Integrated powers of densities of one- or two-multidimensional random variables appear in a variety of problems in mathematical statistics, information theory, and computer science. We study U-statistic estimators for a class of such integral functionals based on the epsilon-close vector observations in the corresponding independent and identically distributed samples. We show some asymptotic properties of these estimators (e.g., consistency and asymptotic normality). The results can be used in a variety of problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).
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