Statistical modelling of time series of counts for a new class of mixture distributions / Khoo Wooi Chen - Khoo, Wooi Chen (2016)

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Integer-valued correlated stochastic processes, which we often meet in the real world, are of major concern in many natural and social sciences. The classical continuous time series models which contain scalar multiaplication are not able to represent count data since the integer nature of the data is not preserved. Therefore, the formulations of discrete-valued time series models for count data are apparently of significance. Much effort has been expended in the past few decades to construct discrete-valued time series models. Nevertheless, the hunt for better models is still ongoing due to the need to improve or sharpen the statistical analysis. This thesis proposes a new mixture model, the mixture of Pegram and thinning integer-valued autoregressive (MPT) processes, which is the combination of current discrete-valued time series operators. The statistical and regression properties, parameter estimation, forecasting, and graphical analysis for the new model have been examined. Model selection based upon the Akaike Information Criterion has been performed. Extensions to the moving average (MA) and autoregressive moving average (ARMA) models have also been considered. Important properties such as reversibility and regression are then discussed. The extension to the qth-order MPT process has also been investigated in the study. Previous studies have emphasized the Poisson sequence as it is an infinitely divisible distribution. In this thesis, it is shown that proposed model is able to deal with infinitely and non-infinitely divisible distributions with simpler expressions. Furthermore, the proposed MPT model is able to handle multimodality and has better performance than the current discretevalued time series models. The available forecasting method based on the conditional expectation may not be appropriate for integer-valued time series models. Thus coherent forecasting, which is based upon the k-step ahead conditional mean, median, mode and distribution, is considered. For low count series the k-step ahead conditional distribution of the MPT model practically exhibits better performance than the other models. The iv score functions and information matrix have been derived to measure the asymptotic standard errors and to analyze the variance-covariance relationship among the parameters. Parameter estimation with the maximum likelihood estimation via the Expectation-Maximization algorithm is discussed and compared with the conditional least squares method. Finally, some real life data sets from different disciplines have been applied to illustrate the analyses. The thesis is concluded with some recommendations for future work.