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In many situations, when dealing with several populations, equality of the covariance operators is assumed.
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Schott, J. (2007). A test for the equality of covariance matrices when the dimension is large relative to the sample sizes
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Fremdt, S., Steinbach, J. G., Horváth, L. and Kokoszka, P. (2013). Testing the equality of covariance operators in functional samples
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Pigoli, D., Aston, J. A., Dryden, I. and Secchi, P. (2014). Distances and inference for covariance operators
2014
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