Goodness-of-fit for distributions on metric spaces

D. Serrano, E. García-Portugués, I. Van Keilegom

We propose a general goodness-of-fit framework for distributions on separable metric spaces. Under suitable identifiability conditions, probability distributions are characterized by distance profiles, which motivates their use in goodness-of-fit testing, for simple and composite null hypotheses. For composite null hypotheses, parameter estimation is incorporated via a Bahadur-type expansion, and the asymptotic distribution of the empirical process for distance profiles is obtained under the null. We define test statistics based on this empirical process and derive their asymptotic null distributions. We further study the behavior of the proposed tests under fixed and local alternatives, establishing consistency results. The methodology is illustrated with simulation studies and real-data experiments.

Keywords: Goodness-of-fit, distance profiles, random objects.

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