Computer simulation procedures have become a staple of research and development in many fields, including statistics. The generation of random number sequences is critical to these procedures. The validity of research results often depend on the underlying validity of the generator being used. In this work we develop the theory for the Empirical Spectral Test (EST). The EST is a class of tests of spatial uniformity based on a multi-dimensional Fourier transform of the empirical probability density function. The test can be applied to sequences from any random number generator, can be adapted to specific user requirements and has the added advantage that its computational complexity is relatively independent of the number of data points being tested.
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