Parameter Estimation and Hypothesis Testing of GTW Compound Correlated Bivariate Poisson Regression Model: A Theoretical Development

Abstract

Each observation location and time possesses distinct characteristics, reflecting heterogeneity at every observation point, both spatially and temporally. This condition renders the Compound Correlated Bivariate Poisson Regression (CCBPR) model inadequate for representing data dynamics that exhibit spatial and temporal heterogeneity. To address this limitation, the Geographically and Temporally Weighted Compound Correlated Bivariate Poisson Regression (GTWCCBPR) model is employed, which allows parameter variation across locations and time periods. This model also incorporates the exposure variable as a weighting factor to adjust for differences in risk across observational units. This study aims to estimate the parameters of the GTWCCBPR model using the Maximum Likelihood Estimation (MLE) approach. Due to the complex structure of the model, the log-likelihood function does not yield a closed-form solution. Therefore, parameter estimation is performed using the iterative Berndt-Hall-Hall-Hausman (BHHH) algorithm. Subsequently, hypothesis testing is conducted to evaluate the parameter similarity between the global model (CCBPR) and the spatiotemporal model (GTWCCBPR), as well as to assess the significance of each predictor variable. Simultaneous testing is carried out using the Maximum Likelihood Ratio Test (MLRT), while partial testing is conducted using the Z-test. The scope of this study is limited to theoretical formulation and methodological development, without empirical or simulation-based validation. Future research may extend this work by applying the GTWCCBPR model to practical datasets exhibiting spatio-temporal heterogeneity, particularly in areas such as public health (e.g., maternal and postneonatal mortality), epidemiology, or regional planning.