Estimation of the Rate-Distortion Function

Motivated by questions in lossy data compression and by theoretical considerations, this paper examines the problem of estimating the rate-distortion function of an unknown (not necessarily discretevalued) source from empirical data. The main focus is the behavior of the so-called “plug-in” estimator, which is simply the rate-distortion function of the empirical distribution of the observed data. Sufficient conditions are given for its consistency, and examples are provided to demonstrate that in certain cases it fails to converge to the true rate-distortion function. The analysis of the performance of the plugin estimator is somewhat surprisingly intricate, even for stationary memoryless sources; the underlying mathematical problem is closely related to the classical problem of establishing the consistency of the maximum likelihood estimator in a parametric family. General consistency results are given for the plugin estimator applied to a broad class of sources, including all stationary and ergodic ones. A more general class of estimation problems is also considered, arising in the context of lossy data compression when the allowed class of coding distributions is restricted; analogous results are developed for the plug-in estimator in that case. Finally, consistency theorems are formulated for modified (e.g., penalized) versions of the plug-in estimator, and for estimating the optimal reproduction distribution.