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regularity index; Lebesgue point; small ball probability
The index of regularity of a measure was introduced by Beirlant, Berlinet and Biau [1] to solve practical problems in nearest neighbour density estimation such as removing bias or selecting the number of neighbours. These authors proved the weak consistency of an estimator based on the nearest neighbour density estimator. In this paper, we study an empirical version of the regularity index and give sufficient conditions for its weak and strong convergence without assuming absolute continuity or other global properties of the underlying measure.
[1] J. Beirlant, A. Berlinet, G. Biau: Higher order estimation at Lebesgue points. Ann. Inst. Statist. Math. 60 (2008), 651-677. DOI 10.1007/s10463-007-0112-x | MR 2434416 | Zbl 1169.62024
[2] A. Berlinet, S. Levallois: Higher order analysis at Lebesgue points. In: G. G. Roussas Festschrift - Asymptotics in Statistics and Probability (M. L. Puri, ed.), 2000, pp. 17-32.
[3] A. Berlinet, R. Servien: Necessary and sufficient condition for the existence of a limit distribution of the nearest neighbour density estimator. J. Nonparametr. Statist. 23 (2011), 633-643. DOI 10.1080/10485252.2011.567334 | MR 2836281
[4] L. Devroye, G. Lugosi: Combinatorial Methods in Density Estimation. Springer, New York 2001. MR 1843146 | Zbl 0964.62025
[5] R. M. Dudley: Real Analysis and Probability. Chapman and Hall, New York 1989. MR 0982264 | Zbl 1023.60001
[6] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York 1987. MR 0924157 | Zbl 1038.00002
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