The Hellinger Distance and its Applications to Hypothesis Testing and Model Uncertainty

The problem of determining the probability model (distribution) that generates observed data commonly arises in econometrics, decision making under ambiguity, robust control, and allied fields. We develop novel procedures that use the Hellinger distance to distinguish distributions based on data. Using the theory of large deviations, contiguity, and local asymptotic normality, we extend our results to accommodate a large class of parametric families, which are frequently used for modeling uncertainty. Our results do not require the assumption of mutual absolute continuity and allow a larger set of competing probability models to influence economic decisions. Consequently, we propose the Hellinger distance and some related quantities, rather than the extensively used Kullback-Leibler (KL) divergence, for comparing probability models and quantifying probabilistic uncertainty