Abstract
In this short chapter, we show how to pick a density estimate from a pre-specified class of densities, F. In particular, the classes we are interested in are totally bounded, that is, for every ε > 0, there exists a finite number N ε of densities in F such that the L 1 balls of radius ε centered at these densities cover F, that is, if these chosen densities are G ε = {g1,…,gN ε }, then
where B g,r = {f : ∫ |f − g| ≤ r}. The smallest such N ε is called the complexity (or covering number) of F (and thus is a function of ε), and log2 N ε is the Kolmogorov entropy of F. G ε is called a skeleton of F.
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§7.9. References
L. Birgé, “On the risk of histograms for estimating decreasing densities,” Annals of Statistics, vol. 15, pp. 1013–1022,1987a.
L. Birgé, “Estimating a density under order restrictions: Nonasymptotic minimax risk,” Annals of Statistics, vol. 15, pp. 995–1012, 1987b.
L. Birgé, “The Grenander estimator: A nonasymptotic approach,” Annals of Statistics, vol. 17, pp. 1532–1549, 1989.
L. Devroye, “A simple algorithm for generating random variates with a log-concave density,” Computing, vol. 33, pp. 247–257, 1984.
L. Devroye, A Course in Density Estimation, Birkhäuser-Verlag, Boston, 1987.
R. Khasminskii and I. Ibragimov, “On density estimation in the view of Kolmogorov’s ideas in approximation theory,” Annals of Statistics, vol. 18, pp. 999–1010, 1990.
A. N. Kolmogorov and V. M. Tikhomirov, “ε-entropy and ε-capacity of sets in function spaces,” Translations of the American Mathematical Society, vol. 17, pp. 277–364, 1961.
G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.
Y. Makovoz, “On the Kolmogorov complexity of functions of finite smoothness,” Journal of Complexity, vol. 2, pp. 121–130, 1986.
R. E. Tarjan, Data Structures and Network Algorithms, CBMS 44, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.
Y. G. Yatracos, “Rates of convergence of minimum distance estimators and Kolmogorov’s entropy,” Annals of Statistics, vol. 13, pp. 768–774, 1985.
Y. G. Yatracos, “A note on L 1 consistent estimation,” Canadian Journal of Statistics, vol. 16, pp. 283–292, 1988.
K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1980.
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© 2001 Springer Science+Business Media New York
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Devroye, L., Lugosi, G. (2001). Skeleton Estimates. In: Combinatorial Methods in Density Estimation. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0125-7_7
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DOI: https://doi.org/10.1007/978-1-4613-0125-7_7
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