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Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.44 no.1 Bogotá Jan./June 2021 Epub Feb 25, 2021
https://doi.org/10.15446/rce.v44n1.84779
Original articles of research
On Some Statistical Properties of the Spatio-Temporal Product Density
Sobre algunas propiedades estadísticas de la densidad producto espacio-temporal
1Ingeniería Ambiental, Facultad de Ingeniería, Universidad de Antioquia, Medellín, Colombia
2Escuela de Estadística, Facultad de Ciencias, Universidad Nacional de Colombia, Medellín, Colombia
3Departamento de Matemáticas, Escuela Superior de Tecnología y Ciencias Experimentales, Universitat Jaume i, Castellón, España
4Dipartimento di Scienze Economiche, Aziendali e Statistiche, Università degli Studidi Palermo, Palermo, Italy
We present an extension of the non-parametric edge-corrected Ohser-type kernel estimator for the spatio-temporal product density function. We derive the mean and variance of the estimator and give a closed-form approximation for a spatio-temporal Poisson point process. Asymptotic properties of this second-order characteristic are derived, using an approach based on martingale theory. Taking advantage of the convergence to normality, confidence surfaces under the homogeneous Poisson process are built. A simulation study is presented to compare our approximation for the variance with Monte Carlo estimated values. Finally, we apply the resulting estimator and its properties to analyse the spatio-temporal distribution of the invasive meningococcal disease in the Rhineland Regional Council in Germany.
Key words: Envelope; Invasive meningococcal disease; Lindeberg condition; Ohser-type estimator; Second-order product density
En este artículo, presentamos un estimador para la función de densidad producto de un patrón de puntos en espacio-tiempo. Este estimador es una extensión del estimador no paramétrico de Ohser, el cuál está basado en una función Kernel y ponderado por un corrector de borde. Deducimos la media y la varianza del estimador y, a su vez, damos una aproximación analítica para el caso de un patrón Poisson (completamente aleatorio). Adicionalmente, estudiamos ciertas propiedades asintóticas de nuestro estimador utilizando un enfoque basado en la teoría de martingalas y construimos superficies de confianza para el caso de aleatoriedad completa. Presentamos un estudio de simulación para comparar nuestra aproximación de la varianza con los valores estimados a través del método Monte Carlo. Finalmente, utilizamos nuestro estimador para analizar la distribución espacio-temporal de los registros de una enfermedad meningocócica invasiva en la provincia del Rin en Alemania.
Palabras clave: Condición de Lindeberg; Densidad de producto de segundo orden; Envoltura; Enfermedad meningocócica invasiva; Estimador de tipo Ohser
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