arXiv subjects

Philip Dörr

Publications and source records attributed to Philip Dörr.

7 recordsLinked to original sources

arXiv2026

Joint Asymptotic Normality and Extremes of Generalized Inversions and Descents

Generalized inversions $X_\mathrm{inv}^{(d)}$ and generalized descents $X_\mathrm{des}^{(d)}$ are an interesting combinatorial extension of the common inversion and descent statistics on permutation groups. By means of the root poset, they can be defined on all classical Weyl groups (i.e., symmetric groups, signed and even-signed permutation groups). In this paper, we investigate the bivariate normality of $(X_\mathrm{inv}^{(d_1)}, X_\mathrm{des}^{(d_2)})^\top$ for different regimes $d_1, d_2$ that vary with the group size. Depending on the regimes, we classify the existence of the asymptotic correlation and provide explicit formulas. We also draw conclusions on the extreme value behavior of $X_\mathrm{inv}^{(d_1)}$, $X_\mathrm{des}^{(d_2)},$ and $(X_\mathrm{inv}^{(d_1)}, X_\mathrm{des}^{(d_2)})^\top$ in suitable triangular arrays, discussing bounds on the number of i.i.d.\ samples $k_n$ from a group of size $n$ so that the extreme values of these statistics are attracted to the Gumbel distribution.

math.CO↗

arXiv2026

Testing for correct model specification in copula regression models

We propose a goodness-of-fit test for semiparametric copula regression models. Such models express the regression function in terms of marginal distribution functions and copula densities and therefore provide a flexible way to avoid fully nonparametric estimation in high-dimensional regression problems. Their performance, however, depends crucially on the specification of the parametric copula family. Instead of testing the copula model itself, we assess misspecification directly at the level of the induced regression function. To this end, we introduce a weighted $L^2$-distance between the true regression function and its best approximation within the postulated copula regression model. A kernel-based estimator of this distance is proposed and shown to be consistent and asymptotically normal under both the null hypothesis of correct specification and fixed alternatives. We derive a classical specification test and, using a self-normalized sequential statistic, construct pivotal confidence intervals and tests for relevant deviations from the model. Finite-sample simulations demonstrate accurate level approximation and good power properties of the proposed procedures.

math.ST↗

arXiv2025

Extreme Values of Permutation Statistics

We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements of finite Coxeter groups. To this end, we construct a triangular array of either distribution from a sequence of Coxeter groups with increasing ranks. To avoid degeneracy of extreme values, the number of i.i.d. samples $k_n$ in each row must be asymptotically bounded. We employ large deviations theory to prove the Gumbel attraction of Mahonian and Eulerian distributions. It is shown that for the two classes, different bounds on $k_n$ ensure this.

math.CO↗

arXiv2024

Joint extremes of inversions and descents of random permutations

We provide asymptotic theory for the joint distribution of $X_{\mathrm{inv}}$ and $X_{\mathrm{des}}$, the numbers of inversions and descents of random permutations. Recently, Dörr & Kahle (2022) proved that $X_{\mathrm{inv}}$, respectively, $X_{\mathrm{des}}$ is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that $(X_{\mathrm{inv}}, X_{\mathrm{des}})$ is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $(X_{\mathrm{inv}}, X_{\mathrm{des}})$ and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.

math.PR↗

arXiv2019

A new test of multivariate normality by a double estimation in a characterizing PDE

This paper deals with testing for nondegenerate normality of a $d$-variate random vector $X$ based on a random sample $X_1,\ldots,X_n$ of $X$. The rationale of the test is that the characteristic function $ψ(t) = \exp(-\|t\|^2/2)$ of the standard normal distribution in $\mathbb{R}^d$ is the only solution of the partial differential equation $Δf(t) = (\|t\|^2-d)f(t)$, $t \in \mathbb{R}^d$, subject to the condition $f(0) = 1$. By contrast with a recent approach that bases a test for multivariate normality on the difference $Δψ_n(t)-(\|t\|^2-d)ψ(t)$, where $ψ_n(t)$ is the empirical characteristic function of suitably scaled residuals of $X_1,\ldots,X_n$, we consider a weighted $L^2$-statistic that employs $Δψ_n(t)-(\|t\|^2-d)ψ_n(t)$. We derive asymptotic properties of the test under the null hypothesis and alternatives. The test is affine invariant and consistent against general alternatives, and it exhibits high power when compared with prominent competitors.

math.ST↗

arXiv2019

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrödinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure.

stat.ME↗

arXiv2018

Induced and Weak Induced Arboricities

We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as the smallest $k$ such that the edges of $G$ can be covered with $k$ induced forests in $G$. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class $\mathcal{F}$ of graphs and a graph parameter $p$, let $p(\mathcal{F}) = \sup\{p(G) \mid G\in \mathcal{F}\}$. We show that ${\rm ia}(\mathcal{F})$ is bounded from above by an absolute constant depending only on $\mathcal{F}$, that is ${\rm ia}(\mathcal{F})\neq\infty$ if and only if $χ(\mathcal{F} \nabla \frac{1}{2}) \neq\infty$, where $\mathcal{F} \nabla \frac{1}{2}$ is the class of $\frac{1}{2}$-shallow minors of graphs from $\mathcal{F}$ and $χ$ is the chromatic number. Further, we give bounds on ${\rm ia}(\mathcal{F})$ when $\mathcal{F}$ is the class of planar graphs, the class of $d$-degenerate graphs, or the class of graphs having tree-width at most $d$. Specifically, we show that if $\mathcal{F}$ is the class of planar graphs, then $8 \leq {\rm ia}(\mathcal{F}) \leq 10$. In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.

math.CO↗