ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60, 62 Koltchinskii, Vladimir I., University of New Mexico, USA Empirical geometry of multivariate data: a deconvolution approach Let {Y_{j} : j = 1, …, n} be independent observations in R^{m} with common distribution Q. Suppose that Y_{j} = X_{j} + ξ_{j}, j = 1, …, n, where X_{j}, ξ_{j}, j = 1, …, n are independent, X_{j}, j = 1, …, n have common distribution P and ξ_{j}, j = 1, …, n have common distribution μ, so that Q = P ∗ μ. The problem is to recover hidden geometric structure of the support of P based on the independent observations Y_{j}, contaminated by random errors ξ_{j}. We suggest deconvolution methods of statistical estimation of the boxcounting dimension and the hierarchical cluster tree of the support of P with exponential convergence rates. The exponential rate holds even in the case of the spherically symmetric Gaussian noise ξ_{j} with unknown variance of the components.