Abstract
The asymptotic behavior of the quantization error allows the definition of a dimension for a probability distribution P, the quantization dimension. This concept fits into standard geometric measure theory, as the quantization dimension is always between the Hausdorff and the box-counting dimension. It is operational in the sense that the empirical quantization error may be computed and used as an estimator of the quantization error of P. We study the empirical quantization error in case the number of prototypes increases with the size of the sample. Results of the consistency of the empirical quantization error and estimators of the quantization dimensions of distributions are given. The results depend on geometrical properties of optimal partitions, such as the eccentricity of the cells and the number of neighbors.
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