Poisson Innovations Research Articles

A zero-inflated Poisson integer-valued autoregressive model with time-varying coefficients covariates

For integer-valued time series, we may be interested in how a covariate affects the series. In the existing models, it is commonly assumed that the coefficients of the covariates are constant, and the innovation process follows a Poisson distribution. However, in practice, the influence of covariates may vary with time, and there may be a large number of zeros. To address this situation, this paper proposes an integer-valued autoregressive model with time-varying coefficients on covariates, incorporating a zero-inflated Poisson innovation process. The binomial thinning parameters and unknown functions of the proposed model are estimated based on approximate conditional least squares and the cubic spline method. The consistency and asymptotic properties of the binomial thinning parameters are investigated. Simulation studies and real data analysis support the flexibility and good performance of the proposed model.

On Comparing and Assessing Robustness of Some Popular Non-Stationary BINAR(1) Models

Intra-day transactions of stocks from competing firms in the financial markets are known to exhibit significant volatility and over-dispersion. This paper proposes some bivariate integer-valued auto-regressive models of order 1 (BINAR(1)) that are useful to analyze such financial series. These models were constructed under both time-variant and time-invariant conditions to capture features such as over-dispersion and non-stationarity in time series of counts. However, the quest for the most robust BINAR(1) models is still on. This paper considers specifically the family of BINAR(1)s with a non-diagonal cross-correlation structure and with unpaired innovation series. These assumptions relax the number of parameters to be estimated. Simulation experiments are performed to assess both the consistency of the estimators and the robust behavior of the BINAR(1)s under mis-specified innovation distribution specifications. The proposed BINAR(1)s are applied to analyze the intra-day transaction series of AstraZeneca and Ericsson. Diagnostic measures such as the root mean square errors (RMSEs) and Akaike information criteria (AICs) are also considered. The paper concludes that the BINAR(1)s with negative binomial and COM–Poisson innovations are among the most suitable models to analyze over-dispersed intra-day transaction series of stocks.

Inference for integer-valued AR model with zero-inflated Poisson innovations

Inference for integer-valued AR model with zero-inflated Poisson innovations

A New INAR(1) Model for ℤ-Valued Time Series Using the Relative Binomial Thinning Operator

Abstract A new first-order integer-valued autoregressive process (INAR(1)) with extended Poisson innovations is introduced based on a signed version of the thinning operator, called relative binomial thinning operator, which can be considered as an extension of standard binomial thinning operator introduced by Steutel, F.W. and van Harn, K. (1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7: 893–899). It is appropriate for modeling Z $\mathbb{Z}$ -valued time series and either positive or negative correlations. Some properties of the process are established. Conditional least squares, Yule–Walker and conditional maximum likelihood methods are considered for the parameter estimation of the model. Moreover, simulation experiments are carried out to attest to the performance of the estimation methods. The applicability of the proposed model is investigated through a practical data set of the Saudi stock market.

Statistical modelling of COVID-19 and drug data via an INAR(1) process with a recent thinning operator and cosine Poisson innovations.

In this paper, we propose the first-order stationary integer-valued autoregressive process with the cosine Poisson innovation, based on the negative binomial thinning operator. It can be equi-dispersed, under-dispersed and over-dispersed. Therefore, it is flexible for modelling integer-valued time series. Some statistical properties of the process are derived. The parameters of the process are estimated by two methods of estimation and the performances of the estimators are evaluated via some simulation studies. Finally, we demonstrate the usefulness of the proposed model by modelling and analyzing some practical count time series data on the daily deaths of COVID-19 and the drug calls data.

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations

Abstract We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x) β , x ∈ (0, 1), with β ∈ (−1, ∞) and lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) $\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$ , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

Modelling and monitoring of INAR(1) process with geometrically inflated Poisson innovations

ABSTRACT To analyse count time series data inflated at the r + 1 values , we propose a new first-order integer-valued autoregressive process with r-geometrically inflated Poisson innovations. Some statistical properties together with conditional maximum likelihood estimate are provided. For the purpose of statistical monitoring, we focus on the cumulative sum chart, exponentially weighted moving average chart and combined jumps chart towards the proposed process. Numerical simulations indicate that the conditional maximum likelihood estimator is unbiased. Moreover, the cumulative sum chart is the best choice to monitor our model in practice. Some applications about telephone complaints data are provided to illustrate the proposed methods.

Parameter Estimation of a Seasonal Poisson INAR(1) Model with Different Monthly Means

Analysing seasonality in count time series is an essential application of statistics to predict phenomena in different fields like economics, agriculture, healthcare, environment, and climatic change. However, the information in the existing literature is scarce regarding the performances of relevant statistical models. This study provides the Yule-Walker (Y-W), Conditional Least Squares (CLS), and Maximum Likelihood Estimation (MLE) for First-order Non-negative Integer-valued Autoregressive, INAR(1), process with Poisson innovations with different monthly means. The performance of Y-W, CLS, and MLE are assessed by the Monte Carlo simulation method. The performance of this model is compared with another seasonal INAR(1) model by reproducing the monthly number of rainy days in the Blackwater River watershed located in coastal Virginia. Two forecast-coherent methods in terms of mode and probability function are applied to make predictions. The models’ performances are assessed using the Root Mean Square Error and Index of Agreement criteria. The results reveal the similar performance of Y-W, CLS, and MLE for estimating the parameters of data sets with larger sample size and values of α close to unite root. Moreover, the results indicate that INAR(1) with different monthly Poisson innovations is more appropriate for modelling and predicting seasonal count time series.

An alternative test for zero modification in the INAR(1) model with Poisson innovations

Several methods have been proposed for detecting zero modification in the first-order integer-valued autoregressive (INAR(1)) process. A basic problem of these tests is that they rely upon asymptotic results. In this paper, an alternative test is introduced which makes direct use of the approximate distribution of the number of zeros, which can be described by a beta-binomial distribution. A hybrid estimator of the mean parameter of the marginal distribution of the Poisson INAR(1) process is given. A simulation study shows that power and size of the proposed test are competitive. Finally, real data examples are provided.

A new mixed first-order integer-valued autoregressive process with Poisson innovations

Integer-valued time series, seen as a collection of observations measured sequentially over time, have been studied with deep notoriety in recent years, with applications and new proposals of autoregressive models that broaden the field of study. This work proposes a new mixed integer-valued first-order autoregressive model with Poisson innovations, denoted POMINAR(1), mixing two operators known as binomial thinning and Poisson thinning. The proposed process presents some advantages in relation to the most common Poisson innovation processes: (1) this new process allows to capture structural changes in the data; (2) if there are no structural changes, the most common processes with Poisson innovations are particular cases of POMINAR(1). Another important contribution of this work is the establishment of the POMINAR(1) theoretical results, such as the marginal expectation, marginal variance, conditional expectation, conditional variance, transition probabilities. Moreover, the Conditional Maximum Likelihood (CML) and Yule-Walker (YW) estimators for the process parameters are studied. We also present three techniques for one-step-ahead forecasting, the nearest integer of the conditional expectation, conditional median and mode. A simulation study of the forecasting procedures, considering the two estimators, CML and YW methods, is performed, and prediction intervals are presented. Finally, we show an application of the proposed process to a real dataset, referred here as larceny data, including a residual analysis.

Bayesian analysis of the p-order integer-valued AR process with zero-inflated Poisson innovations

In recent years, there has been a considerable interest to study count time series with a dependence structure and appearance of excess of zeros values. Such series are commonly encountered in diverse disciplines, such as economics, financial research, environmental science, public health, among others. In this paper, we propose a stationary p-order integer-valued autoregressive process with zero-inflated Poisson innovations, called the ZINAR(p) times series model. We study some of its theoretical properties and develop a Markov chain Monte Carlo (MCMC) algorithm for inferring parameters from Bayesian perspectives. Finally, we demonstrate the utility of proposed ZINAR(p) model through simulated and real data examples.

Inference for bivariate integer-valued moving average models based on binomial thinning operation

Time series of (small) counts are common in practice and appear in a wide variety of fields. In the last three decades, several models that explicitly account for the discreteness of the data have been proposed in the literature. However, for multivariate time series of counts several difficulties arise and the literature is not so detailed. This work considers Bivariate INteger-valued Moving Average, BINMA, models based on the binomial thinning operation. The main probabilistic and statistical properties of BINMA models are studied. Two parametric cases are analysed, one with the cross-correlation generated through a Bivariate Poisson innovation process and another with a Bivariate Negative Binomial innovation process. Moreover, parameter estimation is carried out by the Generalized Method of Moments. The performance of the model is illustrated with synthetic data as well as with real datasets.

Inferential aspects of the zero-inflated Poisson INAR(1) process

• Provide Yule–Walker and two-step Conditional Least Squares estimation methods. • Comparison with the Conditional Maximum Likelihood approach under misspecification. • Obtain the asymptotic distribution of the Yule–Walker estimators. • Two real count time series data are analysed based on our methodology. A first-order INteger-valued AutoRegressive (INAR) process with zero-inflated Poisson distributed innovations was proposed by Jazi, Jones and Lai (2012) [First-order integer valued AR processes with zero inflated Poisson innovations. Journal of Time Series Analysis. 33, 954–963.] , which is able for dealing with zero-inflated/deflated count time series data. The inferential aspects of this model were not well explored by the authors, only a conditional maximum likelihood approach was briefly discussed. In this paper, we explore the inferential aspects of this zero-inflated Poisson INAR(1) process. We propose parameter estimation through Two-Step Conditional Least Squares and Yule–Walker methods. The asymptotic properties of the estimators are provided. Simulation results about the finite-sample behavior of both estimation methods and comparisons with the conditional maximum likelihood approach are presented under correct model specification and misspecification. Two empirical applications to real data sets are considered in order to illustrate the usefulness of the proposed methodology in practical situations.

Testing for INAR effects

In this article, we focus on the integer valued autoregressive model, INAR (1), with Poisson innovations. We test the null of serial independence, where the INAR parameter is zero, versus the alternative of a positive INAR parameter. To this end, we propose different explicit approximations of the likelihood ratio (LR) statistic. We derive the limiting distributions of our statistics under the null. In a simulation study, we compare size and power of our tests with the score test, proposed by Sun and McCabe [2013. Score statistics for testing serial dependence in count data. Journal of Time Series Analysis 34 (3):315–29]. The size is either asymptotic or derived via response surface regressions of critical values. We find that our statistics are superior to score in terms of power and work just as well in terms of size. Another finding is that the powers of our approximate LR statistics compare well with the power of the numerical LR statistic. Power simulations are also performed under an INAR(2) framework, with similar outcome.

Modeling time series of count with excess zeros and ones based on INAR(1) model with zero-and-one inflated Poisson innovations

The first-order non-negative integer-valued autoregressive (INAR(1)) process has been applied to model the counts of events for a long time, and the Poisson model provides a standard and popular framework. However, this model may not be suitable for a data set with excess zeros and excess ones. We introduce a new stationary INAR(1) process with zero-and-one inflated Poisson (ZOIP) innovations. The proposed model is assumed to be a mixture of three separate data generation processes: one generates only zeros, one generates only ones, and the last one is a Poisson data-generating process. We present some structural properties such as the mean, variance, marginal and joint distribution functions of the process. We develop a method to test whether zero and one inflated under a Poisson INAR(1) model, which is based on dispersion index, zero index and one index. We also give the asymptotic distribution of the resulting test statistics under the null hypothesis of a Poisson INAR(1) model. Conditional maximum likelihood estimators are given, and the asymptotic properties of the estimators are established. In addition, the forecasting problem is addressed. Finally, a simulation study shows that the estimation method is accurate and reliable as long as the sample size is reasonably large. Two real data examples lead to superior performances of the proposed model compared with other competitive models in the literature.

Modeling longitudinal INMA(1) with COM–Poisson innovation under non-stationarity: application to medical data

This paper introduces an observation-driven (OD) longitudinal integer-valued moving average model of order 1 (INMA(1)) with COM–Poisson innovations under non-stationary moment conditions. This new longitudinal model provides lot of practical flexibility in terms of modeling a wide range of over-, under-dispersion and any mixed level of dispersion. In this set-up, the model parameters of primary interest consist of the regression and dispersion effects while the serial autocorrelation parameters are treated as nuisance. A robust Generalized Quasi-Likelihood approach is formulated to estimate the different set of parameters. The performance of the estimating algorithm is assessed via Monte Carlo experiments under various combination of the serial and dispersion values and is compared with the existing adaptive generalized method of moments. Application to two medical data: epileptic seizures and polyposis counts are also illustrated.

Extended Poisson INAR(1) processes with equidispersion, underdispersion and overdispersion

ABSTRACTReal count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.

How Compressible Are Innovation Processes?

The sparsity and compressibility of finite-dimensional signals are of great interest in fields, such as compressed sensing. The notion of compressibility is also extended to infinite sequences of independent identically distributed or ergodic random variables based on the observed error in their nonlinear $k$ -term approximation. In this paper, we use the entropy measure to study the compressibility of continuous-domain innovation processes (alternatively known as white noise). Specifically, we define such a measure as the entropy limit of the doubly quantized (time and amplitude) process. This provides a tool to compare the compressibility of various innovation processes. It also allows us to identify an analogue of the concept of “entropy dimension” which was originally defined by Renyi for random variables. Particular attention is given to stable and impulsive Poisson innovation processes. Here, our results recognize Poisson innovations as the more compressible ones with an entropy measure far below that of stable innovations. While this result departs from the previous knowledge regarding the compressibility of impulsive Poisson laws compared with continuous fat-tailed distributions, our entropy measure ranks $\alpha$ -stable innovations according to their tail.

A BINAR(1) time-series model with cross-correlated COM–Poisson innovations

ABSTRACTThis article proposes a bivariate integer-valued autoregressive time-series model of order 1 (BINAR(1) with COM–Poisson marginals to analyze a pair of non stationary time series of counts. The interrelation between the series is induced by the correlated innovations, while the non stationarity is captured through a common set of time-dependent covariates that influence the count responses. The regression and dependence effects are estimated using generalized quasi-likelihood (GQL) approach. Simulation experiments are performed to assess the performance of the estimation algorithms. The proposed BINAR(1) process is applied to analyze a real-life series of day and night accidents in Mauritius.

Iterated limits for aggregation of randomized INAR(1) processes with Poisson innovations

We discuss joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α∈(0,1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x)(1−x)β, x∈(0,1), with limx↑1⁡ψ(x)=ψ1∈(0,∞), different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β∈(−1,0), β=0, β∈(0,1) or β∈(1,∞), when taking first the limit as N→∞ and then the time scale n→∞, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaitė and Surgailis [13] by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.