INAR Process Research Articles

Existence of a Periodic and Seasonal INAR Process

A spectral criterion involving the model parameters is given for the existence and uniqueness of a periodically correlated and seasonal non‐negative integer‐valued autoregressive process. The structure of the mean and covariance functions of the periodically stationary distribution of the model is derived using its implicit state‐space representation. Two infinite series representations for the process, the moving average, and the immigrant generation, are established. Based on the latter representation, a novel and parallelizable simulation method is proposed to generate the process.

Poisson-Lindley minification INAR process with application to financial data

<p>This paper introduces the Poisson-Lindley minification integer-valued autoregressive (PL-MINAR) process, a novel statistical model for analyzing count time series data. The modified negative binomial thinning and the Poisson-Lindley (PL) marginal distribution served as the foundation for the model. The proposed model was examined in terms of its basic stochastic properties, especially related to conditional stochastic measures (e.g., transition probabilities, conditional mean and variance, autocorrelation function). Through comprehensive simulations, the effectiveness of various parameter estimation techniques was validated. The PL-MINAR model's practical utility was demonstrated in analyzing the number of Bitcoin transactions and stock trades, showing its superior or comparable performance to the established INAR model. By offering a robust tool for financial time series analysis, this research holds potential for significant improvements in forecasting and understanding market dynamics.</p>

Some developments on seasonal INAR processes with application to influenza data

Influenza epidemic data are seasonal in nature. Zero-inflation, zero-deflation, overdispersion, and underdispersion are frequently seen in such number of cases of disease (count) data. To explain these counts’ features, this paper introduces a flexible model for nonnegative integer-valued time series with a seasonal autoregressive structure. Some probabilistic properties of the model are discussed for general seasonal INAR(p) model and three estimation methods are used to estimate the model parameters for its special case seasonal INAR(1) model. The performance of the estimation procedures has been studied using simulation. The proposed model is applied to analyze weekly influenza data from the Breisgau- Hochschwarzwald county of Baden–Württemberg state, Germany. The empirical findings show that the suggested model performs better than existing models.

A new bivariate INAR(1) model with paired Poisson-weighted exponential distributed innovations

This paper proposes a novel Bivariate integer-valued auto-regressive model of order 1 with paired Poisson Weighted Exponential (PWE) distributed innovations which is denoted by INAR(1)-PWE with two Sarmanov and classical versions. The CML and CLS estimators of the parameters are obtained and the performance of the proposed models are assessed through some Monte Carlo simulation experiments. Also, the BINAR(1)-PWE is applied to the two real data sets and is compared with some bivariate INAR processes. The research findings commend the BINAR(1)-PWE as another suitable alternative to analyze bivariate series of counts and open the avenues to explore the Sarmanov-based bivariate models.

Statistical Inference for Self-Exciting Threshold INAR Processes with Missing Values

Statistical Inference for Self-Exciting Threshold INAR Processes with Missing Values

Asymptotic inference for moderate deviations from a unit root of nearly unstable INAR(1) processes

ABSTRACT The INAR(1) processes with coefficients , where c>0 is a fixed constant and is a deterministic sequence growing to infinity at a slower rate than n, which are often referred to as nearly unstable INAR processes with moderate deviations from a unit root. We consider some basic properties of the processes and obtain the conditional least squares estimation of the coefficient , which converges to a normal distribution at speed . The simulation study provides numerical support for the theoretical results. The practical utility is illustrated in the data sets about liquor offences, claims of short-term disability and COVID-19, respectively.

Re-visiting the COVID-19 analysis using the class of high ordered integer-valued time series models with harmonic features

The COVID-19 series is obviously one of the most volatile time series with lots of spikes and oscillations. The conventional integer-valued auto-regressive time series models (INAR) may be limited to account for such features in COVID-19 series such as severe over-dispersion, excess of zeros, periodicity, harmonic shapes and oscillations. This paper proposes alternative formulations of the classical INAR process by considering the class of high-ordered INAR models with harmonic innovation distributions. Interestingly, the paper further explores the bivariate extension of these high-ordered INARs. South Africa and Mauritius’ COVID-19 series are re-scrutinized under the optic of these new INAR processes. Some simulation experiments are also executed to validate the new models and their estimation procedures.

Alternative procedures in dependent counting INAR process with application on COVID-19

Alternative estimation and forecasting procedures for the integer-valued autoregressive process with dependent geometric counting series are investigated. Some nonparametric estimation methods are applied to estimate the parameters of the model. The performance of the estimators is assessed via simulation study. We investigate an application of the process using COVID-19 data set and illustrate the best performance of the proposed model among some competitive INAR(1) models via the residual analysis and model adequacy. Consequently, the selected model is applied to predict the number of cases of COVID-19 based on the classic and Bayesian approaches.

Efficient estimation in semiparametric self-exciting threshold INAR processes

This paper focuses on the efficient estimation problem of a more realistic semiparametric SETINAR model of order one with two regimes based on binomial thinning operator. Unlike parametric framework, we do not suppose that the distribution of the innovation process belongs to a parametric family. Instead, the innovation distribution is totally unspecified and is supposed to satisfy only some mild technical assumptions. We, therefore, provide efficient estimators for both parameters of the model, namely a vector of auto-regression parameters and the innovation distribution which is considered as a parameter of infinite-dimension. The performances of these efficient estimators are shown through an intensive simulation study and an application on rotary rig count data in the U.S.A.

First-order random coefficient INAR process with dependent counting series

In this paper, we propose a first-order random coefficient integer-valued autoregressive process with dependent counting series. Some moments and stationary ergodicity of the process are established. The maximum-likelihood estimators of the parameters of interest are presented. We conduct some simulation studies to assess the performance of our method. An example about crime data is provided for practical application.

On the theory of periodic multivariate INAR processes

In this paper a multivariate integer-valued autoregressive model of order one with periodic time-varying parameters, and driven by a periodic innovations sequence of independent random vectors is introduced and studied in detail. Emphasis is placed on models with periodic multivariate negative binomial innovations. Basic probabilistic and statistical properties of the novel model are discussed. Aiming to reduce computational burden arising from the use of the conditional maximum likelihood method, a composite likelihood-based approach is adopted. The performance of such method is compared with that of some traditional competitors, namely moment estimators and conditional maximum likelihood estimators. Forecasting is also addressed. Furthermore, an application to a real data set concerning the monthly number of fires in three counties in Portugal is presented.

Correction to: A seasonal geometric INAR process based on negative binomial thinning operator

Correction to: A seasonal geometric INAR process based on negative binomial thinning operator

Bootstrap-based bias corrections for INAR count time series

ABSTRACTInteger-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.

A seasonal geometric INAR process based on negative binomial thinning operator

In this article, we propose a new seasonal geometric integer-valued autoregressive process based on the negative binomial thinning operator with seasonal period s. Some basic probabilistic and statistical properties of the model are discussed. Conditional maximum likelihood estimators are obtained, and the asymptotic properties of the estimators are established. Some theoretical results of point forecasts are obtained. Numerical results are presented. At the end, two real data examples are investigated to assess the performance of our new model.

Bivariate INAR Processes with Application to Mutual Fund Share Purchase/Redemption Counts

We propose a new bivariate nonnegative integer-autoregressive (BINAR) model for count process data. We first generalize the existing BINAR(1) model by allowing for dependence between different thinning operators. The extended family allows for intuitive interpretation, as well as tractable properties in terms of aggregation and stationarity. Then we also introduce higher order BINAR(p) and BINAR(∞) dynamics allowing for more flexible serial correlation pattern. So far, the literature has considered the standard BINAR(1) model and its potential extensions to be rather computationally intractable. Our biggest contribution is to show that the proposed extended BINAR family allows for (quasi-)closed-form predictive densities, which significantly facilitates non-linear forecasting and likelihood based estimation. Finally, a BINAR(∞) model with memory persistence is applied to mutual fund purchase/redemption order counts.