First-order random coefficients integer-valued threshold autoregressive processes

In this paper, we introduce a first-order random coefficient integer-valued threshold autoregressive process, which is based on binomial thinning. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators are derived for both the cases that the threshold variable is known or not. The asymptotic properties of the estimators are established. Moreover, forecasting problem is addressed. Finally, some numerical results of the estimates and a real data example are presented.     

A new geometric first-order integer-valued autoregressive (NGINAR(1)) process

A new stationary first-order integer-valued autoregressive process with geometric marginal distributions is introduced based on negative binomial thinning. Some properties of the process are established. Estimators of the parameters of the process are obtained using the methods of conditional least squares, Yule–Walker and maximum likelihood. Also, the asymptotic properties of the estimators are derived involving their distributions. Some numerical results of the estimators are presented with a discussion to the obtained results. Real data are used and a possible application is discussed.     

An extension on INAR models with discrete Laplace marginal distributions

The 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.     

First-order random coefficient integer-valued autoregressive processes

A first-order random coefficient integer-valued autoregressive (RCINAR(1)) model is introduced. Ergodicity of the process is established. Moments and autocovariance functions are obtained. Conditional least squares and quasi-likelihood estimators of the model parameters are derived and their asymptotic properties are established. The performance of these estimators is compared with the maximum likelihood estimator via simulation.     

A bivariate first-order signed integer-valued autoregressive process

Bivariate 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.     

A new geometric INAR(1) process based on counting series with deflation or inflation of zeros

In this paper, we introduce a new non-negative integer-valued autoregressive time series model based on a new thinning operator, so called generalized zero-modified geometric (GZMG) thinning operator. The first part of the paper is devoted to the distribution, GZMG distribution, which is obtained as the convolution of the zero-modified geometric (ZMG) distributed random variables. Some properties of this distribution are derived. Then, we construct a thinning operator based on the counting processes with ZMG distribution. Finally, an INAR(1) time series model is introduced and its properties including estimation issues are derived and discussed. A small Monte Carlo experiment is conducted to evaluate the performance of maximum likelihood estimators in finite samples. At the end of the paper, we consider an empirical illustration of the introduced INAR(1) model.     

Estimating monotonic change in the rate and dependence parameters of INAR(1) process (Case study: IP counts data)

In this article, we consider a first-order integer-valued autoregressive (INAR(1)) model. Then, we propose change point estimators for the rate and dependence parameters in INAR(1) model using maximum likelihood estimation method when the type of change belongs to a family of monotonic changes. To monitor the process, a combined EWMA and c control chart is considered. The results show that the proposed change point estimators provide efficient estimates of the change time. At the end, to illustrate the application of the proposed estimators, a real case related to IP counts data is investigated.     

Generalized integer-valued random coefficient for a first order structure autoregressive (RCINAR) process

A random coefficient autoregressive process for count data based on a generalized thinning operator is presented. Existence and weak stationarity conditions for these models are established. For the particular case of the (generalized) binomial thinning, it is proved that the necessary and sufficient conditions for weak stationarity are the same as those for continuous-valued AR(1) processes. These kinds of processes are appropriate for modelling non-linear integer-valued time series. They allow for over-dispersion and are appropriate when including covariates. Model parameters estimators are calculated and their properties studied analytically and/or through simulation.     

Generalized RCINAR(p) Process with Signed Thinning Operator

We propose a new integer-valued time series process, called generalized pth-order random coefficient integer-valued autoregressive process with signed thinning operator. This kind of process is appropriate for modeling negative integer-valued time series; strict stationarity and ergodicity of the process are established. Estimators of the model's parameters are derived and their properties are studied via simulation. We apply our process to a real data example.     

Generalized RCINAR(1) Process with Signed Thinning Operator

A generalized random coefficient first-order integer-valued autoregressive process with signed thinning operator is introduced, this kind of process is appropriate for modeling negative integer-valued time series. Strict stationarity and ergodicity of process are established. Estimators of the parameters of interest are derived and their properties are studied via simulation. At last, we use bootstrap method in the real data analysis.     

Some asymptotic properties in INAR(1) processes with Poisson marginals

The first-order integer-valued autoregressive (INAR(1)) process with Poisson marginal distributions is considered. It is shown that the sample autocovariance function of the model is asymptotically normally distributed. We derive asymptotic distribution of Yule-Walker type estimators of parameters. It turns out that our Yule-Walker type estimators are better than the conditional least squares estimators proposed by Klimko and Nelson (1978) and Al-Osh and Alzaid (1987). also, we study the relationship between the model andM/M/∞ queueing system.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

Innovational Outliers in INAR(1) Models

We consider integer-valued autoregressive models of order one contaminated with innovational outliers. Assuming that the time points of the outliers are known but their sizes are unknown, we prove that Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, CLS estimators of the outliers' sizes are not strongly consistent. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the values of the process at time points preceding the outliers' occurrences, the joint CLS estimator of the sizes of the outliers is asymptotically normal.     

Additive outliers in INAR(1) models

In this paper the integer-valued autoregressive model of order one, contaminated with additive outliers is studied in some detail. Moreover, parameter estimation is also addressed. Supposing that the timepoints of the outliers are known but their sizes are unknown, we prove that the conditional least squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, however, the CLS estimators of the outliers’ sizes are not strongly consistent, although they converge to a random limit with probability 1. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the values of the process at the timepoints neighboring to the outliers’ occurrences, the joint CLS estimator of the sizes of the outliers is also asymptotically normal.     

Higher-order moments,cumulants and spectral densities of the NGINAR(1) process

The motivation for time series with geometric marginal distributions arises from noting that the Poisson distribution is not always suitable for the modeling and analysis of integer-valued time series. The NGINAR(1) process that has been introduced by Risti? et al. (2009) represents a class of such time series. Joint higher-order (factorial) moments and cumulants with some other related statistical measures of the NGINAR(1) process are constructed. Also, the spectral and bispectral density functions of this process are investigated, including their nonparametric estimators, using the multitapering method. A real data example of the nonparametric multitaper spectral estimates is investigated, with a discussion of the results obtained.     

Integer-valued moving average (INMA) process

A simple model for a stationary sequence of dependent integer-valued random variables {X_n} is given. The sequence to be called integer-valued moving average (INMA) process, is taken as the “survivals” of i.i.d. non-negative integervalued random variables. It is argued that the model’s structure reflects to some extent the mechanism generating real life data for many counting process and consequently it is useful for modelling such processes. Various properties for the special case in which {X_n} is Poisson INMA (1) process, such as the joint distribution, regression, time reversibility, along with the conditional and partial correlations, are discussed in details. Extension of the INMA of first order to higher order moving average is considered.     

A non-stationary integer-valued autoregressive model

It is frequent to encounter a time series of counts which are small in value and show a trend having relatively large fluctuation. To handle such a non-stationary integer-valued time series with a large dispersion, we introduce a new process called integer-valued autoregressive process of order p with signed binomial thinning (INARS(p)). This INARS(p) uniquely exists and is stationary under the same stationary condition as in the AR(p) process. We provide the properties of the INARS(p) as well as the asymptotic normality of the estimates of the model parameters. This new process includes previous integer-valued autoregressive processes as special cases. To preserve integer-valued nature of the INARS(p) and to avoid difficulty in deriving the distributional properties of the forecasts, we propose a bootstrap approach for deriving forecasts and confidence intervals. We apply the INARS(p) to the frequency of new patients diagnosed with acquired immunodeficiency syndrome (AIDS) in Baltimore, Maryland, U.S. during the period of 108 months from January 1993 to December 2001.     

Periodic integer-valued bilinear time series model

This paper deals with the study of some probabilistic and statistical properties of a periodic integer-valued diagonal bilinear model. The existence of a periodically strict stationary integer-valued process is shown. Sufficient conditions for the periodically stationary, both in the first and second orders, are established. The closed-forms of the mean and the second moment are obtained. The closed-form of the periodic autocovariance function is established. The Yule–Walker estimations of the underlying parameters are obtained. A simulation study is provided.     

Fractional approaches for the distribution of innovation sequence of INAR(1) processes

Abstract

In this paper, we present a fractional decomposition of the probability generating function of the innovation process of the first-order non-negative integer-valued autoregressive [INAR(1)] process to obtain the corresponding probability mass function. We also provide a comprehensive review of integer-valued time series models, based on the concept of thinning operators with geometric-type marginals. In particular, we develop two fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. These approaches are discussed in detail and illustrated through a few examples.     

Integer-Valued Self-Exciting Threshold Autoregressive Processes

In this article, we introduce a class of self-exciting threshold integer-valued autoregressive models driven by independent Poisson-distributed random variables. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the least squares-type and likelihood-based ones. Their performance is compared through a simulation study.