Conference Agenda

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.

Integer-valued autoregressive (INAR) models provide a flexible framework for modeling count time series through thinning operators that emulate autoregressive dynamics while respecting the discrete nature of the data. These models naturally accommodate both equidispersion and overdispersion, features commonly observed in count-valued processes.

This paper investigates INAR models with structural breaks, with particular emphasis on the detection and estimation of parameter changes over time—an issue of critical importance in dynamic settings such as epidemics, policy interventions, and other regime-shifting phenomena. We consider both classical and Bayesian inferential approaches for identifying change points and estimating model parameters.

The classical framework is based on maximum likelihood estimation, where structural changes are detected using a CUSUM-based procedure, followed by a focused grid search within a window centered around the candidate breakpoint. The Bayesian approach employs advanced Markov Chain Monte Carlo (MCMC) techniques, incorporating hidden Markov chains to model latent regimes and infer structural shifts probabilistically.

A comprehensive simulation study is conducted under a variety of scenarios, including differing regime lengths and sample size proportions, and distributional characteristics. Finally, the proposed methodologies are illustrated through an application to real-world health indicator data, demonstrating their practical effectiveness in capturing complex dynamics and structural changes in count time series.

For modeling the serial dependence in discrete-valued time series, various approaches have been proposed in the literature. In particular, models based on a recursive, autoregressive-type structure such as the integer-valued autoregressive (INAR) models for count time series are very popular in practice. While their estimation typically relies on purely parametric approaches that impose restrictive assumptions on the innovation distribution, we consider semi-parametric estimation techniques that jointly estimate the autoregressive coefficients and the innovation distribution without requiring parametric specification. Building on this, we propose a general semi-parametric bootstrap procedure for INAR models and prove its consistency for general classes of statistics that are functions of the estimated model coefficients and the estimated innovation distribution. This semi-parametric bootstrap approach can be leveraged for various statistical tasks such as goodness-of-fit testing, predictive inference, and dispersion analysis. Additionally, we introduce novel semi-parametric goodness-of-fit tests 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 parametric family of the innovation distribution. We derive the limiting null distribution of our proposed test statistics, prove consistency under fixed alternatives and discuss its asymptotic behavior under local alternatives. Moreover, when it comes to predictive inference for discrete-valued time series, this task cannot be implemented through the construction of prediction intervals as they are generally not able to retain a desired coverage level neither in finite samples nor asymptotically. To address this problem, we propose to reverse the construction principle by considering preselected sets of interest and estimating the corresponding predictive probability. The accuracy of this prediction is then evaluated by quantifying the uncertainty associated with the estimation of these predictive probabilities. In this context, we consider parametric and non-parametric approaches and derive asymptotic as well as bootstrap theory, which also covers the practically important case of model misspecification.

During the last years, there have been many proposals for modelling integer-valued time series. We propose tests of hypotheses related to certain symmetry and antisymmetry properties. For example, we consider the hypotheses that the conditional mean is an odd function or that the conditional variance is an even function. The proposed test statistics are nonparametric and have non-standard limit distributions. We show that the wild bootstrap offers a simple method of generating asymptotically correct critical values. The talk is based on joint work with Paul Doukhan and Christian Weiß.