arXiv subjects

Henrik Kaiser

Publications and source records attributed to Henrik Kaiser.

4 recordsLinked to original sources

arXiv2026

Generalizing Laplace's method by means of Mellin transforms

A well-known procedure for the asymptotic evaluation of Laplace transforms is Laplace's method. Despite its wide applicability, however, it is easy to find relevant examples where the technique is infeasible, because the integrand admits no power series expansion near the critical point, e.g., due to exponential behaviour there. We circumvent this issue through an extension of the method of Mellin transforms, based on an integral representation for the kernel and a generalization of the Taylor expansion. Our main result then extends the Laplacian method in the sense that it provides asymptotic expansions for a wider scope of Laplace transforms in terms of known special functions, rather than only in powers of the asymptotic parameter. The results are illustrated in two examples.

math.CV↗

arXiv2025

Deconvolution of distribution functions without integral transforms

We study the recovery of the distribution function $F_X$ of a random variable $X$ that is subject to an independent additive random error $\varepsilon$. To be precise, it is assumed that the target variable $X$ is available only in the form of a blurred surrogate $Y = X + \varepsilon$. The distribution function $F_Y$ then corresponds to the convolution of $F_X$ and $F_\varepsilon$, so that the reconstruction of $F_X$ is some kind of deconvolution problem. Those have a long history in mathematics and various approaches have been proposed in the past. Most of them use integral transforms or matrix algorithms. The present article avoids these tools and is entirely confined to the domain of distribution functions. Our main idea relies on a transformation of a first kind to a second kind integral equation. Thereof, starting with a right-lateral discrete target and error variable, a representation for $F_X$ in terms of available quantities is obtained, which facilitates the unbiased estimation through a $Y$-sample. It turns out that these results even extend to cases in which $X$ is not discrete. Finally, in a general setup, our approach gives rise to an approximation for $F_X$ as a certain Neumann sum. The properties of this sum are briefly examined theoretically and visually. The paper is concluded with a short discussion of operator theoretical aspects and an outlook on further research. Various plots underline our results and illustrate the capabilities of our functions with regard to estimation.

math.ST↗

arXiv2025

Deconvolution of arbitrary distribution functions and densities

In this article we propose a novel approach for the deconvolution of the distribution function associated with an arbitrary probability measure (and possibly existing density). We first show that the initial convolution equation always can be transformed to a convolution equation that involves a symmetric distribution function, whose characteristic function has its values in the unit interval. As a consequence, the characteristic function of the target measure turns out as the limit of a geometric series. By truncation of this series, approximations for distribution function and density are established. The convergence properties of these approximations are examined in detail across diverse setups.

math.ST↗

arXiv2025

Mean and quantile regression in the copula setting: properties, sharp bounds and a note on estimation

Driven by the interest on how uniformity of marginal distributions propa\-gates to properties of regression functions, in this contribution we tackle the following questions: Given a $(d-1)$-dimensional random vector $\textbf{X}$ and a random variable $Y$ such that all univariate marginals of $(\textbf{X},Y)$ are uniformly distributed on $[0,1]$, how large can the average absolute deviation of the mean and the quantile regression function of $Y$ given $\textbf{X}$ from the value $\frac{1}{2}$ be, and how much mass may sets with large deviation have? We answer these questions by deriving sharp inequalities, both in the mean as well as in the quantile setting, and sketch some cautionary consequences to nowadays quite popular pair copula constructions involving the so-called simplifying assumption. Rounding off our results, working with the so-called empirical checkerboard estimator in the bivariate setting, we show strong consistency for both regression types and illustrate the speed of convergence in terms of a simulation study.

math.ST↗