Minimum Proper Loss Estimators for Parametric Models

In this paper, we propose a methodology for systematically deriving estimators minimizing proper loss functions defined on parametric statistical models, by restricting ourselves to losses taking a functional form which allows for straightforward proofs of key properties. Proving propriety is tantamount to deriving a pairwise divergence quantity between probability measures, admitting a natural interpretation of minimum proper loss estimators as divergence minimizers. We show that proper losses of varying complexity can be readily constructed given propriety of losses taking a rudimentary form, and that for many important models, verifying desired properties in the rudimentary case is immediate. As a special case, we derive computationally tractable estimators requiring only the first and second moments, verify strict propriety properties, and empirically confirm their utility though parameter estimation tasks using both controlled simulations and real-world meteorological network data sets. Comparisons against numerous standard estimators show that the proposed estimators are at least competitive with, and often markedly superior to all standard references, uniformly across tasks, data sets, and model classes, suggesting a strong alternative to standard benchmarks in a wide variety of estimation problems.