Abstract: We present a class of estimators for the Shannon and Renyi information of multi-dimensional probability density, based on the k-th nearest distances in a sample of i.i.d. vectors (see Leonenko, Pronzato and Savani (2008)). The method can be extended to the estimation of the statistical distances between two distributions using one i.i.d. sample from each. An applications of different entropies (å-entropy and quadratic Renyi entropy, see Leonenko and Seleznev (2009)) are also studied. The other approaches to estimation of entropy are discussed.
This is a joint results with Luc Pronzato (University of Nice-Sophia), and Oleg Seleznev (Umea University, Sweden).

Brief Biography of the Speaker:
Nikolai Leonenko MD PhD is a Professor of Statistics at Cardiff School of Mathematics, Cardiff University, Wales, UK. His areas of expertise are: statistical estimation of Shannon and Renyi information and statistical distances; statistical analysis of stochastic processes and random fields; spectral theory of random fields; statistical inference with higher-order information; fractional differential equations and PDE with random data; multifractal processes and fields; finance and stochastic; first passage distribution problem of Pearson jump-diffusion. Besides being an author and co-author of 180 papers, he wrote 2 books. He was awarded by N. M. Krylov medal of Academy of Science of Ukraine (1993), the highest annual award for mathematicians in Ukraine, and he is a Member of the American Mathematical Society, London Mathematical Society and Kyiv Mathematical Society.