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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.54 no.2 Bogotá jul./dez. 2020 Epub 05-Mar-2021
https://doi.org/10.15446/recolma.v54n2.93845
Artículos originales
A Glivenko-Cantelli Bootstrap Theorem for the Foster-Greer-Thorbecke Poverty Index
Un Teorema Glivenko-Cantelli Bootstrap para la Medida de Pobreza de Foster-Greer-Thorbecke
1 Universidad Austral, Rosario, Argentina
2 Universidad Industrial de Santander, Bucaramanga, Colombia
3 Basque Center for Applied Mathematics, Bilbao, España
We assume the Foster-Greer-Thorbecke (FGT) poverty index as an empirical process indexed by a particular Glivenko-Cantelli class or collection of functions and define this poverty index as a functional empirical process of the bootstrap type, to show that the outer almost sure convergence of the FGT empirical process is a necessary and sufficient condition for the outer almost sure convergence of the FGT bootstrap empirical process; that is: both processes are asymptotically equivalent respect to this type of convergence.
Keywords: Foster-Greer-Thorbecke poverty index; convergence of empirical processes; Glivenko-Cantelli classes; bootstrap empirical processes
Asumimos el indicador de pobreza de Foster-Greer-Thorbecke (FGT) como un proceso empírico indexado por una particular clase o colección de funciones Glivenko-Cantelli y definimos este indicador de pobreza como un proceso empírico funcional del tipo bootstrap, para probar que la convergencia casi segura exterior del proceso empírico FGT es una condición necesaria y suficiente para la convergencia casi segura exterior del proceso empírico bootstrap FGT; esto es: ambos procesos son asintóticamente equivalentes respecto de este tipo de convergencia.
Palabras clave: Indicador de pobreza de Foster-Greer-Thorbecke; convergencia de procesos empíricos; clases Glivenko-Cantelli; procesos empíricos bootstrap
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Received: February 06, 2020; Accepted: August 01, 2020