Copula-based Analysis for Dependent Count Data
Open AccessThe analysis of count time series is challenging due to the nonuniqueness of dependence measure caused by the existence of ties. Various methods have been developed but most of them rely on restrictive parametric distributional assumptions. In this dissertation, we aim to develop copula-based methods for analyzing dependent count data, including count time series, multivariate count time series and spatio-temporal count data. In the first project, we propose a copula-based Markov model for analyzing count time series, where the temporal dependence is captured by a parametric copula. One challenge in using copula to analyze count data is due to the identifiability issue, which arises from the discrepancy of using a continuous copula function to characterize discrete distributions. We show that the identifiability can be ensured in the regression setup under one sufficient condition. Resolving the identifiability issue allows us to develop a cross-validation procedure to select the appropriate copula to capture different types of temporal dependence, leading to more flexibility in modeling. We propose an estimation procedure and establish the asymptotic properties of the proposed estimator. The proposed method is data adaptive and computationally efficient to capture the temporal dependence. It also provides a convenient way to construct both point and interval predictions at a future time point. Through a simulation study and the analyses of a baseball data and a COVID-19 daily death data, we show that our method produces more stable point and interval predictions than existing methods based on Gaussian copula and autoregression. In the second project, we focus on the analysis of multivariate count time series. We consider using the copula-linked Dvine (CDvine) to simultaneously capture the temporal and cross-sectional dependence. To reduce the computational burden, a sequential selection approach is used to select the order and bivariate copulas of each univariate Dvine, and a two-stage maximum likelihood estimator is proposed for estimation. We conduct simulation studies to examine the performance of the sequential model selection procedure and demonstrate the flexibility of the CDvine. The application to a Covid-19 death data shows that the proposed method leads to better prediction over existing approaches.The third project focuses on the analysis of spatio-temporal count data with zero-inflations. In the first two projects, the marginal distributions of the count data are estimated by fitting parametric regressions such as Poisson regression, which may be restrictive in some applications. To gain more flexibility, we extend the copula-based idea in the first two projects, and develop a new procedure based on nonparametric estimation of the marginal distribution through gradient boosting model (GBM). We consider zero-inflated generalized Poisson (ZIGP) distribution to account for zero-inflation and potential over-dispersion. Instead of specifying the link functions, we model the parameters non-parametrically using GBM. The spatial and temporal dependence is captured by CDvine and the Mat\'{e}rn correlation function. Simulation study is conducted to assess the performance of the proposed nonparametric estimation of the marginal distribution and the prediction accuracy of the proposed procedure over existing methods. This proposed method is applied to a Covid-19 mortality data from Northeastern US, and we observe significant improvement over the method based on parametric estimation of the marginal distribution.
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