In this paper we investigate the dimensional structure of probability distributions on Euclidean space and characterize a class of regular distributions. We obtain a consistent estimator of dimension based on a nearest neighbor statistic and in addition obtain asymptotic confidence intervals for dimension in the case of regular distributions. Although many examples of point estimation of dimension have recently appeared in the literature on chaotic attractors in dynamical systems, questions of consistency and interval estimation have not previously been addressed systematically.

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Keywords asymptotic confidence intervals, extreme value distribution, Hausdorff dimension, nearest neighbor statistic, point estimation
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Journal Journal of Multivariate Analysis
Cutler, C.D. (C.D), & Dawson, D.A. (1989). Estimation of dimension for spatially distributed data and related limit theorems. Journal of Multivariate Analysis, 28(1), 115–148. doi:10.1016/0047-259X(89)90100-0