Sequential Functional Quantization

We consider the problem of lossy estimation of an arbitrary smooth function of correlated data in a stream. In this problem, a user sequentially observes correlated random variables and wants to construct an estimate of the specified function so that the mean squared estimation error is small. Techniques from high resolution quantization theory are applied and expanded for this problem, and the optimal distortionrate exponent for companding quantization is determined. In the process, connections are established to sufficient statistics and to sensitivity matrices, as introduced by Linder et al. in the context of companding quantization under non-difference distortion measures. These results are applied to several example statistical functions, including the sample mean, sample variance, and the p-th order statistic.