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Behavior of Some Hypothesis Tests for the Covariance Matrix of High Dimensional Data
Comportamiento de algunas pruebas de hipótesis para la matriz de covarianza de datos de dimensión alta
DOI:
https://doi.org/10.15446/rce.v45n2.98550Keywords:
Hypothesis test, covariance matrix, high dimensional data, Tracy-Widom law, multivariate Gaussian data (en)Pruebas de hipótesis, matriz de covarianza, datos de dimensión alta, ley Tracy-Widom, datos Gaussianos multivariados (es)
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The study of the structure of the covariance matrix when the dimension of the data is much greater than the sample size (high dimensional data) is a complicated problem, since we have many unknown parameters and few data. Several hypothesis tests for the covariance matrix, in the high dimensional context and in the classical case (where the dimension of the data is less than the sample size), can be found in the literature. It has been of interest the tests for the null hypothesis that the covariance matrix of Gaussian data is equal or proportional to the identity matrix, considering the classical case as well as the high dimensional context. Since it is important to have a wide comparison between these tests found in the literature, and for some of them it is difficult to have theoretical results about their powers, in this work we compare several tests by simulations, in terms of the size and power of the test. We also present some examples of application with real high dimensional data found in the literature.
El estudio de la matriz de covarianza cuando la dimensión de los datos es mucho más grande que el tamaño de la muestra (datos de dimensión alta) es un problema complicado, ya que se tienen muchos parámetros desconocidos y pocos datos. Se pueden encontrar en la literatura varias pruebas de hipótesis para la matriz de covarianza, en el contexto de datos de dimensión alta y en el caso clásico (donde la dimensión de los datos es menor que el tamaño de la muestra). Han sido de interés las pruebas para la hipótesis nula de que la matriz de covarianza de datos Gaussianos es igual o proporcional a la matriz identidad, considerando el contexto clásico como el de dimensión alta. Ya que es importante tener una amplia comparación entre estas pruebas encontradas en la literatura, y para algunas de ellas es difícil tener resultados teóricos acerca de sus potencias, en este trabajo comparamos varias pruebas mediante simulaciones, en términos del tamaño y la potencia de la prueba. También presentamos algunos ejemplos de aplicación con datos de dimensión alta reales encontrados en la literatura.
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