INAR Models Research Articles

Local stationarity and time-inhomogeneous Markov chains

A primary motivation of this contribution is to define new locally stationary Markov models for categorical or integer-valued data. For this initial purpose, we propose a new general approach for dealing with time-inhomogeneity that extends the local stationarity notion developed in the time series literature. We also introduce a probabilistic framework which is very flexible and allows us to consider a much larger class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes studied in the literature. We consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting and slowly-varying Markov kernels. The finite-dimensional distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and some mixing properties are also available for these triangular arrays. As a consequence of our results, some classical geometrically ergodic homogeneous Markov chain models have a locally stationary version, which lays the theoretical foundations for new statistical modeling. Statistical inference of finite-state Markov chains can be based on kernel smoothing and we provide a complete and fast implementation of such models, directly usable by the practitioners. We also illustrate the theory on a real data set. A central limit theorem for Markov chains on more general state spaces is also provided and illustrated with the statistical inference in INAR models, Poisson ARCH models and binary time series models. Additional examples such as locally stationary regime-switching or SETAR models are also discussed.

Model-based INAR bootstrap for forecasting INAR(p) models

In this paper we analyse some bootstrap techniques to make inference in INAR(p) models. First of all, via Monte Carlo experiments we compare the performances of these methods when estimating the thinning parameters in INAR(p) models; we state the superiority of model-based INAR bootstrap approaches on block bootstrap in terms of low bias and Mean Square Error. Then we adopt the model-based bootstrap methods to obtain coherent predictions and confidence intervals in order to avoid difficulty in deriving the distributional properties. Finally, we present an empirical application.

INARMA Modeling of Count Time Series

While most of the literature about INARMA models (integer-valued autoregressive moving-average) concentrates on the purely autoregressive INAR models, we consider INARMA models that also include a moving-average part. We study moment properties and show how to efficiently implement maximum likelihood estimation. We analyze the estimation performance and consider the topic of model selection. We also analyze the consequences of choosing an inadequate model for the given count process. Two real-data examples are presented for illustration.

Statistical inference for the new INAR(2) models with random coefficient

In this paper, we investigate a random coefficient INAR(2) process which may model the number of traded stocks, the number of infected people, the number of birds in some area, etc. We show that this process is a stationary and ergodic process under some mild conditions. Adopting the two-step conditional least-square estimation method, we give consistent estimations of the unknown parameters. Furthermore, the asymptotic distributions of the estimators are obtained and a simulation study is conducted for the evaluation of the developed approach.

On Some Stationary Inar Models with Discrete Laplace Marginals

On Some Stationary Inar Models with Discrete Laplace Marginals

Random Environment INAR Models of Higher Order

Random Environment INAR Models of Higher Order

Goodness-of-fit testing of a count time series’ marginal distribution

Popular goodness-of-fit tests like the famous Pearson test compare the estimated probability mass function with the corresponding hypothetical one. If the resulting divergence value is too large, then the null hypothesis is rejected. If applied to i. i. d. data, the required critical values can be computed according to well-known asymptotic approximations, e. g., according to an appropriate \(\chi ^2\)-distribution in case of the Pearson statistic. In this article, an approach is presented of how to derive an asymptotic approximation if being concerned with time series of autocorrelated counts. Solutions are presented for the case of a fully specified null model as well as for the case where parameters have to be estimated. The proposed approaches are exemplified for (among others) different types of CLAR(1) models, INAR(p) models, discrete ARMA models and Hidden-Markov models.

Outliers detection in INAR (1) time series

This article considers the problem of detecting outliers of discrete time series. The underlying process is modeled as Poisson INAR (1) models, a special class of integer-valued autoregressive process. Based on this model, we develop a Gibbs sampling algorithm to estimate the model parameters and the size of outliers. This current method can detect outliers and specify the type of outliers simultaneously, without any additional computation to distinguish between additive outliers (AO) and innovational outliers (IO). The approach is illustrated with simulation studies as well as a real data analysis.

Residual Analysis with Bivariate INAR(1) Models

Residual Analysis with Bivariate INAR(1) Models

Estimation and Forecasting in INAR(P) Models Using Sieve Bootstrap

In this paper we analyse some bootstrap techniques to make inference in INAR(p) models. First of all, via Monte Carlo experiments we compare the performances of these methods when estimating the thinning parameters in INAR(p) models. We state the superiority of sieve bootstrap approaches on block bootstrap in terms of low bias and Mean Square Error (MSE). Then we apply the sieve bootstrap methods to obtain coherent predictions and confidence intervals in order to avoid difficulty in deriving the distributional properties.

English

Many real-life count data are frequently characterized by overdispersion, excess zeros and autocorrelation. Zero-inflated count time series models can provide a powerful procedure to model this type of data. In this paper, we introduce a new stationary first-order integer-valued autoregressive process with random coefficient and zero-inflated geometric marginal distribution, named ZIGINARRC(1) process, which contains some sub-models as special cases. Several properties of the process are established. Estimators of the model parameters are obtained and their performance is checked by a small Monte Carlo simulation. Also, the behavior of the inflation parameter of the model is justified. We investigate an application of the process using a real count climate data set with excessive zeros for the number of tornados deaths and illustrate the best performance of the proposed process as compared with a set of competitive INAR(1) models via some goodness-of-fit statistics. Consequently, forecasting for the data is discussed with estimation of the transition probability and expected run length at state zero. Moreover, for the considered data, a test of the random coefficient for the proposed process is investigated.

Generalized Random Environment INAR Models of Higher Order

The random environment integer-valued autoregressive models of higher orders introduced by Nastic et al. (J Time Ser Anal 37: 267–287, 2016) are generalized. Generalizations are made by relaxing two assumptions: the assumption about the equality of negative binomial thinning operators and the assumption that the maximal orders are equal for all the process random states. Some statistical properties of the introduced models are derived and discussed. Their unknown parameters are estimated by the Yule–Walker method and the performances of the obtained estimators are checked using simulated data sets. A possible application of the models on the real-life data is discussed at the end of the paper.

A New Geometric INAR(1) Model with Mixing Pegram and Generalized Binomial Thinning Operators

In this paper, we introduce a new stationary integer-valued autoregressive process of the first order with geometric marginal based on mixing Pegram and generalized binomial thinning operators. The count series of the process consists of dependent Bernoulli count variables. Various properties of the process are obtained, including the distribution of its innovation process. Maximum likelihood estimation by EM algorithm is applied to estimate the parameters of the process and the performance of the estimates is checked by Monte Carlo simulation. We investigate applicability of the process using a real count data set and compare the process to many competitive INAR(1) models via some goodness-of-fit statistics. As a result, forecasting of the data is discussed under the proposed process.

Testing the constancy of the thinning parameter in a random coefficient integer autoregressive model

Integer-valued autoregressive models are widely used for modeling the time dependent count data. Many of the inference problems related to these types of models are not yet addressed due to the complexities of the related distribution theory. In this paper, we consider one such inference problem associated with these types of models. For a random coefficient integer-valued autoregressive model, we develop a locally most powerful-type test for testing the hypothesis that the thinning parameter is constant across the time. The asymptotic distribution of the suggested test statistic is derived. The Poisson and geometric INAR(1) models are considered for the illustration of the suggested methodology. Simulation studies indicate that the suggested test performs quite well. We have applied our methods to count time series data sets, where the thinning parameter is suspected to be varying.

A mixed thinning based geometric INAR(1) model

In this article a geometrically distributed integer-valued autoregressive model of order one based on the mixed thinning operator is introduced. This new thinning operator is defined as a probability mixture of two well known thinning operators, binomial and negative binomial thinning. Some model properties are discussed. Method of moments and the conditional least squares are considered as possible approaches in model parameter estimation. Asymptotic characterization of the obtained parameter estimators is presented. The adequacy of the introduced model is verified by its application on a certain kind of real-life counting data, while its performance is evaluated by comparison with two other INAR(1) models that can be also used over the observed data.

Testing for zero inflation and overdispersion in INAR(1) models

The marginal distribution of count data processes rarely follows a simple Poisson model in practice. Instead, one commonly observes deviations such as overdispersion or zero inflation. To express the extend of such deviations from a Poisson model, one can compute an appropriately defined dispersion index or zero index. In this article, we develop several tests based on such indexes, including joint tests being based on an index combination. The asymptotic distribution of the resulting test statistics under the null hypothesis of a Poisson INAR(1) model is derived, and the finite-sample performance of the resulting tests is analyzed. Real data examples illustrate the application of these tests in practice.

On Mixing Properties of Some INAR Models

Strictly stationary INAR(1) processes (“integer-valued autoregressive processes of order 1”) with Poisson innovations are “interlaced ρ-mixing.” Bibliography: 20 titles.

A bivariate first-order signed integer-valued autoregressive process

ABSTRACTBivariate integer-valued time series occur in many areas, such as finance, epidemiology, business etc. In this article, we present bivariate autoregressive integer-valued time-series models, based on the signed thinning operator. Compared to classical bivariate INAR models, the new processes have the advantage to allow for negative values for both the time series and the autocorrelation functions. Strict stationarity and ergodicity of the processes are established. The moments and the autocovariance functions are determined. The conditional least squares estimator of the model parameters is considered and the asymptotic properties of the obtained estimators are derived. An analysis of a real dataset from finance and a simulation study are carried out to assess the performance of the model.

An extension on INAR models with discrete Laplace marginal distributions

ABSTRACTThe main theme considered in this article is an integer-valued thinning operator with both positive and negative values, its properties, and a new time series with skew discrete Laplace marginals. Some properties of this model are discussed, as well as estimators of unknown parameters, similarities and differences with some other existing models, applications in real-life situations, and identification and approximation of latent processes affecting the concerning process.

A geometric time series model with a new dependent Bernoulli counting series

ABSTRACTNew generalized binomial thinning operator with dependent counting series is introduced. An integer valued time series model with geometric marginals based on this thinning operator is constructed. Main features of the process are analyzed and determined. Estimation of the parameters are presented and some asymptotic properties of the obtained estimators are discussed. Behavior of the estimators is described through the numerical results. Also, model is applied on the real data set and compared to some relevant INAR(1) models.