High-Ordered Integer-Valued Time Series Models with Harmonic Features

Examples of integer-valued series with periodic structure can include the monthly counts of claims of short-term disability benefits Freeland [1998 ], Fokianos [2012 ], the day-time and night-time road accidents Pedeli and Karlis [2011 ], Brijs et al. [2008 ], Karlis et al. [2008 ], the number of infected cases due to the outbreak of a virus Ferland et al. [2006 ], Chan et al. [2021 ] and, the monthly number of short-term unemployed people Monteiro et al. [2015 ]. In this book, in Chapter 1 , it can clearly be observed that in Figure 1.3 , that is in the COVID-19 new infection and death series in South Africa, there are huge fluctuations with a number of ups and downs causing oscillations and with repeated trend causing periodicity. On the other hand, in Figure 1.1 , the COVID-19 new infection and death series in Mauritius exhibit huge variability over the mean, thus causing severe over-dispersion due to the preponderance of zeros in the early stage of detection. The question of interest is therefore— How to model such series in the presence of the phenomena of periodicity, over-dispersion, presence of excess zeros, covariate specifications and others? Up to this extent, there exists integer-valued time series models of auto-regressive nature that handles the modeling of counting time series. However, these auto-regressive models are restricted as they cannot accommodate for the above features in an unified framework. Thus, it is proposed to bring forward a flexible high-ordered integer-valued auto-regressive time series process that has the ability to simultaneously consider periodicity via harmonic specification and all sources of over-dispersion through the proper specification of the components of the time series model, while considering the simple binomial thinning with constant coefficient.