Target-Oriented Statistical Compression: Sufficiency, Reverse Martingales, and Sequential Monitoring

Abstract

Statistical procedures rarely retain all features of the observed data. A sufficient statistic removes information irrelevant to a parameter; a maximum likelihood estimate compresses an empirical objective into an optimizing point; and a hidden state in a sequential model compresses past observations into a learned representation. This article develops these practices under the unified notion of \emph{target-oriented statistical compression}: a useful summary preserves what matters for an inferential, predictive, or decision-relevant target, rather than every detail of the realized data path. The central object is the conditional target process \(M_n=\E(Z\given\G_n)\), where \(Z\) is the target and \(\G_n=\sigma(T_n)\) is the information retained by the compression map \(T_n\). When \((\G_n)\) is a decreasing filtration, \((M_n)\) is a reverse martingale with limit \(M_\infty=\E(Z\given\G_\infty)\). Exact sufficiency corresponds to lossless compression, while approximate summaries such as penalized estimators, principal components, and neural-network hidden states produce reverse quasi-martingale defects measuring coherence loss across compression levels. The diagnostic \(r_n=|M_n-M_{n-1}|\) is treated as an observable stability proxy, not as an unbiased estimator of the theoretical defect. Boundary degeneracy in sequential binary problems is developed as a central application. Practical boundary claims require joint assessment of boundary closeness, uncertainty control, and trajectory stability. The companion paper \citet{chang2025rm} develops the corresponding stopping procedures, finite-sample bounds, and numerical evidence; the present paper provides the broader theoretical infrastructure and extends the framework to Gaussian, Poisson, and quasi-martingale monitoring problems.