Weiß, Christian H.

The non-parametric testing of spatial dependence is considered, where the data are generated in a regular two-dimensional grid. Recently, spatial ordinal patterns (SOPs) and corresponding types (which imply a three-part partition of the set of all SOPs) have been proposed for this purpose, where the test statistics are linear expressions of the three type frequencies. In order to use more information from the original data while keeping the tests non-parametric, three versions of refined types are proposed that always lead to six classes of SOPs, namely rotation types, direction types, and diagonal types. In this context, we also present a novel visual representation of SOPs that allows for an intuitive understanding of their characteristics. Furthermore, to incorporate the full frequency distribution of types and refined types, our novel tests for spatial dependence rely on entropy-like statistics instead of the existing linear statistics. Their asymptotic distributions under the null of spatial independence are derived, and the finite-sample performance is analyzed within an extensive simulation study covering various unilateral and bilateral spatial processes. It is shown that the novel entropy tests have appealing power properties and help to recognize how spatial dependence propagates across the data. To illustrate possible applications in practice, two real-world data examples are analyzed, namely one from agricultural science and another one about population changes in a region of Finland.

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. In supplementary material, we provide an application to three real-world economic data sets.

The linear autoregressive models are among the most popular models in the practice of time series analysis, which constitutes an incentive to adapt them to ordinal time series as well. Our starting point for modeling ordinal time series data is the latent variable approach to define a generalized linear model. This method, however, typically leads to a non-linear relationship between the past observations and the current conditional cumulative distribution function (cdf). To overcome this problem, we use the soft-clipping link to obtain an approximately linear model structure and propose a wide and flexible class of soft-clipping autoregressive (scAR) models. The constraints imposed on the model parameters allow us to identify relevant special cases of the scAR model family. We study the calculation of transition probabilities as well as approximate formulae for the CDF. Our proposals are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation as well as model selection is analyzed. The novel model family is successfully applied to a real-world ordinal time series from finance.

Analyzing the cross-dependence within sequentially observed pairs of random variables is an interesting mathematical problem that also has several practical applications. Most of the time, classical dependence measures like Pearson's correlation are used to this end. This quantity, however, only measures linear dependence and has other drawbacks as well. Different concepts for measuring cross-dependence in sequentially observed random vectors, which are based on so-called ordinal patterns or multivariate generalizations of them, are described. In all cases, limiting distributions of the corresponding test statistics are derived. In a simulation study, the performance of these statistics is compared with three competitors, namely, classical Pearson's and Spearman's correlation as well as the rank-based Chatterjee's correlation coefficient. The applicability of the test statistics is illustrated by using them on two real-world data examples.

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.

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.

A common approach for modeling categorical time series is Hidden-Markov models (HMMs), where the actual observations are assumed to depend on hidden states in their behavior and transitions. Such categorical HMMs are even applicable to nominal data but suffer from a large number of model parameters. In the ordinal case, however, the natural order among the categorical outcomes offers the potential to reduce the number of parameters while improving their interpretability at the same time. The class of ordinal HMMs proposed in this article link a latent-variable approach with categorical HMMs. They are characterized by parametric parsimony and allow the easy calculation of relevant stochastic properties, such as marginal and bivariate probabilities. These points are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation is analyzed in finite samples. The developed methodology is applied to real-world data from a health application.