Michael Kohler (Mathematisches Institut A, UniversitätStuttgart)

Given a large set of problems and their individual solutions case based reasoningseeks to solve a new problem by refering to the solution of that problem which is“most similar” to the new problem. Crucial in case based reasoning is the decision,which problem “most closely” matches a given new problem. In this talk we proposea new method for deciding this question. The basic idea is to define a familyof distance functions and to use these distance functions as parameters of localaveraging regression estimates of the final result. Then one chooses that distancefunction for which the resulting estimate is optimal with respect to a certain errormeasure used in regression estimation. We illustrate our method by simulationsand apply it to breast cancer data.

Grand Canonical Simulations of Hard-Disk Systems bySimulated Tempering

Sec. 5: Monte-Carlo-Methods, Simulation and Image Processing

Gunter Döge (Institut für Stochastik, TU Bergakademie Freiberg)

The simulation of hard core Gibbs point processes in the two-dimensional space isa difficult task for high area fractions of disks, i. e., greater than 0.65. Commonlyused Markov Chain Monte Carlo algorithms (Gibbs sampler, Metropolis Hastings),which work fine for low area fractions, have very low acceptance rates for high areafractions, they are no longer applicable in this case.

The method of Simulated Tempering is shown to be an efficient alternative tothese algorithms. It is based on a modified Metropolis Hastings algorithm whichleads to usable acceptance rates even for higher are fractions between 0.65 and 0.75.This interval is especially important for statistical physics (“phase transition”).Previous simulations in statistical physics were mostly performed with canonicaldisk ensembles, i. e., with a fixed number of disks, for instance by Molecular Dy-namics. Simulated Tempering permits the direct simulation of grand canonicalensembles, i. e., with a variable number of disks but constant chemical potential.

Various spatial characteristics of the hard core process are studied using sim-ulated samples of point patterns, for instance pair correlation function and someorder characteristics.

Point- and Interval Estimators for Nonparametric TreatmentEffects in Designs with Repeated Measures


Sebastian Domhof (Abteilung Medizinische Statistik, Universität Göttingen)

Several groups of independent subjects are repeatedly observed at t time points. Itis neither assumed that the observations are coming from distributions belongingto certain parametric family nor that the distributions are continuous. To de-scribe the outcome of the trial, we use the so-called relative treatment effects in anonparametric marginal model. These effects can be estimated by using differenttypes of rankings. The estimators are asymptotically unbiased and consistent andtheir asymptotic distribution is derived. Moreover, L2-consistent estimators for


the unknown asymptotic variances are given which can also easily be computed byusing different types of rankings. By means of a simulation study, the accuracy ofthe approximation by the asymptotic distribution is investigated. The proceduresare applied to a data set from a clinical trial where ordered categorical data wererepeatedly observed for two groups of patients at six time points.

Extreme Quantile Estimation with Applications to FinancialRisk

Sec. 9: Statistics of Extremes and Subexponential Distributions

Holger Drees (Mathematisches Institut, Universität zu Köln)

Estimators of extreme quantiles that are based on extreme value approximationsare well known. However, in literature almost all results about the asymptoticbehavior of such estimators are restricted to the case of iid observations, wherease. g. consecutive returns on share prices or foreign exchange rates usually exhibit anon-negligible dependence.

Starting from a new approximation of the tail empirical quantile function, weestablish the asymptotic normality of extreme quantile estimators in quite generalnonparametric models of stationary time series. The consequences for estimatorsof the so-called Value at Risk of a financial portfolio are discussed.

Some Properties of Model Selection and Related Methods inRegression Analysis

Sec. 7: Model Choice in Statistics

Bernd Droge (Sonderforschungsbereich 373, Humboldt-Universität zu Berlin)

In practice, there is seldom sure evidence on the validity of a certain model, sothat one has to choose a good one from those being tentatively proposed. Modelselesction belongs therefore to the main tasks of applied statisticians. In a regressionsituation, this problem occurs since some of the observed explanatory variables maynot be related to the response variable. Then it is required to select an appropriatesubset of explanatory variables in accordance with the intended use of the model.For this selection process, many data-driven procedures have been proposed.

In assessing the results of analyzing experimental data we have to take intoaccount that the obtained predictions and estimates may depend heavily on theunderlying model, which in turn has been chosen by data-driven automated meth-ods. Therefore it is hard to establish finite sample properties of model selectionprocedures in general settings, and consequently most results in this field are as-ymptotic in character.

In the literature there are mainly two notions for describing the asymptotic be-haviour of model selection procedures: consistency and asymptotic optimality. Herewe present a short review of some recent results in this field. We discuss the condi-tions required for the optimality of the different methods in various situations. Inparticular, we show the asymptotic optimality of the so-called full cross-validationcriterion, see Droge (1999).

In the special case of orthogonal regressors and normally distributed observa-tions, we study the finite sample properties of model selection procedures within adecision theoretic framework. The focus is here on the minimax-regret optimalityof the procedures. Moreover, we consider related methods as competitors to modelselection. One example is a certain shrinkage approach, which corresponds to thesoft thresholding technique in the context of wavelet based estimation. Employing


the minimax regret principle shows the superiority of the optimal data-dependentshrunk estimator over its selection-type analogue, compare Droge (1998).

References

Droge, B. (1998). Minimax regret analysis of orthogonal series regression estimation:Selection versus shrinkage. Biometrika 85, 631-643.Droge, B. (1999). Asymptotic optimality of full cross-validation for selecting linearregression models. Statistics & Probability Letters 44, 351-357.

P -Values for Discriminant AnalysisSec. 6: Data Analysis and Design of Experiments

Lutz Dümbgen (Mathematical Institute, Med. University at Lübeck)

In this talk I will discuss a new approach to discriminant analysis and classificationprocedures. In the simplest case one observes a random vector X with distributionPθ, where θ is an unknown parameter in {1, 2, . . . , L} while the distributions Pjare specified. The classical (frequentist) approach to discriminant analysis is toestimate θ by θ̂(X), and to estimate misclassification probabilities Pj(θ̂ 6= j). Butthis does not quantify the uncertainty when classifying one particular observation.Thus I propose to replace θ̂(·) with a confidence set for θ. In the basic settingwith known distributions Pj this leads to a standard problem. Things becomemore involved and interesting in case of unknown distributions Pj which have to beestimated from training samples. I will describe possible solutions in a parametricand nonparametric context.

Power comparison of multivariate rank tests for locationalternatives


Rainer Dyckerhoff (Seminar für Wirtschafts- und Sozialstatistik, Universitätzu Köln)

In the univariate Wilcoxon-Mann-Whitney test the linear ranking of points fromsmallest to largest is used to construct a two-sample test for location alternatives.The problem in generalizing this or other rank tests to the multivariate case is theabsence of a natural linear order in d-space. Thus, there is no natural rankingof points. However, in recent years, there has been a lot of work on orderingmultidemsional data. One idea is to order points according to their so-called datadepth. Data depth is a concept that measures the “depth” of a point y in a givencloud of data points x1, x2, . . . , xn in Rd. Any sensible notion of data depthshould be affine-invariant. Many notions of data depth have been proposed, e.g.,the Mahalanobis depth (Mahalanobis 1936), the halfspace depth (Tukey 1975),and the zonoid depth (Koshevoy and Mosler 1997). Ordering points with respectto depth yields an ordering where points far from the center of the data cloud havelow ranks and points near to the center have high ranks.

In the univariate Wilcoxon test ranks are assigned with respect to their positionin the pooled sample. If one uses the same idea in the multivariate case, i. e., ifranks are assigned according to the depth in the pooled sample, the resulting ranksum test is able to detect changes in scale but not in location. To overcome thisproblem and develop a rank test that is able to detect shifts in location, a differentscheme to assign the ranks has to be used. Liu and Singh (1993) have proposedsuch a test based on the notion of data depth. For each point of the second sample


they compute the rank according to its depth in the first sample. If the sum ofthese ranks is low, this means that most of the points in the second sample are farfrom the center of the first sample, which indicates a shift in location.

We simulate the power of these rank tests for the Mahalanobis depth, the half-space depth and the zonoid depth on different alternatives. Further, the powerfunctions are compared with the power function of a nonparametric rank test, pro-posed by Puri and Sen (1971), which uses univariate ranks of the components, anda parametric test for the normal distribution, the two-sample Hotelling T 2 test.

Spectral gaps on loop spaces: A counterexampleSec. 14: Stochastic Analysis

Andreas Eberle (Fakultät für Mathematik, Universität Bielefeld)

It is shown that the Ornstein-Uhlenbeck type operators on a certain class of loopspaces over compact simply connected Riemannian manifolds do not have a spectralgap above 0. This is contrary to the situation on path spaces where a spectral gaphas been derived from a non-flat version of the Clark-Ocone formula by S. Fang.

Exact asymptotic risk in a density reconstruction problemSec. 3: Asymptotic Statistics, Nonparametrics and Resampling

Werner Ehm (Institut für Grenzgebiete der Psychologie Freiburg)

For a statistical inverse problem related to positron emission tomo-graphy, asymp-totically exact estimates are obtained for the mean square error of kernel estimatorsof the filtered backprojection type. In contrast to the density estimation problemwith “direct” observations, the approximation of the variance term contains a factordepending on the unknown object. That factor has a geometric significance. Alsoaddressed is the question of an asymptotically o

abstracts of hamburger stochastik-tage 2000...6 plenary lecture functional limit laws in statistics...

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