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Abstract: Count time series are by now widely encountered in practice. Motivated by recent advances in latent Gaussian count time series modeling in Jia et al. , we aim to establish some asymptotic theory and more general modeling frameworks for latent Gaussian count times series models.
Our first task, which constitutes the bulk of this writeup, constructs seasonal/periodic count time series by replacing the stationary latent Gaussian process and marginal distribution in the stationary count construction with a seasonal/periodic Gaussian process and a seasonal marginal distribution requirement. The mean and covariance structure of the resulting periodic count series is derived. Connections to periodic latent Gaussian process, including PARMA and SARMA models, are used to drive the discourse. The behavior of statistics estimated by particle filtering approximations of the model's likelihood is given. Application to the number of rainy days in successive weeks in Seattle, Washington over 20 years is detailed.
We then move to proposing two additional projects to complete this dissertation. In particular, the first projects aims to prove an asymptotic limit theory for the case where the latent Gaussian process is a causal autoregression. Here, we seek to show that the likelihood estimators are consistent and asymptotically normally.
Finally, we extend the methods to multivariate settings. Here, we propose to develop multivariate particle filtering methods for parameter inference.