Negative Binomial Thinning Research Articles

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.

Generalized Random Environment INAR Models of Higher Order

The random environment integer-valued autoregressive models of higher orders introduced by Nastic et al. (J Time Ser Anal 37: 267–287, 2016) are generalized. Generalizations are made by relaxing two assumptions: the assumption about the equality of negative binomial thinning operators and the assumption that the maximal orders are equal for all the process random states. Some statistical properties of the introduced models are derived and discussed. Their unknown parameters are estimated by the Yule–Walker method and the performances of the obtained estimators are checked using simulated data sets. A possible application of the models on the real-life data is discussed at the end of the paper.

A generalised NGINAR(1) process with inflated‐parameter geometric counting series

SummaryIn this paper we propose a new stationary first‐order non‐negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer‐valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer‐valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.

A mixed thinning based geometric INAR(1) model

In this article a geometrically distributed integer-valued autoregressive model of order one based on the mixed thinning operator is introduced. This new thinning operator is defined as a probability mixture of two well known thinning operators, binomial and negative binomial thinning. Some model properties are discussed. Method of moments and the conditional least squares are considered as possible approaches in model parameter estimation. Asymptotic characterization of the obtained parameter estimators is presented. The adequacy of the introduced model is verified by its application on a certain kind of real-life counting data, while its performance is evaluated by comparison with two other INAR(1) models that can be also used over the observed data.

An integer-valued threshold autoregressive process based on negative binomial thinning

In this paper, we introduce an integer-valued threshold autoregressive process, which is driven by independent negative-binomial distributed random variables and based on negative binomial thinning. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators and corresponding iterative algorithms are investigated for both the cases that the threshold variable is known or not. Also, the asymptotic properties of the estimators are obtained. Finally, some numerical results of the estimates and a real data example are presented.

Estimation in a bivariate integer-valued autoregressive process

ABSTRACTA bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.

Random environment integer‐valued autoregressive process

An r states random environment integer‐valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non‐stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the k‐step‐ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real‐life counting data.

Some geometric mixed integer-valued autoregressive (INAR) models

In this paper, we introduce some mixed integer-valued autoregressive models of orders 1 and 2 with geometric marginal distributions, denoted by MGINAR(1) and MGINAR(2), using a mixture of the well-known binomial and the negative binomial thinning. The distributions of the innovation processes are derived and several properties of the model are discussed. Conditional least squares and Yule–Walker estimators are obtained, and some numerical results of the estimations are presented. A real-life data example is investigated to assess the performance of the models.

A combined geometric INAR(p) model based on negative binomial thinning

In this paper, we introduce a new (combined) integer-valued autoregressive model of order p with geometric marginal distributions, denoted by CGINAR(p), using the negative binomial thinning introduced by Ristić et al. [19]. Several properties of the model are constructed and discussed, including one-step ahead conditional statistical measures. Some methods for estimating the model parameters are considered and the asymptotic properties of the obtained estimators are derived. A real-life data example is investigated to assess the performance of the model.

A bivariate INAR(1) time series model with geometric marginals

In this paper we introduce a simple bivariate integer-valued time series model with positively correlated geometric marginals based on the negative binomial thinning mechanism. Some properties of the model are considered. The unknown parameters of the model are estimated using the modified conditional least squares method.

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.