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I need to do the similar kind of test as Kruskal-Wallis Test but for the multidimensional vector. Would you please kindly suggest such kind of test or provide some references for N-d case.

Thank you for your help.

Best Regards.

Tom
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If we assume that the component distributions vary only in their location (as per Kruskal-Wallis) then under the null hypothesis that the two vectors are equal, you can perform the Kruskal-Wallis procedure on the difference in the component sample means, since under the null, each such difference should have the same distribution.

If you don't want to rely on just location differences, you can use the Rust-Fligner modification of the Kruskal-Wallis, where you only need to assume that the difference in means statistics are all part of the same location-scale family.