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The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings. [ABSTRACT FROM AUTHOR]

This article considers the modeling of count data time series with a finite range having extra-binomial variation. We propose a beta-binomial autoregressive model using the concept of random coefficient thinning. We discuss the stationarity conditions, derive the moments and autocovariance function and consider approaches for parameter estimation. Furthermore, we develop two new tests for detecting extra-binomial variation, and we derive the asymptotic distributions of the test statistics under the null hypothesis of a binomial autoregressive model. The size and power performance of the two tests are analyzed under various alternatives taken from a beta-binomial autoregressive model with Monte Carlo experiments. The article ends with a real-data example about the Harmonised Index of Consumer Prices of the European Union. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces new classes of bivariate time series models being useful to fit count data time series with a finite range of counts. Motivation comes mainly from the comparison of schemes for monitoring tourism demand, stock data, production and environmental processes. All models are based on the bivariate binomial distribution of Type II. First, a new family of bivariate integer-valued GARCH models is proposed. Then, a new bivariate thinning operation is introduced and explained in detail. The new thinning operation has a number of advantages including the fact that marginally it behaves as the usual binomial thinning operation and also that allows for both positive and negative cross-correlations. Based upon this new thinning operation, a bivariate extension of the binomial autoregressive model of order one is introduced. Basic probabilistic and statistical properties of the model are discussed. Parameter estimation and forecasting are also covered. The performance of these models is illustrated through an empirical application to a set of rainy days time series collected from 2000 up to 2010 in the German cities of Bremen and Cuxhaven. [Copyright &y& Elsevier]

We present an elaboration of the usual binomial AR(1) process on {0,1, ... , N}that allows the thinning probabilities to depend on the current state N only through the 'density' n ∕ N, a natural assumption in many real contexts. We derive some basic properties of the model and explore approaches to parameter estimation. Some special cases are considered that allow for overdispersion and underdispersion, as well as positive and negative autocorrelations. We derive a law of large numbers and a central limit theorem, which provide useful large- N approximations for various quantities of interest. [ABSTRACT FROM AUTHOR]

Count data processes are often encountered in manufacturing and service industries. To describe the autocorrelation structure of such processes, a Poisson integer-valued autoregressive model of order 1, namely, Poisson INAR(1) model, might be used. In this study, we propose a two-sided cumulative sum control chart for monitoring Poisson INAR(1) processes with the aim of detecting changes in the process mean in both positive and negative directions. A trivariate Markov chain approach is developed for exact evaluation of the ARL performance of the chart in addition to a computationally efficient approximation based on bivariate Markov chains. The design of the chart for an ARL-unbiased performance and the analyses of the out-of-control performances are discussed. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The categorical time series analysis has become an area of emerging interest both in research and professional practice. In this article, we propose empirical measures of signed serial dependence, which are particularly important for identifying and fitting an appropriate model to a given categorical time series. We derive asymptotic properties of these measures and a rule for identifying significant dependence. We investigate the finite-sample performance of the proposed measures in a simulation study and show that they are sensitive to both positive and negative serial dependence even for rather short time series. [ABSTRACT FROM AUTHOR]

The time series of counts observed in practice often exhibit overdispersion. The INGARCH( p, q) models are able to describe integer-valued processes with overdispersion. Known properties of these models, however, are nearly exclusively restricted to the special case p = q = 1. In this article, we derive a set of equations from which the variance and the autocorrelation function of the general case can be obtained. We investigate the purely autoregressive INGARCH( p, 0) models and show that they are closely related to the standard AR( p) models. For p = 1, we determine the marginal distribution in terms of its cumulants. A real-data example highlights potential fields of application of the INGARCH( p, 0) models. [ABSTRACT FROM AUTHOR]

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 κ ϕ 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. [ABSTRACT FROM AUTHOR]

The empirical likelihood ratio (ELR) test is proposed for uncovering a structural change in integer-valued autoregressive (INAR) processes. The limiting distribution is derived under the null hypothesis that the parameter did not change at the anticipated change points. To evaluate the finite-sample performance of the proposed ELR test, the empirical sizes and powers are investigated in a simulation study. The ELR test is also applied to real data on infectious disease and crime counts. [ABSTRACT FROM AUTHOR]

Not a single time series, but a large collection of univariate time series is considered by Lindstrom et al. [[5]], who use functional kernel density estimation for uncovering anomalous time series within such a collection. While time series data have been collected for a relatively long time in history (one may recall the famous time series on sunspot numbers), the development of methods and stochastic models for such time series is more recent. Keywords: time series; models EN time series models 1163 1163 1 10/01/21 20210901 NES 210901 Time series consist of data observed sequentially in time, and they are assumed to stem from an underlying stochastic process. The second half of contributed articles deals with time series having a discrete range, or more precisely, with count time series, where the observations are count values from the set of non-negative integers. [Extracted from the article]

Volatility, which represents the magnitude of fluctuating asset prices or returns, is used in the problems of finance to design optimal asset allocations and to calculate the price of derivatives. Since volatility is unobservable, it is identified and estimated by latent variable models known as volatility fluctuation models. Almost all conventional volatility fluctuation models are linear time-series models and thus are difficult to capture nonlinear and/or non-Gaussian properties of volatility dynamics. In this study, we propose an entropy based Student's t-process Dynamical model (ETPDM) as a volatility fluctuation model combined with both nonlinear dynamics and non-Gaussian noise. The ETPDM estimates its latent variables and intrinsic parameters by a robust particle filtering based on a generalized H-theorem for a relative entropy. To test the performance of the ETPDM, we implement numerical experiments for financial time-series and confirm the robustness for a small number of particles by comparing with the conventional particle filtering. [ABSTRACT FROM AUTHOR]

In this research, we consider monitoring mean and correlation changes from zero-inflated autocorrelated count data based on the integer-valued time series model with random survival rate. A cumulative sum control chart is constructed due to its efficiency, the corresponding calculation methods of average run length and the standard deviation of the run length are given. Practical guidelines concerning the chart design are investigated. Extensive computations based on designs of experiments are conducted to illustrate the validity of the proposed method. Comparisons with the conventional control charting procedure are also provided. The analysis of the monthly number of drug crimes in the city of Pittsburgh is displayed to illustrate our current method of process monitoring. [ABSTRACT FROM AUTHOR]

In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the minimum density power divergence estimator. We demonstrate the proposed estimator is consistent and asymptotically normal under certain regularity conditions. Monte Carlo simulations are conducted to evaluate the performance of the estimator in the presence of outliers. Finally, a real data analysis using monthly count series of crimes in New South Wales and an artificial data example are provided as an illustration. [ABSTRACT FROM AUTHOR]

Unemployment has risen as the economy has shrunk. The coronavirus crisis has affected many sectors in Romania, some companies diminishing or even ceasing their activity. Making forecasts of the unemployment rate has a fundamental impact and importance on future social policy strategies. The aim of the paper is to comparatively analyze the forecast performances of different univariate time series methods with the purpose of providing future predictions of unemployment rate. In order to do that, several forecasting models (seasonal model autoregressive integrated moving average (SARIMA), self-exciting threshold autoregressive (SETAR), Holt–Winters, ETS (error, trend, seasonal), and NNAR (neural network autoregression)) have been applied, and their forecast performances have been evaluated on both the in-sample data covering the period January 2000–December 2017 used for the model identification and estimation and the out-of-sample data covering the last three years, 2018–2020. The forecast of unemployment rate relies on the next two years, 2021–2022. Based on the in-sample forecast assessment of different methods, the forecast measures root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE) suggested that the multiplicative Holt–Winters model outperforms the other models. For the out-of-sample forecasting performance of models, RMSE and MAE values revealed that the NNAR model has better forecasting performance, while according to MAPE, the SARIMA model registers higher forecast accuracy. The empirical results of the Diebold–Mariano test at one forecast horizon for out-of-sample methods revealed differences in the forecasting performance between SARIMA and NNAR, of which the best model of modeling and forecasting unemployment rate was considered to be the NNAR model. [ABSTRACT FROM AUTHOR]

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay τ ∗ and embedding dimension p for phase space reconstruction. The value of τ ∗ can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that τ ∗ should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay τ w = (p − 1) τ ∗ , which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally τ w and τ ∗ . In this paper, we suggest a simple method for estimating τ ∗ and τ w based on symbolic analysis and symbolic entropy. As in the C-C method, τ ∗ is estimated as the first local optimal time delay and τ w as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Motivated by a large dataset containing time series of weekly number of rainy days collected over two thousand locations across Europe and Russia for the period 2000-2010, we propose a new class of ARMA-like model for time series of bounded counts, which can also handle extra-binomial variation. We abbreviate this model as bvARMA, as it is based upon a novel operation referred to as binomial variation. After having discussed important stochastic properties and proposed a model-fitting approach relying on maximum likelihood estimation, we apply the bvARMA model family to the rainy-days time series. Results show that both bvAR and bvMA models are adequate and exhibit a similar performance. Furthermore, bvARMA results outperform those obtained by fitting ordinary discrete ARMA (NDARMA) models of the same order. [ABSTRACT FROM AUTHOR]

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.

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., (© 2022 The Authors. Biometrical Journal published by Wiley-VCH GmbH.)

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.

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.

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.

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., (© 2021 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.)

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., Competing Interests: No potential conflict of interest was reported by the author(s)., (© 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.)

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., (© 2011, The International Biometric Society.)