Testing Equality of Several Correlation Matrices

Prueba de igualdad de varias matrices de correlación

In this article we show that the Kullback’s statistic for testing equality of several correlation matrices may be considered a modified likelihood ratio statistic when sampling from multivariate normal populations. We derive the asymptotic null distribution of L∗ in series involving independent chisquare variables by expanding L∗ in terms of other random variables and then inverting the expansion term by term. An example is also given to exhibit the procedure to be used when testing the equality of correlation matrices using the statistic L∗.

igualdad de varias matrices de correlación, puede ser considerado como un estadístico modificado del test de razón de verosimilitud cuando se muestrean poblaciones normales multivariadas. Derivamos la distribución asintótica nula de L* en series que involucran variables independientes chi-cuadrado, mediante la expansión de L* en términos de otras variables aleatorias y luego invertir la expansión término a término. Se da también un ejemplo para mostrar el procedimiento a ser usado cuando se prueba igualdad de matrices de correlación mediante el estadístico L*.

Testing Equality of Several Correlation Matrices Prueba de igualdad de varias matrices de correlación ARJUN K. GUPTA¹, BRUCE E. JOHNSON², DAYA K. NAGAR³

¹Bowling Green State University, Department of Mathematics and Statistics, Bowling Green, USA. Professor. Email: gupta@bgsu.edu
²Experient Research Group, Severna Park, USA. Researcher. Email: bruce.johnson@experientresearch.com
³Universidad de Antioquia, Facultad de Ciencias Exactas y Naturales, Instituto de Matemáticas, Medellín, Colombia. Professor. Email: dayaknagar@yahoo.com

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Abstract

In this article we show that the Kullbacks statistic for testing equality of several correlation matrices may be considered a modified likelihood ratio statistic when sampling from multivariate normal populations. We derive the asymptotic null distribution of L^* in series involving independent chi-square variables by expanding L^* in terms of other random variables and then inverting the expansion term by term. An example is also given to exhibit the procedure to be used when testing the equality of correlation matrices using the statistic L^\ast.

Key words: Asymptotic null distribution, Correlation matrix, Covariance matrix, Cumulants, Likelihood ratio test.

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Resumen

En este artículo se muestra que el estadístico L^* de Kullback, para probar la igualdad de varias matrices de correlación, puede ser considerado como un estadístico modificado del test de razón de verosimilitud cuando se muestrean poblaciones normales multivariadas. Derivamos la distribución asintótica nula de L^* en series que involucran variables independientes chi-cuadrado, mediante la expansión de L^* en términos de otras variables aleatorias y luego invertir la expansión término a término. Se da también un ejemplo para mostrar el procedimiento a ser usado cuando se prueba igualdad de matrices de correlación mediante el estadístico L^*.

Palabras clave: distribución asintótica nula, matriz de correlación, matriz de covarianza, razón de verosimilitud.

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References

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[Recibido en junio de 2012. Aceptado en julio de 2013]

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv36n2a04,
    AUTHOR  = {Gupta, Arjun K. and Johnson, Bruce E. and Nagar, Daya K.},
    TITLE   = {{Testing Equality of Several Correlation Matrices}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {237-258}
}

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