Estimation of the variance-covariance matrix for the Maximum Likelihood Estimator in Space-Time autoregressive model in the presence of missing data.
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Date
2025
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Universidad de Concepción
Abstract
Los datos espacio-temporales han cobrado una creciente relevancia en diversas disciplinas como la epidemiología, las ciencias sociales, los estudios ambientales, la meteorología y la agricultura. El análisis preciso de estas bases de datos es fundamental para una contar con modelos e inferencia confiables, especialmente en presencia de datos faltantes, lo que requiere el uso de algoritmos eficientes capaces de manejar escenarios complejos. El objetivo principal de esta investigación es estimar la matriz de varianza-covarianzas del estimador máximo verosímil en un modelo autorregresivo espacio-temporal con datos faltantes. Los errores estándar se obtuvieron invirtiendo la Matriz de información de Fisher (FIM) observada, calculada utilizando el método de Louis con aproximaciones Monte Carlo para evaluar esperanzas incalculables. Continuando el trabajo de Padilla et al. (2020), quienes utilizaron el filtro de Kalman y un algoritmo de Esperanza-maximización Generalizado para estimar y predecir bajo datos incompletos, evaluamos la precisión de los estimadores de los parámetros y comparamos los resultados obtenidos mediante el método de Louis con los errores estándar derivados del bootstrap espacio-temporal. Nuestros resultados destacan la precisión, estabilidad y ventajas computacionales del enfoque propuesto, especialmente bajo niveles moderados de datos faltantes. Esta contribución metodológica permite comprender las fortalezas y limitaciones de ambos enfoques, y ofrece una base sólida para futuras investigaciones y aplicaciones en modelos espacio temporales.
Space-time data have become increasingly important in diverse fields, including epidemiology, so cial sciences, environmental studies, meteorology, and agriculture. Accurate analysis of these types of databases is crucial for reliable modeling and inference, particularly when dealing with miss ing data, and therefore requires efficient algorithms that can handle complex scenarios. The main objective of this research is to estimate the variance-covariance matrix for the Maximum Likeli hood Estimator in a Space-Time Autoregressive model in the presence of missing data. Standard errors were obtained by inverting the observed Fisher Information Matrix (FIM), which is com puted via Louis’ method using Monte Carlo approximations to evaluate intractable expectations. Building on the research of Padilla et al. (2020), who employed the Kalman filter and the General ized Expectation-Maximization algorithm for accurate parameter estimation and prediction under incomplete data, we asses the precision of parameter estimates and compare the results obtained via Louis’ method with those using space-time bootstrap standard errors. Our findings highlight the accuracy, stability, and computational advantages of the proposed approach, particularly in the presence of moderate levels of missing data. This methodological contribution provides insight into the respective strengths and limitations of these two approaches, offering a foundation for fu ture research and applications in space-time models.
Space-time data have become increasingly important in diverse fields, including epidemiology, so cial sciences, environmental studies, meteorology, and agriculture. Accurate analysis of these types of databases is crucial for reliable modeling and inference, particularly when dealing with miss ing data, and therefore requires efficient algorithms that can handle complex scenarios. The main objective of this research is to estimate the variance-covariance matrix for the Maximum Likeli hood Estimator in a Space-Time Autoregressive model in the presence of missing data. Standard errors were obtained by inverting the observed Fisher Information Matrix (FIM), which is com puted via Louis’ method using Monte Carlo approximations to evaluate intractable expectations. Building on the research of Padilla et al. (2020), who employed the Kalman filter and the General ized Expectation-Maximization algorithm for accurate parameter estimation and prediction under incomplete data, we asses the precision of parameter estimates and compare the results obtained via Louis’ method with those using space-time bootstrap standard errors. Our findings highlight the accuracy, stability, and computational advantages of the proposed approach, particularly in the presence of moderate levels of missing data. This methodological contribution provides insight into the respective strengths and limitations of these two approaches, offering a foundation for fu ture research and applications in space-time models.
Description
Tesis presentada para optar al grado de Magíster en Estadística.
Keywords
Mathematical statistics, Expectation-maximization algorithms, Multivariate analysis