Point processes and integer-valued time series

This thesis explores two primary themes across five scientific papers: Integer-value time series and their relationship with classical point processes. The first part of the thesis focuses on the development and application of Integer-valued autoregressive (INAR) models, extending from univariate to multivariate cases, with applications in financial and insurance count data. In Paper A, we introduce a new family of binomial-mixed Poisson INAR model of order one, INAR(1), by incorporating a mixed Poisson component to the innovation of the classical Poisson INAR(1). This allows for the capture of overdispersion and serial correlation evident in financial count data. Furthermore, we explore its distributional properties, estimation procedure and asymptotic properties and apply the model to iceberg count data from financial system. In Paper B, we extending beyond univariate case, introducing a novel family of multivariate mixed Poisson-Generalized Inverse Gaussian INAR(1), MMPGIG-INAR(1), regression models for modelling multivariate count time series. This family of models can accommodate a wide range of dispersion and cross-sectional correlation structures due to the flexibility in the parameter setting of the Generalized Inverse Gaussian. We then illustrate different members of the MMPGIG-INAR(1) through applying the model to Local Government Property Insurance Fund data from the state of Wisconsin. In Paper C, we develop novel Expectation-Maximization estimation algorithm for maximum likelihood estimation of bivariate mixed Poisson INAR(1) model. This method is readily extensible to the multivariate case. We examine three different mixing densities, univariate gamma, bivariate Lognormal and bivariate copula and demonstrate the algorithm through fitting the same used in Paper B. The second part of the thesis shifts focus to integer-valued approximation of classical point processes and applications of point process on Covid data modelling. In Paper D, we represent the Cox process and the dynamic contagion process, which is a Hawkes process whose immigration part is a Cox process, as limit of timeseries based point processes, namely integer-valued moving average model (INMA) and Integer-valued Autoregressive Moving Average model (INARMA). This would potentially facilitate the statistical inference of classical point processes. In Paper E, we propose a new type of univariate and bivariate Integer-valued autoregressive model of order one, INAR(1), to approximate univariate and bivariate linear birth and death process with constant rates. Due to the simplicity of Markov structure of INAR model, we demonstrate through simulation study that the parameters of linear birth and death process can be estimated through Quasi-likelihood function of INAR model

Multivariate integer-valued autoregressive models applied to earthquake counts

In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer-valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial contagion is significant in both directions. Furthermore, ignoring cross autocorrelation can underestimate the potential for high numbers of occurrences over the short-term. Our overall findings seem to further confirm Parsons & Velasco (2001)

First-order multivariate integer-valued autoregressive model with multivariate mixture distributions

The univariate integer-valued time series has been extensively studied, but literature on multivariate integer-valued time series models is quite limited and the complex correlation structure among the multivariate integer-valued time series is barely discussed. In this study, we proposed a first-order multivariate integer-valued autoregressive model to characterize the correlation among multivariate integer-valued time series with higher flexibility. Under the general conditions, we established the stationarity and ergodicity of the proposed model. With the proposed method, we discussed the models with multivariate Poisson-lognormal distribution and multivariate geometric-logitnormal distribution and the corresponding properties. The estimation method based on EM algorithm was developed for the model parameters and extensive simulation studies were performed to evaluate the effectiveness of proposed estimation method. Finally, a real crime data was analyzed to demonstrate the advantage of the proposed model with comparison to the other models

Bivariate binomial autoregressive models

This paper introduces new classes of bivariate time series models being useful to fit count data time series with a finite range of counts. Motivation comes mainly from the comparison of schemes for monitoring tourism demand, stock data, production and environmental processes. All models are based on the bivariate binomial distribution of Type II. First, a new family of bivariate integer-valued GARCH models is proposed. Then, a new bivariate thinning operation is introduced and explained in detail. The new thinning operation has a number of advantages including the fact that marginally it behaves as the usual binomial thinning operation and also that allows for both positive and negative cross-correlations. Based upon this new thinning operation, a bivariate extension of the binomial autoregressive model of order one is introduced. Basic probabilistic and statistical properties of the model are discussed. Parameter estimation and forecasting are also covered. The performance of these models is illustrated through an empirical application to a set of rainy days time series collected from 2000 up to 2010 in the German cities of Bremen and Cuxhaven.publishe

Inference for bivariate integer-valued moving average models based on binomial thinning operation

Time series of (small) counts are common in practice and appear in a wide variety of fields. In the last three decades, several models that explicitly account for the discreteness of the data have been proposed in the literature. However, for multivariate time series of counts several difficulties arise and the literature is not so detailed. This work considers Bivariate INteger-valued Moving Average, BINMA, models based on the binomial thinning operation. The main probabilistic and statistical properties of BINMA models are studied. Two parametric cases are analysed, one with the cross-correlation generated through a Bivariate Poisson innovation process and another with a Bivariate Negative Binomial innovation process. Moreover, parameter estimation is carried out by the Generalized Method of Moments. The performance of the model is illustrated with synthetic data as well as with real datasets.publishe