Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

The asymptotic behavior of threshold-based classification rules constructed from a sample from a mixture with varying concentrations


Authors: Yu. Ivan'ko and R. Maiboroda
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 37-47
MSC (2000): Primary 62H30; Secondary 62G07
Published electronically: June 25, 2007
MathSciNet review: 2336777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a problem on finding the best threshold-based classification rule constructed from a sample from a mixture with varying concentrations. We show that the rate of convergence of the minimal empirical risk estimators to the optimal threshold is of order $ N^{-1/3}$ for smooth distributions, while the rate of convergence of the Bayes empirical estimators is of order $ N^{-2/5}$ where $ N$ is the size of a sample.


References [Enhancements On Off] (What's this?)

  • 1. Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
  • 2. V. N. Vapnik, Inductive principles for the search for empirical laws, Pattern recognition. Classification. Prediction (Russian), “Nauka”, Moscow, 1989, pp. 17–82 (Russian). MR 1074215
  • 3. Luc Devroye and László Györfi, Nonparametric density estimation, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons, Inc., New York, 1985. The 𝐿₁ view. MR 780746
  • 4. R. E. Ma{\u{\i\/}}\kern.15emboroda, Statistical Analysis of Mixtures. A course of lectures, ``Kyiv University'', Kyiv, 2003. (Ukrainian)
  • 5. Yu. O. Īvan′ko and R. Ē. Maĭboroda, Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations, Ukraïn. Mat. Zh. 54 (2002), no. 10, 1421–1428 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 10, 1722–1731. MR 2015493, 10.1023/A:1023792522291
  • 6. Yu. O. Ivan'ko, The asymptotic behavior of kernel estimators and their derivatives constructed from observations from a mixture with varying concentrations, Visnyk KNU, Ser. Matematika. Mekhanika (2003), no. 9-10, 29-35.
  • 7. O. V. Sugakova, Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentration, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 156–166 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 161–171 (2000). MR 1793776
  • 8. Herman Chernoff, Estimation of the mode, Ann. Inst. Statist. Math. 16 (1964), 31–41. MR 0172382
  • 9. Leila Mohammadi and Sara van de Geer, On threshold-based classification rules, Mathematical statistics and applications: Festschrift for Constance van Eeden, IMS Lecture Notes Monogr. Ser., vol. 42, Inst. Math. Statist., Beachwood, OH, 2003, pp. 261–280. MR 2138297
  • 10. JeanKyung Kim and David Pollard, Cube root asymptotics, Ann. Statist. 18 (1990), no. 1, 191–219. MR 1041391, 10.1214/aos/1176347498
  • 11. Vladimir N. Vapnik, The nature of statistical learning theory, 2nd ed., Statistics for Engineering and Information Science, Springer-Verlag, New York, 2000. MR 1719582

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 62H30, 62G07

Retrieve articles in all journals with MSC (2000): 62H30, 62G07


Additional Information

Yu. Ivan'ko
Affiliation: SK Lemma-Vite, Brats’ka Street, Kyiv, 6, 04070 Ukraine
Email: ivanko@lemma-insur.com.ua

R. Maiboroda
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email: mre@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-07-00696-5
Keywords: Minimization of the empirical risk, kernel estimators of densities, Bayes empirical classification rule, estimates of components of a mixture, mixtures with varying concentrations
Received by editor(s): December 20, 2004
Published electronically: June 25, 2007
Article copyright: © Copyright 2007 American Mathematical Society