On first-order integer-valued autoregressive process with Katz family innovations

This paper considers the first-order integer-valued autoregressive (INAR) process with Katz family innovations. This family of INAR processes includes a broad class of INAR(1) processes with Poisson, negative binomial, and binomial innovations, respectively, featuring equi-, over-, and under-dispersion. Its probabilistic properties such as ergodicity and stationarity are investigated and the formula of the marginal mean and variance is provided. Further, a statistical process control procedure based on the cumulative sum control chart is considered to monitor autocorrelated count processes. A simulation and real data analysis are conducted for illustration.     

Flexible Bivariate INAR(1) Processes Using Copulas

Multivariate count time series data occur in many different disciplines. The class of INteger-valued AutoRegressive (INAR) processes has the great advantage to consider explicitly both the discreteness and autocorrelation characterizing this type of data. Moreover, extensions of the simple INAR(1) model to the multi-dimensional space make it possible to model more than one series simultaneously. However, existing models do not offer great flexibility for dependence modelling, allowing only for positive correlation. In this work, we consider a bivariate INAR(1) (BINAR(1)) process where cross-correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations. We mainly emphasize on the parametric case that arises under the assumption of Poisson marginals. Other marginal distributions are also considered. A short application on a bivariate financial count series illustrates the model.     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.     

Integer autoregressive models with structural breaks

Even though integer-valued time series are common in practice, the methods for their analysis have been developed only in recent past. Several models for stationary processes with discrete marginal distributions have been proposed in the literature. Such processes assume the parameters of the model to remain constant throughout the time period. However, this need not be true in practice. In this paper, we introduce non-stationary integer-valued autoregressive (INAR) models with structural breaks to model a situation, where the parameters of the INAR process do not remain constant over time. Such models are useful while modelling count data time series with structural breaks. The Bayesian and Markov Chain Monte Carlo (MCMC) procedures for the estimation of the parameters and break points of such models are discussed. We illustrate the model and estimation procedure with the help of a simulation study. The proposed model is applied to the two real biometrical data sets.     

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

ABSTRACT

New 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.     

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.     

Process capability analysis for serially dependent processes of Poisson counts

Process capability indices evaluate the actual compliance of a process with given external specifications in a single number. For the case of a process of independent and identically distributed Poisson counts, two types of index have been proposed and investigated in the literature. The assumption of serial independence, however, is quite unrealistic for practice. We consider the case of an underlying Poisson INAR(1) process which has an AR(1)-like autocorrelation structure. We show that the performance of the estimated indices is degraded heavily if serial dependence is ignored. Therefore, we develop approaches for estimating the process capability (both for the observation and innovation process), which explicitly consider the observed degree of autocorrelation. For this purpose, we introduce a new unbiased estimator of the innovations’ mean of a Poisson INAR(1) process and derive its exact as well as asymptotic stochastic properties. In this context, we also present new explicit expressions for the third- and fourth-order moments of a Poisson INAR(1) process. Then the capability indices and the performance of their estimators are analysed and recommendations for practice are given.     

Zero-modified power series distribution and its Hurdle distribution version

This paper presents a new family of distributions for count data, the so called zero-modified power series (ZMPS), which is an extension of the power series (PS) distribution family, whose support starts at zero. This extension consists in modifying the probability of observing zero of each PS distribution, enabling the new zero-modified distribution to appropriately accommodate data which have any amount of zero observations (for instance, zero-inflated or zero-deflated data). The Hurdle distribution version of the ZMPS distribution is presented. PS distributions included in the proposed ZMPS family are the Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial and Generalized Negative Binomial distributions. The paper also describes the properties and particularities of the new distribution family for count data. The distribution parameters are estimated via maximum likelihood method and the use of the new family is illustrated in three real data sets. We emphasize that the new distribution family can accommodate sets of count data without any previous knowledge on the characteristic of zero-inflation or zero-deflation present in the data.     

A Note on an Approximation of the Mobile Communications Expenditures Distribution Function Using a Mixture Model

Approximating the distribution of mobile communications expenditures (MCE) is complicated by zero observations in the sample. To deal with the zero observations by allowing a point mass at zero, a mixture model of MCE distributions is proposed and applied. The MCE distribution is specified as a mixture of two distributions, one with a point mass at zero and the other with full support on the positive half of the real line. The model is empirically verified for individual MCE survey data collected in Seoul, Korea. The mixture model can easily capture the common bimodality feature of the MCE distribution. In addition, when covariates were added to the model, it was found that the probability that an individual has non-expenditure significantly varies with some variables. Finally, the goodness-of-fit test suggests that the data are well represented by the mixture model.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

Modelling a non-stationary BINAR(1) Poisson process

ABSTRACT

Non-stationarity in bivariate time series of counts may be induced by a number of time-varying covariates affecting the bivariate responses due to which the innovation terms of the individual series as well as the bivariate dependence structure becomes non-stationary. So far, in the existing models, the innovation terms of individual INAR(1) series and the dependence structure are assumed to be constant even though the individual time series are non-stationary. Under this assumption, the reliability of the regression and correlation estimates is questionable. Besides, the existing estimation methodologies such as the conditional maximum likelihood (CMLE) and the composite likelihood estimation are computationally intensive. To address these issues, this paper proposes a BINAR(1) model where the innovation series follow a bivariate Poisson distribution under some non-stationary distributional assumptions. The method of generalized quasi-likelihood (GQL) is used to estimate the regression effects while the serial and bivariate correlations are estimated using a robust moment estimation technique. The application of model and estimation method is made in the simulated data. The GQL method is also compared with the CMLE, generalized method of moments (GMM) and generalized estimating equation (GEE) approaches where through simulation studies, it is shown that GQL yields more efficient estimates than GMM and equally or slightly more efficient estimates than CMLE and GEE.     

On Fisher’s dispersion test for integer-valued autoregressive Poisson models with applications

The integer-valued autoregressive (INAR) model has been widely used in diverse fields. Since the task of identifying the underlying distribution of time-series models is a crucial step for further inferences, we consider the goodness-of-fit test for the Poisson assumption on first-order INAR models. For a test, we employ Fisher’s dispersion test due to its simplicity and then derive its null limiting distribution. As an illustration, a simulation study and real data analysis are conducted for the counts of coal mining disasters, the monthly crime data set from New South Wales, and the annual numbers of worldwide earthquakes.     

On Shifted Geometric INAR(1) Models Based on Geometric Counting Series

Two types of shifted geometric integer valued autoregressive models of order one (SGINAR(1)) are proposed. Both are based on the thinning operator generated by counting series of i.i.d. geometric random variables. Their correlation properties are derived and compared. Also, regression and conditional variance are considered. Nonparametric estimators of model parameters are obtained and their asymptotic characterizations are given. Finally, these two models are applied to a real-life data set and they are compared to some referent INAR(1) models.     

Simultaneous confidence regions for the parameters of a Poisson INAR(1) model

The INAR(1) model (integer-valued autoregressive) is commonly used to model serially dependent processes of Poisson counts. We propose several asymptotic simultaneous confidence regions for the two parameters of a Poisson INAR(1) model, and investigate their performance and robustness for finite-length time series in a simulation study. Practical recommendations are derived, and the application of the confidence regions is illustrated by a real-data example.     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

Improved estimation for Poisson INAR(1) models

We consider the first-order Poisson autoregressive model proposed by McKenzie [Some simple models for discrete variate time series. Water Resour Bull. 1985;21:645–650] and Al-Osh and Alzaid [First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275], which may be suitable in situations where the time series data are non-negative and integer valued. We derive the second-order bias of the squared difference estimator [Weiß. Process capability analysis for serially dependent processes of Poisson counts. J Stat Comput Simul. 2012;82:383–404] for one of the parameters and show that this bias can be used to define a bias-reduced estimator. The behaviour of a modified conditional least-squares estimator is also studied. Furthermore, we access the asymptotic properties of the estimators here discussed. We present numerical evidence, based upon Monte Carlo simulation studies, showing that the here proposed bias-adjusted estimator outperforms the other estimators in small samples. We also present an application to a real data set.     

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.     

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.     

Diagnostic checks for integer-valued autoregressive models using expected residuals

Integer-valued time series models make use of thinning operators for coherency in the nature of count data. However, the thinning operators make residuals unobservable and are the main difficulty in developing diagnostic tools for autocorrelated count data. In this regard, we introduce a new residual, which takes the form of predictive distribution functions, to assess probabilistic forecasts, and this new residual is supplemented by a modified usual residuals. Under integer-valued autoregressive (INAR) models, the properties of these two residuals are investigated and used to evaluate the predictive performance and model adequacy of the INAR models. We compare our residuals with the existing residuals through simulation studies and apply our method to select an appropriate INAR model for an over-dispersed real data.