Estimation of the Rate–Distortion Function

Motivated by questions in lossy data compression and by theoretical considerations, the problem of estimating the rate–distortion function of an unknown (not necessarily discrete-valued) source from empirical data is examined. The 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 demonstrating that in certain cases it fails to converge to the true rate–distortion function. The analysis of its performance is complicated by the fact that the rate–distortion function is not continuous in the source distribution; the underlying mathematical problem is closely related to the classical problem of establishing the consistency of maximum-likelihood estimators (MLEs). General consistency results are given for the plug-in 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, and for estimating the optimal reproduction distribution.