A sphericity test in high-dimensional data using random projections

J. A. Cuesta Albertos, P. Navarro Esteban, A. Nieto Reyes

Testing sphericity, which refers to the situation when the covariance matrix is proportional to the identity, is a problem which has received some attention in the literature. Indeed, various tests of sphericity for high-dimensional data have been recently proposed. Some of them require to estimate the covariance matrix. The higher the dimension, the more complex the estimation of the matrix due to the increase of the number of parameters which have to be estimated in comparison with the sample size. In order to avoid estimating this matrix, a sequential procedure to test the sphericity of the data based on random projections is proposed. At each stage, the method consists in:
1.Select a random basis and project the data on each vector of it where a measure of dispersion is computed
2.Calculate the quotient between the largest and the smallest measure of dispersion
The decision of stopping the sequential process or to get an additional random basis depends on the value of the quotient.

Palabras clave: Sphericity Test, High-dimensional, Random Projections

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