Asymptotically efficient quantizing

It is shown, under weak assumptions on the density function of a random variable and under weak assumptions on the error criterion, that uniform quantizing yields an output entropy which asymptotically is smaller than that for any other quantizer, independent of the density function or the error criterion. The asymptotic behavior of the rate distortion function is determined for the class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> th law loss functions, and the entropy of the uniform quantizer is compared with the rate distortion function for this class of loss functions. The extension of these results to the quantizing of sequences is also given. It is shown that the discrepancy between the entropy of the uniform quantizer and the rate distortion function apparently lies with the inability of the optimal quantizing shapes to cover large dimensional spaces without overlap. A comparison of the entropies of the uniform quantizer and of the minimum-alphabet quantizer is also given.