Open Access
Issue
ESAIM: PS
Volume 26, 2022
Page(s) 171 - 207
DOI https://doi.org/10.1051/ps/2022003
Published online 01 March 2022
  1. C. Aaron and O. Bodart, Local convex hull support and boundary estimation. J. Multivar. Anal. 147 (2016) 82–101. [CrossRef] [Google Scholar]
  2. C.C. Aggarwal and S. Sathe, Theoretical foundations and algorithms for outlier ensembles. ACM Sigkdd Explor. Newsl. 17 (2015) 24–47. [CrossRef] [Google Scholar]
  3. N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68 (1950) 337–404. [CrossRef] [Google Scholar]
  4. M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces: Manifolds, Curves, and Surfaces, Vol. 115, Springer Science & Business Media (2012). [Google Scholar]
  5. A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics. Springer Science & Business Media (2011). [Google Scholar]
  6. R. Berman, S. Boucksom and D.W. Nyström, Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207 (2011) 1–27. [CrossRef] [MathSciNet] [Google Scholar]
  7. G. Biau, B. Cadre and B. Pelletier, Exact rates in density support estimation. J. Multivar. Anal. 99 (2008) 2185–2207. [CrossRef] [Google Scholar]
  8. L. Bos, Asymptotics for the Christoffel function for Jacobi like weights on a ball in ℝm. New Zealand J. Math. 23 (1994) 109. [Google Scholar]
  9. L. Bos, B. Della Vecchia and G. Mastroianni, On the asymptotics of Christoffel functions for centrally symmetric weight functions on the ball in ℝd. Rend. Cir. Mat. 52 (1998) 277–290. [Google Scholar]
  10. J. Chevalier, Estimation du support et du contour du support d’une loi de probabilité. Ann. Inst. Henri Poincaré Stat. 12 (1976) 339–364. [Google Scholar]
  11. A. Cholaquidis, A. Cuevas and R. Fraiman, On Poincaré cone property. Ann. Stat. 42 (2014) 255–284. [CrossRef] [Google Scholar]
  12. A. Cuevas and R. Fraiman, A plug-in approach to support estimation. Ann. Stat. 25 (1997) 2300–2312. [CrossRef] [Google Scholar]
  13. A. Cuevas, R. Fraiman and B. Pateiro-López, On statistical properties of sets fulfilling rolling-type conditions. Adv. Appl. Prob. 44 (2012) 311–329. [CrossRef] [Google Scholar]
  14. A. Cuevas, W. González-Manteiga and A. Rodríguez-Casal, Plug-in estimation of general level sets. Aust. New Zealand J. Stat. 48 (2006) 7–19. [CrossRef] [MathSciNet] [Google Scholar]
  15. A. Cuevas and A. Rodríguez-Casal, On boundary estimation. Adv. Appl. Prob. 36 (2004) 340–354. [CrossRef] [Google Scholar]
  16. L. Devroye and G.L. Wise, Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38 (1980) 480–488. [CrossRef] [MathSciNet] [Google Scholar]
  17. D. Dua and C. Graff, UCI Machine Learning Repository (2017). [Google Scholar]
  18. C.F. Dunkl and Y. Xu, Vol. 155 of Orthogonal polynomials of several variables. Cambridge University Press (2014). [CrossRef] [Google Scholar]
  19. H. Edelsbrunner, D. Kirkpatrick and R. Seidel, On the shape of a set of points in the plane. IEEE Trans. Inf. Theory 29 (1983) 551–559. [CrossRef] [Google Scholar]
  20. J. Geffroy, Sur un probleme d’estimation géométrique. Publ. Inst. Statist. Univ. Paris 13 (1964) 191–210. [MathSciNet] [Google Scholar]
  21. F. Keller, E. Muller and K. Bohm, HiCS: High contrast subspaces for density-based outlier ranking, in 2012 IEEE 28th international conference on data engineering, IEEE (2012) 1037–1048. [CrossRef] [Google Scholar]
  22. A. Kroó and D. Lubinsky, Christoffel functions and universality in the bulk for multivariate orthogonal polynomials. Can. J. Math. 65 (2013) 600–620. [CrossRef] [Google Scholar]
  23. A. Kroó and D. Lubinsky, Christoffel functions and universality on the boundary of the ball. Acta Math. Hung. 140 (2013) 117–133. [CrossRef] [Google Scholar]
  24. J.B. Lasserre and E. Pauwels, The empirical Christoffel function with applications in data analysis. Adv. Comput. Math. 45 (2019) 1439–1468. [CrossRef] [MathSciNet] [Google Scholar]
  25. E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Stat. 23 (1995) 502–524. [CrossRef] [Google Scholar]
  26. S. Marx, E. Pauwels, T. Weisser, D. Henrion and J. Lasserre, Tractable semi-algebraic approximation using Christoffel-Darboux kernel. Preprint arXiv:1904.01833 (2019). [Google Scholar]
  27. I.S. Molchanov, A limit theorem for solutions of inequalities. Scand. J. Stat. 25 (1998) 235–242. [CrossRef] [Google Scholar]
  28. T. Patschkowski and A. Rohde, Adaptation to lowest density regions with application to support recovery. Ann. Stat. 44 (2016) 255–287. [CrossRef] [Google Scholar]
  29. E. Pauwels and J.B. Lasserre, Sorting out typicality with the inverse moment matrix SOS polynomial, in Advances in Neural Information Processing Systems (2016) 190–198. [Google Scholar]
  30. E. Pauwels, M. Putinar and J.-B. Lasserre, Data analysis from empirical moments and the Christoffel function. Found. Comput. Math. 21 (2021) 243–273. [CrossRef] [MathSciNet] [Google Scholar]
  31. F. Piazzon, Bernstein Markov properties and applications, Ph.D. thesis, Dipartimento di Matematica, Università degli Studi di Padova (2016). [Google Scholar]
  32. W. Polonik, Measuring mass concentrations and estimating density contour clusters-an excess mass approach. Ann. Stat. 23 (1995) 855–881. [CrossRef] [Google Scholar]
  33. A. Rényi and R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten. Prob. Theory Related Fields 2 (1963) 75–84. [Google Scholar]
  34. P. Rigollet and R. Vert, Optimal rates for plug-in estimators of density level sets. Bernoulli 15 (2009) 1154–1178. [CrossRef] [MathSciNet] [Google Scholar]
  35. A. Rodríguez Casal Set estimation under convexity type assumptions. Ann. Inst. Henri Poincaré, Prob. Stat. 43 (2007) 763–774. [CrossRef] [Google Scholar]
  36. H.L. Royden and P. Fitzpatrick, Real analysis, vol. 32. Macmillan New York (1988). [Google Scholar]
  37. A. Singh, C. Scott and R. Nowak, Adaptive Hausdorff estimation of density level sets. Ann. Stat. 37 (2009) 2760–2782. [CrossRef] [Google Scholar]
  38. G. Szegö, Vol. 23 of Orthogonal polynomials. American Mathematical Soc. (1939). [Google Scholar]
  39. V. Totik, Asymptotics for Christoffel functions for general measures on the real line. J. d’Anal. Math. 81 (2000) 283–303. [CrossRef] [Google Scholar]
  40. A.B. Tsybakov, On nonparametric estimation of density level sets. Ann. Stat. 25 (1997) 948–969. [CrossRef] [Google Scholar]
  41. R. Vershynin, Introduction to the non-asymptotic analysis of random matrices. Preprint arXiv:1011.3027 (2010). [Google Scholar]
  42. G. Walther, Granulometric smoothing. Ann. Stat. 25 (1997) 2273–2299. [Google Scholar]
  43. G. Walther, On a generalization of Blaschke’s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22 (1999) 301–316. [CrossRef] [MathSciNet] [Google Scholar]
  44. Y. Xu, Asymptotics of the Christoffel functions on a simplex in ℝd. J. Approx. Theory 99 (1999) 122–133. [CrossRef] [MathSciNet] [Google Scholar]
  45. Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in ℝd. Trans. Am. Math. Soc. 351 (1999) 2439–2458. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.