arXiv subjects

Christian H. Weiß

Publications and source records attributed to Christian H. Weiß.

18 recordsLinked to original sources

arXiv2026

Novel Stein-type Characterizations of Bivariate Count Distributions with Applications

The derivation and application of Stein identities have received considerable research interest in recent years, especially for continuous or discrete-univariate distributions. In this paper, we complement the existing literature by deriving and investigating Stein-type characterizations for the three most common types of bivariate count distributions, namely the bivariate Poisson, binomial, and negative-binomial distribution. Then, we demonstrate the practical relevance of these novel Stein identities by a couple of applications, namely the deduction of sophisticated moment expressions, of flexible goodness-of-fit tests, and of novel tests for the symmetry of bivariate count distributions. The paper concludes with an analysis of real-world data examples.

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arXiv2026

INARMA Models for Count Random Fields -- a Survey

The thinning-based integer-valued autoregressive moving-average (INARMA) models are popular for count time series. Recently, types of INARMA models have also been developed for count random fields, i.e., for spatial count data located on a regular two-dimensional grid. This article provides a comprehensive survey on existing INARMA random fields, covering approaches with different thinning operators, first- and higher-order models, as well as unilateral and multilateral model structures.

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arXiv2026

The Integer-valued Moving-Average Random Field

An integer-valued moving average (INMA) model for count random fields is proposed and investigated. Closed-form expressions are derived for both its marginal distribution and spatial dependence structure, for arbitrary model order and also covering the multilateral case. In particular, general expressions for bivariate distributions and autocovariances are provided. It is shown that the INMA random field can be equipped (among others) with a Poisson marginal distribution. It is also demonstrated that different and well-interpretable dependence structures are possible. For illustration, we discuss a real-world data example and propose an INMA approximation to a given spatial dependence structure.

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arXiv2026

A Class of Higher-Order INAR Random Fields for Poisson Counts and Beyond

Existing integer-valued autoregressive (INAR) models for count random fields suffer from difficulties in characterizing the stationary marginal distribution and in computing conditional probabilities (as required for likelihood inference). To overcome these drawbacks, the novel class of combined INAR (CINAR) models is proposed, which both exhibits the classical autoregressive dependence structure and allows to specify the marginal distribution within the wide class of discrete self-decomposable distributions. In particular, CINAR random fields can be equipped with a Poisson or negative-binomial marginal distribution. The CINAR's key stochastic properties are derived (including a simple expression for conditional probabilities), and special cases as well as possible extensions are discussed. Approaches for parameter estimation are developed and investigated, and the practical relevance of the novel CINAR family is demonstrated by an agricultural data application.

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arXiv2026

Using Transcripts for Nonparametric Monitoring of Serial Dependence

Control charts for process monitoring are widely used in practice. Most control charts require the monitored (residuals) process to be serially independent (and to satisfy specified distributional assumptions), whereas undetected dependence (or violations of distributional assumptions) may severely affect the charts' performances. Therefore, (distribution-free) control charts for monitoring serial dependence are of utmost relevance for practice. Recently, various nonparametric control charts have been proposed for this purpose, which are based on ordinal patterns, and which showed an appealing performance in detecting different types of serial dependence. In this research, we further progress in this direction and develop novel nonparametric control charts being based on transcripts and algebraic distances (as derived from ordinal patterns). The performance of the newly proposed control charts is evaluated in a simulation study, and their application in practice is illustrated with a real-world data example from chemical industry.

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arXiv2026

Transcripts and Algebraic Distances in Time Series: Stochastic Properties and Nonparametric Dependence Tests

The use of ordinal patterns (OPs) for analyzing the dependence structure of univariate and continuously distributed processes has gained popularity in recent years. This research goes one step further and considers the transcripts being computed from successive OPs in the time series. Transcripts constitute a kind of ``difference'' between successive OPs and thus naturally relate to two algebraic distances between OPs, the Cayley and Kendall edit distances. The original time series is transformed into a sequence of transcripts or distances, respectively, and important stochastic properties thereof are derived. It is shown that these properties differ substantially among different types of original processes. This motivates the development of various statistics based on transcripts and edit distances in order to investigate the dependence structure of the original process. In particular, the asymptotic distribution of these statistics under the null hypothesis of serial independence is derived, which is then used to implement nonparametric tests for serial dependence. A simulation study shows that these novel dependence tests have appealing power properties, often outperforming former OP-based dependence tests. A concluding real-world data example illustrates the application and interpretation of the proposed approaches in practice.

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arXiv2026

Asymptotic Inference for Rank Correlations

Kendall's tau and Spearman's rho are widely used tools for measuring dependence. Surprisingly, when it comes to asymptotic inference for these rank correlations, some fundamental results and methods have not yet been developed, in particular for discrete random variables and in the time series case, and concerning variance estimation in general. Consequently, asymptotic confidence intervals are not available. We provide a comprehensive treatment of asymptotic inference for classical rank correlations, including Kendall's tau, Spearman's rho, Goodman-Kruskal's gamma, Kendall's tau-b, and grade correlation. We derive asymptotic distributions for both iid and time series data, resorting to asymptotic results for U-statistics, and introduce consistent variance estimators. This enables the construction of confidence intervals and tests, generalizes classical results for continuous random variables and leads to corrected versions of widely used tests of independence. We analyze the finite-sample performance of our variance estimators, confidence intervals, and tests in simulations and illustrate their use in case studies.

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arXiv2025

Nonparametric Testing of Spatial Dependence in 2D and 3D Random Fields

We propose a flexible and robust nonparametric framework for testing spatial dependence in two- and three-dimensional random fields. Our approach involves converting spatial data into one-dimensional time series using space-filling Hilbert curves. We then apply ordinal pattern-based tests for serial dependence to this series. Because Hilbert curves preserve spatial locality, spatial dependence in the original field manifests as serial dependence in the transformed sequence. The approach is easy to implement, accommodates arbitrary grid sizes through generalized Hilbert (``gilbert'') curves, and naturally extends beyond three dimensions. This provides a practical and general alternative to existing methods based on spatial ordinal patterns, which are typically limited to two-dimensional settings.

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arXiv2025

Nonparametric Monitoring of Spatial Dependence

In process monitoring, it is common for measurements to be taken regularly or randomly from different spatial locations in two or three dimensions. While there are nonparametric methods for process monitoring with such spatial data to detect changes in the mean, there is a gap in the literature for nonparametric control charting methods developed to monitor spatial dependence. This study considers streams of regular, rectangular data sets using spatial ordinal patterns (SOPs) as a nonparametric method to detect spatial dependencies. We propose novel SOP control charts, which are distribution-free and do not require prior Phase-I analysis. To uncover higher-order dependencies, we develop a new class of statistics that combines SOPs with the Box-Pierce approach. An extensive simulation study demonstrates the superiority and effectiveness of our proposed charts over traditional parametric approaches, particularly when the spatial dependence is nonlinear or bilateral or when the spatial data contains outliers. The proposed SOP control charts are illustrated using real-world datasets to detect (i) heavy rainfall in Germany, (ii) war-related fires in (eastern) Ukraine, and (iii) manufacturing defects in textile production. This wide range of applications and findings demonstrates the broad utility of the proposed nonparametric control charts. In addition, all methods in this study are provided as a publicly available \texttt{Julia} package on \href{https://github.com/AdaemmerP/OrdinalPatterns.jl}{GitHub} for further implementations.

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arXiv2024

The Mollified (Discrete) Uniform Distribution and its Applications

The mollified uniform distribution is rediscovered, which constitutes a ``soft'' version of the continuous uniform distribution. Important stochastic properties are presented and used to demonstrate potential fields of applications. For example, it constitutes a model covering platykurtic, mesokurtic and leptokurtic shapes. Its cumulative distribution function may also serve as the soft-clipping response function for defining generalized linear models with approximately linear dependence. Furthermore, it might be considered for teaching, as an appealing example for the convolution of random variables. Finally, a discrete type of mollified uniform distribution is briefly discussed as well.

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arXiv2024

Semi-parametric goodness-of-fit testing for INAR models

Among the various models designed for dependent count data, integer-valued autoregressive (INAR) processes enjoy great popularity. Typically, statistical inference for INAR models uses asymptotic theory that relies on rather stringent (parametric) assumptions on the innovations such as Poisson or negative binomial distributions. In this paper, we present a novel semi-parametric goodness-of-fit test tailored for the INAR model class. Relying on the INAR-specific shape of the joint probability generating function, our approach allows for model validation of INAR models without specifying the (family of the) innovation distribution. We derive the limiting null distribution of our proposed test statistic, prove consistency under fixed alternatives and discuss its asymptotic behavior under local alternatives. By manifold Monte Carlo simulations, we illustrate the overall good performance of our testing procedure in terms of power and size properties. In particular, it turns out that the power can be considerably improved by using higher-order test statistics. We conclude the article with the application on three real-world economic data sets.

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arXiv2024

Tobit models for count time series

Several models for count time series have been developed during the last decades, often inspired by traditional autoregressive moving average (ARMA) models for real-valued time series, including integer-valued ARMA (INARMA) and integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models. Both INARMA and INGARCH models exhibit an ARMA-like autocorrelation function (ACF). To achieve negative ACF values within the class of INGARCH models, log and softplus link functions are suggested in the literature, where the softplus approach leads to conditional linearity in good approximation. However, the softplus approach is limited to the INGARCH family for unbounded counts, i.e. it can neither be used for bounded counts, nor for count processes from the INARMA family. In this paper, we present an alternative solution, named the Tobit approach, for achieving approximate linearity together with negative ACF values, which is more generally applicable than the softplus approach. A Skellam--Tobit INGARCH model for unbounded counts is studied in detail, including stationarity, approximate computation of moments, maximum likelihood and censored least absolute deviations estimation for unknown parameters and corresponding simulations. Extensions of the Tobit approach to other situations are also discussed, including underlying discrete distributions, INAR models, and bounded counts. Three real-data examples are considered to illustrate the usefulness of the new approach.

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arXiv2024

Mean-preserving rounding integer-valued ARMA models

In the past four decades, research on count time series has made significant progress, but research on $\mathbb{Z}$-valued time series is relatively rare. Existing $\mathbb{Z}$-valued models are mainly of autoregressive structure, where the use of the rounding operator is very natural. Because of the discontinuity of the rounding operator, the formulation of the corresponding model identifiability conditions and the computation of parameter estimators need special attention. It is also difficult to derive closed-form formulae for crucial stochastic properties. We rediscover a stochastic rounding operator, referred to as mean-preserving rounding, which overcomes the above drawbacks. Then, a novel class of $\mathbb{Z}$-valued ARMA models based on the new operator is proposed, and the existence of stationary solutions of the models is established. Stochastic properties including closed-form formulae for (conditional) moments, autocorrelation function, and conditional distributions are obtained. The advantages of our novel model class compared to existing ones are demonstrated. In particular, our model construction avoids identifiability issues such that maximum likelihood estimation is possible. A simulation study is provided, and the appealing performance of the new models is shown by several real-world data sets.

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arXiv2024

Stein EWMA Control Charts for Count Processes

The monitoring of serially independent or autocorrelated count processes is considered, having a Poisson or (negative) binomial marginal distribution under in-control conditions. Utilizing the corresponding Stein identities, exponentially weighted moving-average (EWMA) control charts are constructed, which can be flexibly adapted to uncover zero inflation, over- or underdispersion. The proposed Stein EWMA charts' performance is investigated by simulations, and their usefulness is demonstrated by a real-world data example from health surveillance.

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arXiv2023

Generalized Moment Estimators based on Stein Identities

For parameter estimation of continuous and discrete distributions, we propose a generalization of the method of moments (MM), where Stein identities are utilized for improved estimation performance. The construction of these Stein-type MM-estimators makes use of a weight function as implied by an appropriate form of the Stein identity. Our general approach as well as potential benefits thereof are first illustrated by the simple example of the exponential distribution. Afterward, we investigate the more sophisticated two-parameter inverse Gaussian distribution and the two-parameter negative-binomial distribution in great detail, together with illustrative real-world data examples. Given an appropriate choice of the respective weight functions, their Stein-MM estimators, which are defined by simple closed-form formulas and allow for closed-form asymptotic computations, exhibit a better performance regarding bias and mean squared error than competing estimators.

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arXiv2023

Conditional-mean Multiplicative Operator Models for Count Time Series

Multiplicative error models (MEMs) are commonly used for real-valued time series, but they cannot be applied to discrete-valued count time series as the involved multiplication would not preserve the integer nature of the data. Thus, the concept of a multiplicative operator for counts is proposed (as well as several specific instances thereof), which are then used to develop a kind of MEMs for count time series (CMEMs). If equipped with a linear conditional mean, the resulting CMEMs are closely related to the class of so-called integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models and might be used as a semi-parametric extension thereof. Important stochastic properties of different types of INGARCH-CMEM as well as relevant estimation approaches are derived, namely types of quasi-maximum likelihood and weighted least squares estimation. The performance and application are demonstrated with simulations as well as with two real-world data examples.

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arXiv2023

Control Charts for Poisson Counts based on the Stein-Chen Identity

If monitoring Poisson count data for a possible mean shift (while the Poisson distribution is preserved), then the ordinary Poisson exponentially weighted moving-average (EWMA) control chart proved to be a good solution. In practice, however, mean shifts might occur in combination with further changes in the distribution family. Or due to a misspecification during Phase-I analysis, the Poisson assumption might not be appropriate at all. In such cases, the ordinary EWMA chart might not perform satisfactorily. Therefore, two novel classes of generalized EWMA charts are proposed, which utilize the so-called Stein-Chen identity and are thus sensitive to further distributional changes than just sole mean shifts. Their average run length (ARL) performance is investigated with simulations, where it becomes clear that especially the class of so-called "ABC-EWMA charts" shows an appealing ARL performance. The practical application of the novel Stein-Chen EWMA charts is illustrated with an application to count data from semiconductor manufacturing.

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arXiv2019

Penalized estimation of flexible hidden Markov models for time series of counts

Hidden Markov models are versatile tools for modeling sequential observations, where it is assumed that a hidden state process selects which of finitely many distributions generates any given observation. Specifically for time series of counts, the Poisson family often provides a natural choice for the state-dependent distributions, though more flexible distributions such as the negative binomial or distributions with a bounded range can also be used. However, in practice, choosing an adequate class of (parametric) distributions is often anything but straightforward, and an inadequate choice can have severe negative consequences on the model's predictive performance, on state classification, and generally on inference related to the system considered. To address this issue, we propose an effectively nonparametric approach to fitting hidden Markov models to time series of counts, where the state-dependent distributions are estimated in a completely data-driven way without the need to select a distributional family. To avoid overfitting, we add a roughness penalty based on higher-order differences between adjacent count probabilities to the likelihood, which is demonstrated to produce smooth probability mass functions of the state-dependent distributions. The feasibility of the suggested approach is assessed in a simulation experiment, and illustrated in two real-data applications, where we model the distribution of i) major earthquake counts and ii) acceleration counts of an oceanic whitetip shark (Carcharhinus longimanus) over time.

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