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Dietmar Ferger

Publications and source records attributed to Dietmar Ferger.

11 recordsLinked to original sources
arXiv2026

Tests for the mean of high-dimensional data

We consider the problem of testing the mean of high-dimensional data when the dimension may grow without explicit rate restrictions relative to the sample size. The proposed procedure is based on the statistic V_n = n||Xn||^2, which avoids inversion of the covariance matrix and is therefore suitable for high-dimensional settings.We establish asymptotic distributional results for both fixed and increasing dimension by embedding the observations into the Hilbert space l2. Furthermore, we prove the asymptotic validity of a bootstrap approximation for the distribution of the test statistic. The resulting bootstrap test yields asymptotic level-a procedures without requiring sparsity assumptions or structural conditions on the covariance matrix. In all this, a new Central Limit Theorem in l2 is proving to be an extremely useful tool.

math.ST↗
arXiv2025

Extended Argmin-Theorems for multiple nets of multivariate càdlàg stochastic processes

Consider finitely many nets of multivariate càdlàg stochastic processes. We show that the vectors consisting of the respective minimizing points converge in distribution to a random closed set. This set is given as a cartesian product with factors which are equal to the set of all minimizing points of stochastic processes occurring as functional limits of the respective nets. If these limit processes have almost surely exactly one minimizer, then the vectors converge classically in distribution to the vector of these minimizers.

math.PR↗
arXiv2025

Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions

We derive necessary and sufficient conditions for epi-convergence in distribution of normal integrands. As a basic tool for the proof a new characterisation for distributional convergence of random closed sets is used. Our approach via the epi-topology allows us to show that, if a net of normal integrands epiconverges in distribution, then the pertaining sets of epsilon-optimal solutions converge in distribution in the underlying hyperspace endowed with the upper-Fell topology. Under some boundedness and uniquenss assumptions the convergence even holds for the Fell topology. Finally, measurable selections converge weakly to a Choquet-capacity.

math.PR↗
arXiv2025

Minimizers of U-processes and their domains of attraction

In this paper, we study the minimizers of U-processes and their domains of attraction. U-processes arise in various statistical contexts, particularly in M-estimation, where estimators are defined as minimizers of certain objective functions. Our main results establish necessary and sufficient conditions for the distributional convergence of these minimizers, identifying a broad class of normalizing sequences that go beyond the standard square-root asymptotics with normal limits. We show that the limit distribution belongs to exactly one of the four classes introduced by Smirnov. These results do not only extend Smirnov's theory but also generalize existing asymptotic theories for M-estimators, including classical results by Huber and extensions to higher-degree U-statistics. Furthermore, we analyze the domain of attraction for each class, providing alternative characterizations that determine which types of statistical estimators fall into a given asymptotic regime.

math.ST↗
arXiv2024

Weak convergence of probability measures on hyperspaces with the upper Fell-topology

Let E be a locally compact second countable Hausdorff space and F the pertaining family of all closed sets. We endow F respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak convergence of probability measures on the corresponding hyperspaces with a focus on the upper Fell topology. The results can be transferred to distributional convergence of random closed sets in E with applications to the asymptotic behavior of measurable selection.

math.PR↗
arXiv2023

On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics

We prove that the smallest minimizer s(f) of a real convex function f is less than or equal to a real point x if and only if the right derivative of f at x is non-negative. Similarly, the largest minimizer t(f) is greater or equal to x if and only if the left derivative of f at x is non-positive. From this simple result we deduce measurability and semi-continuity of the functionals s and t. Furthermore, if f has a unique minimizing point, so that s(f) = t(f), then the functional is continuous at f. With these analytical preparations we can apply Continuous Mapping Theorems to obtain several Argmin theorems for convex stochastic processes. The novelty here are statements about classical distributional convergence and almost sure convergence, if the limit process does not have a unique minimum point. This is possible by replacing the natural topology on R with the order topologies. Another new feature is that not only sequences but more generally nets of convex stochastic processes are allowed.

math.PR↗
arXiv2022

Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process

The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function $F$ of i.i.d. real random variables is either equal to a given distribution function $F_0$ (hypothesis) or $F \ge F_0$ (one-sided alternative). Since \cite{r5} it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-$α$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $α$ even for sample sizes beyond $10.000$. Now, the standardization consists of the weight-function $1/\sqrt{F_0(x)(1-F_0(x))}$. Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to \cite{r20}. Our methodology also works for the two-sided alternative $F \neq F_0$.

math.ST↗
arXiv2015

Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

We prove the asymptotic independence of the empirical process $α_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $β_n = n (F_n(τ+\frac{\cdot}{n})-F_n(τ))$, where $F$ is an arbitrary cdf, differentiable at some point $τ$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $α_n$ and $β_n$. Precisely, we show that the pair $(α_n,β_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(α_n,β_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $β_n$ to converge and can be further weakened.

math.PR↗