Services on Demand
Journal
Article
English (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mailIndicators
-
Cited by SciELO -
Access statistics
Related links
-
Cited by Google -
Similars in
SciELO -
Similars in Google
Share
Permalink
Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.45 no.2 Bogotá July/Dec. 2022 Epub Feb 01, 2023
https://doi.org/10.15446/rce.v45n2.98550
Artículos originales de investigación
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
1 División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Cunduacán, México
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 to test the null hypothesis that either the covariance matrix of Gaussian data is equal to the identity matrix or proportional to it, considering the cl 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.
Key words: Covariance matrix; High dimensional data; Hypothesis test; Multivariate Gaussian data; Tracy-Widom law
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 tiene una gran cantidad de 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). Ha sido de interés probar la hipótesis nula de que la matriz de covarianza de datos Gaussianos es igual a la matriz identidad o proporcional a ella, considerando el contexto clásico así 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.
Palabras clave: Datos de dimensión alta; Datos Gaussianos multivariados; Ley Tracy-Widom; Matriz de covarianza; Prueba de hipótesis
References
Anderson, T. W. (1984), An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, Inc. [ Links ]
Bai, Z., Jiang, D., Yao, J. F. & Zheng, S. (2009), 'Corrections to LRT on large-dimensional covariance matrix by RMT', The Annals of Statistics 37(6B), 3822-3840. [ Links ]
Cai, T. T. & Ma, Z. (2013), 'Optimal hypothesis testing for high dimensional covariance matrices', Bernoulli 19(5B), 2359-2388. [ Links ]
Chen, S. X., Zhang, L.-X. & Zhong, P.-S. (2010), 'Tests for high-dimensional covariance matrices', Journal of the American Statistical Association 105(490), 810-819. [ Links ]
Cortez-Elizalde, D. (2020), Pruebas de hipótesis para la matriz de covarianza poblacional de datos de dimensión alta, Tesis de Maestría, Universidad Juárez Autónoma de Tabasco, División Académica de Ciencias Básicas, Cunduacán, México. [ Links ]
Hallin, M. & Paindaveine, D. (2006), 'Semiparametrically efficient rank-based inference for shape, i. optimal rank-based tests for sphericity', The annals of statistics 34(6), 2707-2756. [ Links ]
Hastie, T., Tibshirani, R. & Friedman, J. H. (2009), Elements of statistical learning: data mining, inference, and prediction, Springer. [ Links ]
John, S. (1971), 'Some optimal multivariate tests', Biometrika 58(1), 123-127. [ Links ]
John, S. (1972), 'The distribution of a statistic used for testing sphericity of normal distributions', Biometrika 59(1), 169-173. [ Links ]
Johnstone, I. M. (2001), 'On the Distribution of the Largest Eigenvalue in Principal Components Analysis', The Annals of Statistics 29(2), 295-327. [ Links ]
Ledoit, O. & Wolf, M. (2002), 'Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size', The Annals of Statistics 30(4), 1081-1102. [ Links ]
Li, Z. & Yao, J. (2016), 'Testing the sphericity of a covariance matrix when the dimension is much larger than the sample size', Electronic Journal of Statistics 10(2), 2973-3010. [ Links ]
Ma, Z. (2012), 'Accuracy of the tracy-widom limits for the extreme eigenvalues in white wishart matrices', Bernoulli 18(1), 322-359. [ Links ]
Muirhead, R. J. (2005), Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc. [ Links ]
Nagao, H. (1973), 'On some test criteria for covariance matrix', The Annals of Statistics (1), 700-709. [ Links ]
Rosenwald, A., Wright, G., Chan, W. C, Connors, J. M., Campo, E., Fisher, R. I. & et al. (2002), 'The use of molecular profiling to predict survival after chemotherapy for diffuse large-B-cell lymphoma', New England Journal of Medicine 346(25), 1937-1947. [ Links ]
Ross, D. T., Scherf, U., Eisen, M. B., Perou, C. M., Rees, C, Spellman, P. & et al. (2000), 'Systematic variation in gene expression patterns in human cancer cell lines', Nature genetics 24(3), 227-235. [ Links ]
Srivastava, M. S. (2005), 'Some tests concerning the covariance matrix in high dimensional data', Journal of the Japan Statistical Society 35(2), 251-272. [ Links ]
Srivastava, M. S., Yan aginara, H. & Kubokawa, T. (2014), 'Tests for covariance matrices in high dimension with less sample size', Journal of Multivariate Analysis 130, 289-309. [ Links ]
Zou, C, Peng, L., Feng, L. & Wang, Z. (2014), 'Multivariate sign-based high-dimensional tests for sphericity', Biometrika 101(1), 229-236. [ Links ]
Received: September 2021; Accepted: May 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
