The asymptotic behavior of thresholdbased 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), 3747
MSC (2000):
Primary 62H30; Secondary 62G07
Published electronically:
June 25, 2007
MathSciNet review:
2336777
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a problem on finding the best thresholdbased 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 for smooth distributions, while the rate of convergence of the Bayes empirical estimators is of order where is the size of a sample.
 1.
Patrick
Billingsley, Convergence of probability measures, John Wiley
& Sons, Inc., New YorkLondonSydney, 1968. MR 0233396
(38 #1718)
 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
(92e:68170)
 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
(86i:62065)
 4.
R. E. Maboroda, 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
(2004g:62022), http://dx.doi.org/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. 910, 2935.
 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
(30 #2601)
 9.
Leila
Mohammadi and Sara
van de Geer, On thresholdbased 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
(2006b:62106)
 10.
JeanKyung
Kim and David
Pollard, Cube root asymptotics, Ann. Statist.
18 (1990), no. 1, 191–219. MR 1041391
(91f:62059), http://dx.doi.org/10.1214/aos/1176347498
 11.
Vladimir
N. Vapnik, The nature of statistical learning theory, 2nd ed.,
Statistics for Engineering and Information Science, SpringerVerlag, New
York, 2000. MR
1719582 (2001c:68110)
 1.
 P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1968. MR 0233396 (38:1718)
 2.
 V. N. Vapnik, Inductive principles for the search for empirical laws, Pattern Recognition. Classification. Prediction, vol. 1, ``Nauka'', Moscow, 1989, pp. 1781. (Russian) MR 1074215 (92e:68170)
 3.
 L. Devroye and L. Gyorfi, Nonparametric Density Estimation. The View, John Wiley & Sons, Inc., New York, 1985. MR 780746 (86i:62065)
 4.
 R. E. Maboroda, Statistical Analysis of Mixtures. A course of lectures, ``Kyiv University'', Kyiv, 2003. (Ukrainian)
 5.
 Yu. O. Ivan'ko and R. E. Maboroda, Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations, Ukrain. Mat. Zh. 54 (2002), no. 10, 14211428; English transl. in Ukrainian Math. J. 54 (2002), no. 10, 17221731. MR 2015493 (2004g:62022)
 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. 910, 2935.
 7.
 O. V. Sugakova, Asymptotics of a kernel estimate for distribution density constructed from observations of a mixture with varying concentrations, Teor. Imovirnost. Matem. Statist. 59 (1998), 156166; English transl. in Theor. Probability Math. Statist. 59 (1999), 161171. MR 1793776
 8.
 H. Chernoff, Estimation of the mode, Ann. Inst. Statist. Math. 16 (1964), 3141. MR 0172382 (30:2601)
 9.
 L. Mohammadi and S. van de Geer, On thresholdbased classification rules, Lecture Notes Monograph Series, Mathematical Statistics and Applications: Festschrift for Constance van Eeden, vol. 42, Institute of Mathematical Statistics, 2003, pp. 261280. MR 2138297 (2006b:62106)
 10.
 J. Kim and D. Pollard, Cube root asymptotics, Ann. Stat. 18 (1990), no. 1, 191219. MR 1041391 (91f:62059)
 11.
 V. N. Vapnik, The Nature of Statistical Learning Theory, Springer, New York, 1996. MR 1719582 (2001c:68110)
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 LemmaVite, Brats’ka Street, Kyiv, 6, 04070 Ukraine
Email:
ivanko@lemmainsur.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/S0094900007006965
PII:
S 00949000(07)006965
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
