A Statistical Analysis of Adaptive Scalar Quantization based on Quantized Past Data

{ In this paper, we investigate from several angles the adaptive quantization problem based on quantized causal past data (cf. Ortega and Vetterli, 1994). First, when stationarity and ergodicity are assumed, it is shown that the marginal density can be estimated consistently based on quantized past data in the parametric case. Then, in the non-parametric ase, we analyze the piecewise linear approximation algorithm in Ortega and Vetterli (1994). We characterize the approximate density to which the algorithm leads if convergence is assumed and explains why and why not the algorithm works. Based on Bennett's distortion integral, we argue that estimating the interval partition of the optimal quantizer is an easier problem than density estimation. Moreover, the OV algorithm performs well for this interval partition estimation. Finally, we propose six procedures to improve the OV reproduction level estimates in the low resolution case and simulation studies are carried out in three examples to compare the proposals with the OV algorithm.