The Circumstance-Driven Bivariate Integer-Valued Autoregressive Model.

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule-Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated.

Discrete-Valued Time Series.

Time series are sequentially observed data in which important information about the phenomenon under consideration is contained not only in the individual observations themselves, but also in the way these observations follow one another [...].

Partial Autocorrelation Diagnostics for Count Time Series.

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule-Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example.

Optimal Stein-type goodness-of-fit tests for count data.

Common count distributions, such as the Poisson (binomial) distribution for unbounded (bounded) counts considered here, can be characterized by appropriate Stein identities. These identities, in turn, might be utilized to define a corresponding goodness-of-fit (GoF) test, the test statistic of which involves the computation of weighted means for a user-selected weight function f. Here, the choice of f should be done with respect to the relevant alternative scenario, as it will have great impact on the GoF-test's performance. We derive the asymptotics of both the Poisson and binomial Stein-type GoF-statistic for general count distributions (we also briefly consider the negative-binomial case), such that the asymptotic power is easily computed for arbitrary alternatives. This allows for an efficient implementation of optimal Stein tests, that is, which are most powerful within a given class  F $\mathcal {F}$ of weight functions. The performance and application of the optimal Stein-type GoF-tests is investigated by simulations and several medical data examples.

Non-parametric tests for serial dependence in time series based on asymptotic implementations of ordinal-pattern statistics.

Ordinal patterns can be used to construct non-parametric hypothesis tests that aim to discover (possibly non-linear) serial dependence in a real-valued time series. We derive the asymptotic distribution of the vector of sample frequencies of ordinal patterns and that of various corresponding tests statistics such that the targeted tests for serial dependence are easily implemented based on asymptotic approximations. Simulations are used to check the finite-sample performance of these tests as well as their power properties with respect to various alternative scenarios. The application and interpretation of the tests in practice are illustrated by an environmental data example.

Efficient accounting for estimation uncertainty in coherent forecasting of count processes.

Coherent forecasting techniques for count processes generate forecasts that consist of count values themselves. In practice, forecasting always relies on a fitted model and so the obtained forecast values are affected by estimation uncertainty. Thus, they may differ from the true forecast values as they would have been obtained from the true data generating process. We propose a computationally efficient resampling scheme that allows to express the uncertainty in common types of coherent forecasts for count processes. The performance of the resampling scheme, which results in ensembles of forecast values, is investigated in a simulation study. A real-data example is used to demonstrate the application of the proposed approach in practice. It is shown that the obtained ensembles of forecast values can be presented in a visual way that allows for an intuitive interpretation.

Time Series Modelling.

Time series consist of data observed sequentially in time, and they are assumed to stem from an underlying stochastic process [...].

Analyzing categorical time series in the presence of missing observations.

In real applications, time series often exhibit missing observations such that standard analytical tools cannot be applied. While there are approaches of how to handle missing data in quantitative time series, the case of categorical time series seems not to have been treated so far. Both for the case of ordinal and nominal time series, solutions are developed that allow to analyze their marginal and serial properties in the presence of missing observations. This is achieved by adapting the concept of amplitude modulation, which allows to obtain closed-form asymptotic expressions for the derived statistics' distribution (assuming that missingness happens independently of the actual process). The proposed methods are investigated with simulations, and they are applied in a project on migraine patients, where the monitored qualitative time series are often incomplete.

Measuring Dispersion and Serial Dependence in Ordinal Time Series Based on the Cumulative Paired ϕ-Entropy.

The family of cumulative paired ϕ-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen's κ, with the weights being related to the actual choice of ϕ. Again, the asymptotic distribution of the corresponding sample κϕ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel ϕ-entropy-related measures are applied to an ordinal time series on the daily level of air quality.

Regime-Switching Discrete ARMA Models for Categorical Time Series.

For the modeling of categorical time series, both nominal or ordinal time series, an extension of the basic discrete autoregressive moving-average (ARMA) models is proposed. It uses an observation-driven regime-switching mechanism, leading to the family of RS-DARMA models. After having discussed the stochastic properties of RS-DARMA models in general, we focus on the particular case of the first-order RS-DAR model. This RS-DAR ( 1 ) model constitutes a parsimoniously parameterized type of Markov chain, which has an easy-to-interpret data-generating mechanism and may also handle negative forms of serial dependence. Approaches for model fitting are elaborated on, and they are illustrated by two real-data examples: the modeling of a nominal sequence from biology, and of an ordinal time series regarding cloudiness. For future research, one might use the RS-DAR ( 1 ) model for constructing parsimonious advanced models, and one might adapt techniques for smoother regime transitions.

A Binomial Integer-Valued ARCH Model.

We present an integer-valued ARCH model which can be used for modeling time series of counts with under-, equi-, or overdispersion. The introduced model has a conditional binomial distribution, and it is shown to be strictly stationary and ergodic. The unknown parameters are estimated by three methods: conditional maximum likelihood, conditional least squares and maximum likelihood type penalty function estimation. The asymptotic distributions of the estimators are derived. A real application of the novel model to epidemic surveillance is briefly discussed. Finally, a generalization of the introduced model is considered by introducing an integer-valued GARCH model.

Chain binomial models and binomial autoregressive processes.

We establish a connection between a class of chain-binomial models of use in ecology and epidemiology and binomial autoregressive (AR) processes. New results are obtained for the latter, including expressions for the lag-conditional distribution and related quantities. We focus on two types of chain-binomial model, extinction-colonization and colonization-extinction models, and present two approaches to parameter estimation. The asymptotic distributions of the resulting estimators are studied, as well as their finite-sample performance, and we give an application to real data. A connection is made with standard AR models, which also has implications for parameter estimation.