## 1 Introduction
[21] developed an alternative GWBPIGR model, namely the Mixed Geographically Weighted Bivariate Poisson Inverse Gaussian Regression (MCWBPIGR) model. In all those studies, parameter estimation was carried out using the MLE and NR methods and the spatial kernel weighting function. The purpose of this study was to develop a spatial based-point model for multivariate count data with PIG distribution, namely the Geographically Weighted Multivariate Poisson Inverse Gaussian Regression (GWMPIGR) model. The proposed model was developed based on the MPIGR model, which was developed by [10]. There are two main differences in the model that we developed in this study compared to those that have been developed in previous studies. First, by considering the probability of each occurrence in each research unit, several exposures were used for the case study in this study. We have previously studied univariance, via case study, using the PIGR model with exposure, which performed better than the PIGR model without exposure [9]. Second, we used the iterative Berndt-Hall-Hall-Hausman (BHHH) methods for parameter estimation. Compared to the NR method, the BHHH method does not need the second derivative of the log-likelihood function with respect to each parameter in the model. This second derivative has the potential to cause problems such as the Hessian matrix and may not be positively definite. It also typically results in a complex nonlinear function that needs to be solved, meaning that the expected value remains unknown. Therefore, the second derivative was eliminated, and a simpler computation with guaranteed convergence was obtained, the BHHH method [22; 23]. For the real-world application of the proposed model, we analyzed one of the health problems in Indonesia, namely the high number of infant, under-5 and maternal deaths. Numerous studies have been conducted to produce effective policies to reduce the mortality rate of the three groups of individuals. This problem is also one of the targets in the third goal of the SDGs for 2030, living a healthy and prosperous life, which can partially be achieved by reducing the neonatal mortality rate to 12 per 1000 live births and the under-5 mortality rate by 25 per 1000 children under five in the middle of the same year. Furthermore, the maternal mortality rate should also be reduced to less than 70 per 100,000 live births. Indonesia, the fourth most populous country in the world, is struggling to achieve the 2030 SDG targets for maternal and under-5 mortality. Based on the Indonesian Demographic and Health Survey (IDHS) in 2017, the Maternal Mortality Rate (MMR) was 305 per 100,000 live births. In 2020, the Infant Mortality Rate (IMR) in Indonesia was 17.6 per 1000 live births and the under-5 Mortality Rate (U5 MR) was 23 per 1000 live births [24,25]. East Java is one of the cities in Indonesia with the highest number of infant, under-5 and maternal deaths compared to other regions in Indonesia. Therefore, it is necessary to study the causal factors to reduce infant, under-5 and maternal mortality in East Java. In terms of the factors that affect infant, under-5 and maternal mortality, various aspects of life such as socioeconomic status, health facilities, environment, education and culture play equally important roles. Several factors are often the closest factor or the direct cause of the death. These factors include maternal factors, nutrition and disease control in infant and children under five, while bleeding and hypertension are the main causes of death in mothers [26,27]. In 2018, Ref. [28] conducted a study on the factors of neonatal mortality in Indonesia and found that low birth weight (LBW), giving birth to children too close together, medical personnel and postnatal/postpartum visits have a significant effect on neonatal mortality in Indonesia. In this study, we use four GWMPIGR models, the exposure variable and the spatial weighting function.