Histogram estimators of the shape of the concentration function in a two-component mixture
Author:
D. I. Pohyl’ko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 72 (2006), 125-133
MSC (2000):
Primary 62G20; Secondary 62G05
DOI:
https://doi.org/10.1090/S0094-9000-06-00670-3
Published electronically:
September 5, 2006
MathSciNet review:
2168142
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We construct the projection estimators of the shape of the concentration function and obtain their expansions in the basis of indicator functions (the histogram basis) in the case where the data is a sample from a mixture of two components with unknown distributions whose concentrations are varying with observations. We prove that the estimators are consistent and find the rate of the almost sure convergence.
References
- R. Ē. Maĭboroda, An asymptotically efficient probability estimator constructed from observations of a mixture, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 117–124 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 121–128 (2000). MR 1793771
- ---, Nonparametric Statistics of Nonhomogeneous Observations, Doctoral dissertation, Kyiv, 1994. (Ukrainian)
- R. E. Maĭboroda, Projective estimates for changing concentrations of mixtures, Teor. Īmovīr. ta Mat. Statist. 46 (1992), 70–76 (Ukrainian, with Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 46 (1993), 71–75. MR 1196209
- R. Ē. Maĭboroda, Estimation of the distributions of the components of mixtures having varying concentrations, Ukraïn. Mat. Zh. 48 (1996), no. 4, 558–562 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 4, 618–622 (1997). MR 1417019, DOI https://doi.org/10.1007/BF02390622
- ---, Statistical Analysis of Mixtures, Kyiv University, Kyiv, 2003. (Ukrainian)
- Charles K. Chui, Wavelets: a mathematical tool for signal processing, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. With a foreword by Gilbert Strang. MR 1443204
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
- Wolfgang Härdle, Applied nonparametric regression, Econometric Society Monographs, vol. 19, Cambridge University Press, Cambridge, 1990. MR 1161622
- Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, and Alexander Tsybakov, Wavelets, approximation, and statistical applications, Lecture Notes in Statistics, vol. 129, Springer-Verlag, New York, 1998. MR 1618204
- R. E. Maiboroda, Estimation and classification by mixtures with time-dependent concentrations, VI International Vilnius Conference on Probability Theory and Math. Statistics. Abstracts of Communications, vol. 2, 1993, p. 48.
- 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
- Brani Vidakovic, Statistical modeling by wavelets, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1681904
References
- R. E. Maĭboroda, An asymptotically effective probability estimator constructed from observations of a mixture, Teor. Ĭmovir. Mat. Stat. 59 (1998), 108–115; English transl. in Theory Probab. Math. Statist. 59 (1999), 121–128. MR 1793771
- ---, Nonparametric Statistics of Nonhomogeneous Observations, Doctoral dissertation, Kyiv, 1994. (Ukrainian)
- ---, Projection estimates for changing concentrations of mixtures, Teor. Ĭmovir. Mat. Stat. 46 (1992), 70–76; English transl. in Theory Probab. Math. Statist. 46 (1993), 71–73. MR 1196209 (93j:62102)
- ---, An estimate of the distributions of the mixture components with varying concentrations, Ukrain. Mat. Zh. 48 (1996), no. 4, 562–566; English transl. in Ukrainian Math. J. 48 (1997), 618–622. MR 1417019 (97j:62055)
- ---, Statistical Analysis of Mixtures, Kyiv University, Kyiv, 2003. (Ukrainian)
- C. K. Chui, Wavelets: A Mathematical Tool for Signal Processing, SIAM, Philadelphia, PA, 1997. MR 1443204 (99b:42012)
- I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1996. MR 1162107 (93e:42045)
- W. Härdle, Applied Nonparametric Regression, Cambridge University Press, Cambridge, 1990. MR 1161622 (93i:62030)
- W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Springer-Verlag, New York, 1998. MR 1618204 (99f:42065)
- R. E. Maiboroda, Estimation and classification by mixtures with time-dependent concentrations, VI International Vilnius Conference on Probability Theory and Math. Statistics. Abstracts of Communications, vol. 2, 1993, p. 48.
- O. V. Sugakova, Asymptotics of a kernel estimate for distribution density constructed from observations of a mixture with varying concentration, Teor. Ĭmovir. Mat. Stat. 59 (1999), 156–166; English transl. in Theory Probab. Math. Statist. 59 (2000), 161–171. MR 1793776
- B. Vidakovic, Statistical Modeling by Wavelets, John Wiley & Sons, New York, 1999. MR 1681904 (2000f:42023)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
62G20,
62G05
Retrieve articles in all journals
with MSC (2000):
62G20,
62G05
Additional Information
D. I. Pohyl’ko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
pokhid@ukr.net
Keywords:
Projection estimates,
estimates of the concentration,
the shape of the concentration function
Received by editor(s):
May 26, 2004
Published electronically:
September 5, 2006
Article copyright:
© Copyright 2006
American Mathematical Society