arXiv subjects

Jürgen Kampf

Publications and source records attributed to Jürgen Kampf.

7 recordsLinked to original sources

arXiv2022

An epidemic model in inhomogeneous environment

The current work deals with an epidemic model on the complete graph K_n on n vertices in a non-homogeneous setting, where the vertices may have distinct types. Different types differ in the probability of getting infected, and/or in the capacity of infecting other vertices. This generalizes previous models where vertices are all of the same type and have equal probabilities of being infected. We prove laws of large numbers and central limit theorems for the the total duration of the process and for the number of infected vertices, respectively, when n goes to infinity. By coupling the epidemic model with a Poisson process, we also obtain continuous-time counterparts of the above-mentioned limit results. Moreover, we also prove that when all individuals have the same spread capacity, then a population with inhomogeneous susceptibility is less affected by the epidemics than a homogeneous population.

math.PR↗

arXiv2020

Testing for linearity in boundary regression models with application to maximal life expectancies

We consider a regression model with errors that are a.s. negative. Thus the regression function is not the expected value of the observations but the right endpoint of their support. We develop two goodness-of-fit tests for the hypotheses that the regression function is an affine function, study the asymptotic distributions of the test statistics in order to approximately fix the sizes of the tests, derive their finite-sample properties based on simulations and apply them to life expectancy data.

math.ST↗

arXiv2019

Nonparametric estimation of the kernel function of symmetric stable moving average random functions

We estimate the kernel function of a symmetric alpha stable ($SαS$) moving average random function which is observed on a regular grid of points. The proposed estimator relies on the empirical normalized (smoothed) periodogram. It is shown to be weakly consistent for positive definite kernel functions, when the grid mesh size tends to zero and at the same time the observation horizon tends to infinity (high frequency observations). A simulation study shows that the estimator performs well at finite sample sizes, when the integrator measure of the moving average random function is $SαS$ and for some other infinitely divisible integrators.

math.ST↗

arXiv2019

Variances of surface area estimators based on pixel configuration counts

The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting -- we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in $O(t^2)$ as the lattice distance $t$ tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically derived convergence order is optimal in general, but further improvements are possible in special cases.

math.ST↗

arXiv2016

A central limit theorem for Lebesgue integrals of random fields

In this paper we show a central limit theorem for Lebesgue integrals of stationary $BL(θ)$-dependent random fields as the integration domain grows in Van Hove-sense. Our method is to use the (known) analogue result for discrete sums. As applications we obtain various multivariate versions of this central limit theorem.

math.PR↗

arXiv2015

A functional central limit theorem for integrals of stationary mixing random fields

We prove a functional central limit theorem for integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbb{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the function $f$, as the integration domain $W$ grows in Van Hove-sense. We discuss properties of the covariance function of the asymptotic Gaussian process.

math.PR↗

arXiv2010

On the convex hull of symmetric stable processes

Let alpha \in (1, 2] and X be an R^d-valued alpha-stable process with independent and symmetric components starting in 0. We consider the closure S_t of the path described by X on the interval [0, t] and its convex hull Z_t. The first result of this paper provides a formula for certain mean mixed volumes of Z_t and in particular for the expected first intrinsic volume of Z_t. The second result deals with the asymptotics of the expected volume of the stable sausage Z_t+B (where B is an arbitrary convex body with interior points) as t \to 0.

math.PR↗