Modelling of low count heavy tailed time series data consisting large number of zeros and ones

In this paper, we construct a new mixture of geometric INAR(1) process for modeling over-dispersed count time series data, in particular data consisting of large number of zeros and ones. For some real data sets, the existing INAR(1) processes do not fit well, e.g., the geometric INAR(1) process overestimates the number of zero observations and underestimates the one observations, whereas Poisson INAR(1) process underestimates the zero observations and overestimates the one observations. Furthermore, for heavy tails, the PINAR(1) process performs poorly in the tail part. The existing zero-inflated Poisson INAR(1) and compound Poisson INAR(1) processes have the same kind of limitations. In order to remove this problem of under-fitting at one point and over-fitting at others points, we add some extra probability at one in the geometric INAR(1) process and build a new mixture of geometric INAR(1) process. Surprisingly, for some real data sets, it removes the problem of under and over-fitting over all the observations up to a significant extent. We then study the stationarity and ergodicity of the proposed process. Different methods of parameter estimation, namely the Yule-Walker and the quasi-maximum likelihood estimation procedures are discussed and illustrated using some simulation experiments. Furthermore, we discuss the future prediction along with some different forecasting accuracy measures. Two real data sets are analyzed to illustrate the effective use of the proposed model.     

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.     

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.     

Detecting mean increases in Poisson INAR(1) processes with EWMA control charts

Processes of serially dependent Poisson counts are commonly observed in real-world applications and can often be modeled by the first-order integer-valued autoregressive (INAR) model. For detecting positive shifts in the mean of a Poisson INAR(1) process, we propose the one-sided s exponentially weighted moving average (EWMA) control chart, which is based on a new type of rounding operation. The s-EWMA chart allows computing average run length (ARLs) exactly and efficiently with a Markov chain approach. Using an implementation of this procedure for ARL computation, the s-EWMA chart is easily designed, which is demonstrated with a real-data example. Based on an extensive study of ARLs, the out-of-control performance of the chart is analyzed and compared with that of a c chart and a one-sided cumulative sum (CUSUM) chart. We also investigate the robustness of the chart against departures from the assumed Poisson marginal distribution.     

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.     

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.     

Integer-valued autoregressive models for counts showing underdispersion

The Poisson distribution is a simple and popular model for count-data random variables, but it suffers from the equidispersion requirement, which is often not met in practice. While models for overdispersed counts have been discussed intensively in the literature, the opposite phenomenon, underdispersion, has received only little attention, especially in a time series context. We start with a detailed survey of distribution models allowing for underdispersion, discuss their properties and highlight possible disadvantages. After having identified two model families with attractive properties as well as only two model parameters, we combine these models with the INAR(1) model (integer-valued autoregressive), which is particularly well suited to obtain auotocorrelated counts with underdispersion. Properties of the resulting stationary INAR(1) models and approaches for parameter estimation are considered, as well as possible extensions to higher order autoregressions. Three real-data examples illustrate the application of the models in practice.     

Inference for random coefficient INAR(k) with the occasional level shift random noise based on dual empirical likelihood

The INAR(k) model has been widely used in various kinds of fields. However, there are little discussions about the INAR(k) model with the occasional level shift random noise. In this paper, the maximum likelihood estimation of parameter based on martingale difference sequence is given, the log empirical likelihood ratio test statistic is obtained and the test statistic converges to chi-square distribution, we prove that the confidence region of the parameter is convex. Furthermore, the numerical simulation of the proposed INAR(k) model is given, which illustrates the effectiveness of the model. Then, the proofs of asymptotic results are given in the Appendix.     

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.     

Risk aggregation with dependence and overdispersion based on the compound Poisson INAR(1) process

Abstract

This paper considers an extension of the classical discrete time risk model for which the claim numbers are assumed to be temporal dependence and overdispersion. The risk model proposed is based on the first-order integer-valued autoregressive (INAR(1)) process with discrete compound Poisson distributed innovations. The explicit expression for the moment generating function of the discounted aggregate claim amount is derived. Some numerical examples are provided to illustrate the impacts of dependence and overdispersion on related quantities such as the stop-loss premium, the value at risk and the tail value at risk.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

Serial dependence and regression of Poisson INARMA models

Time series of counts occur in many fields of practice, with the Poisson distribution as a popular choice for the marginal process distribution. A great variety of serial dependence structures of stationary count processes can be modelled by the INARMA family. In this article, we propose a new approach to the INMA(q) family in general, including previously known results as special cases. In the particular case of Poisson marginals, we will derive new results concerning regression properties and the serial dependence structure of INAR(1) and INMA(q) models. Finally, we present explicit expressions for the distribution of jumps in such processes.     

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.     

A New Estimator of a Circular Median

The INARCH(1) model for overdispersed time series of counts has a simple structure, a parsimonious parametrization, and a great potential for applications in practice. We analyze two approaches to approximate the marginal process distribution: a Markov chain approach and the Poisson–Charlier expansion. Then approaches for estimating the two model parameters are discussed. We derive explicit expressions for the asymptotic distribution of the maximum likelihood and conditional least squares estimators. They are used for constructing simultaneous confidence regions, the finite-sample performance of which is analyzed in a simulation study. A real-data example from economics illustrates the application of the INARCH(1) model.     

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.     

Empirical bayesInterval estimates: An application to geographical epidemiology

Summary Empirical Bayes estimates have been advocated as an improvement for mapping rare diseases or health events aggregated in small areas. In particular different parametric approaches have been proposed for dealing with non-normal data, assuming that disease occurrencies follow non-homogeneous Poisson law, whose parameters are treated as random variables. This paper shows how to conduct a complete Empirical Bayes analysis under an exchangeable model in the context of Geographical Epidemiology. Three different approaches for defining confidence limits obtained using a parametric bootstrap are compared: method 1 relies only on the first and second moment of the bootstrapped posterior distributions; method 2 computes the centiles of the bootstrapped posteriors; method 3 equates to α the average of the probabilities derived from the estimated bootstrapped cumulative posterior distributions. The simple Poisson-Gamma formulation was used to model mortality data on Larynx Cancer in the Local Health Units of Tuscany (1980–82 males). Two areas of significant elevated risk are identified.     

Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models

Summary.  Integer-valued auto-regressive (INAR) processes have been introduced to model non-negative integer-valued phenomena that evolve over time. The distribution of an INAR( p ) process is essentially described by two parameters: a vector of auto-regression coefficients and a probability distribution on the non-negative integers, called an immigration or innovation distribution. Traditionally, parametric models are considered where the innovation distribution is assumed to belong to a parametric family. The paper instead considers a more realistic semiparametric INAR( p ) model where there are essentially no restrictions on the innovation distribution. We provide an (semiparametrically) efficient estimator of both the auto-regression parameters and the innovation distribution.     

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.     

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.     

Bootstrap-based bias corrections for INAR count time series

Integer-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.