On The Poisson Difference Distribution Inference and Applications

The distribution of the difference between two independent Poisson random variables involves the modified Bessel function of the first kind. Using properties of this function, maximum likelihood estimates of the parameters of the Poisson difference were derived. Asymptotic distribution property of the maximum likelihood estimates is discussed. Maximum likelihood estimates were compared with the moment estimates in a Monte Carlo study. Hypothesis testing using likelihood ratio tests was considered. Some new formulas concerning the modified Bessel function of the first kind were provided. Alternative formulas for the probability mass function of the Poisson difference (PD) distribution are introduced. Finally, two new applications for the PD distribution are presented. The first is from the Saudi stock exchange (TASI) and the second is from Dallah hospital

Periodic INAR(1) Models with Skellam-Distributed Innovations

In this paper, an integer-valued autoregressive model of order one (INAR(1)) with time-varying parameters and driven by a periodic sequence of innovations is introduced. The proposed INAR(1) model is based on the signed thinning operator defined by Kachour and Truquet (2011) and conveniently adapted to the periodic case. Basic notations and definitions concerning the periodic signed thinning operator are provided. Based on this thinning operator, Chesneau and Kachour (2012) established a signed INAR(1) model. Motivated by the work of Chesneau and Kachour (2012), we introduce a periodic model, denoted by S-PINAR(1), with period s. In contrast to conventional INAR(1) models, these models are defined in Z allowing for negative values both for the series and its autocorrelation function. For a proper Z-valued time series, a distribution for the innovation term defined on Z is required. The S-PINAR(1) model assumes a specific innovation distribution, the Skellam distribution. Regarding parameter estimation, two methods are considered: conditional least squares and conditional maximum likelihood. The performance of the S-PINAR(1) model is assessed through a simulation study.publishe

Generalized seasonal autoregressive integrated moving average models for count data with application to malaria time series with low case numbers

With the renewed drive towards malaria elimination, there is a need for improved surveillance tools. While time series analysis is an important tool for surveillance, prediction and for measuring interventions' impact, approximations by commonly used Gaussian methods are prone to inaccuracies when case counts are low. Therefore, statistical methods appropriate for count data are required, especially during "consolidation" and "pre-elimination" phases.; Generalized autoregressive moving average (GARMA) models were extended to generalized seasonal autoregressive integrated moving average (GSARIMA) models for parsimonious observation-driven modelling of non Gaussian, non stationary and/or seasonal time series of count data. The models were applied to monthly malaria case time series in a district in Sri Lanka, where malaria has decreased dramatically in recent years.; The malaria series showed long-term changes in the mean, unstable variance and seasonality. After fitting negative-binomial Bayesian models, both a GSARIMA and a GARIMA deterministic seasonality model were selected based on different criteria. Posterior predictive distributions indicated that negative-binomial models provided better predictions than Gaussian models, especially when counts were low. The G(S)ARIMA models were able to capture the autocorrelation in the series.; G(S)ARIMA models may be particularly useful in the drive towards malaria elimination, since episode count series are often seasonal and non-stationary, especially when control is increased. Although building and fitting GSARIMA models is laborious, they may provide more realistic prediction distributions than do Gaussian methods and may be more suitable when counts are low

A new INARMA(1,1) model with Poisson Marginals

We suggest an INARMA(1, 1) model with Poisson marginals which extends the INAR(1) in a similar way as the INGARCH(1, 1) does for the INARCH(1) model. The new model is equivalent to a binomially thinned INAR(1) process. This allows us to obtain some of its stochastic properties and use inference methods for hidden Markov models. The model is compared to various other models in two case studies.Comment: This is a pre-print (submitted version before peer review) of a contribution in Steland, A., Rafajlowicz, E., Okhrin, O. (Eds.): Stochastic Models, Statistics and Their Applications, p. 323-333, published by Springer Nature Switzerland, 2019. The final authenticated version is available at https://doi.org/10.1007/978-3-030-28665-1_2

Zero truncated Poisson integer-valued AR(1) model

Zero truncated Poisson distribution, Integer-valued autoregressive processes, Estimation,

Some characterizations of discrete distributions based on weak records

weak records, geometric distribution, partial independence, identical distribution, characterizations of discrete distributions, difference equations,