Optimal designs for nonparametric estimation of zeros of regression functions

Zdeněk Hlávka

doi:10.2478/v10127-012-0006-z

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Optimal designs for nonparametric estimation of zeros of regression functions

Tatra Mountains Mathematical Publications

Volume 51 (2012): Issue 1 (April 2012)

By: Zdeněk Hlávka

Open Access

|Nov 2012

Abstract

We investigate nonparametric estimators of zeros of a regression function and its derivatives and we derive the distribution of design points minimizing the expected width of a confidence interval and the expected variance of the proposed estimator.

References

[1] BIEDERMANN, S.-DETTE, H.: Minimax optimal designs for nonparametric regres- sion-a further optimality property of the uniform distribution, in: mODa 6-Advances in Model-Oriented Design and Analysis (A. C. Atkinson et al., eds.), Puchberg/Schneeberg, Austria, 2001, Contrib. Statist., Physica-Verlag, Heidelberg, 2001, pp. 13-20.10.1007/978-3-642-57576-1_2
Search in Google Scholar
[2] , Optimal designs for testing the functional form of a regression via nonparametric estimation techniques, Statist. Probab. Lett. 52 (2001), 215-224.10.1016/S0167-7152(00)00244-3
Search in Google Scholar
[3] CHENG, M.-Y.-HALL, P.-TITTERINGTON, D. M.: Optimal design for curve esti- mation by local linear smoothing, Bernoulli 4 (1998), 3-14.10.2307/3318529
Search in Google Scholar
[4] FARAWAY, J. J.: Sequential design for the nonparametric regression of curves and sur- faces, in: Proc. of the 22nd Symposium on the Interface, Comput. Sci. Statist., Vol. 22, Springer, 1990, pp. 104-110.10.1007/978-1-4612-2856-1_13
Search in Google Scholar
[5] FEDOROV, V. V.-MONTEPIEDRA,G.-NACHTSHEIM, C. J.: Design of experiments for locally weighted regression, J. Statist. Plann. Inference 81 (1999), 363-382.10.1016/S0378-3758(99)00018-X
Search in Google Scholar
[6] GASSER, T.-M¨ULLER, H.-G.: Estimating regression functions and their derivatives by the kernel method, Scand. J. Stat. 11 (1984), 171-185.
Search in Google Scholar
[7] HAINES, H.-G.: Discussion (Titterington, D. M.: Optimal design in flexible models, including feed-forward networks and nonparametric regression), Optimum Design 2000 (A. Atkinson, B. Bogacka and A. Zhigljavsky, eds.), Nonconvex Optim. Appl., Vol. 51, Kluwer, Dordrecht, 2001, pp. 272-273.
Search in Google Scholar
[8] HL´AVKA, Z.: On nonparametric estimators of location of maximum, Acta Univ. Carolin. Math. Phys. 52 (2011), 5-13.
Search in Google Scholar
[9] M¨ULLER, H.-G.: Boundary effects in nonparametric curve estimation models, in: 6th Symposium-COMPSTAT ’84, (T. Havr´anek, Z. ˇ Sid´ak and M. Nov´ak, eds.), Prague, 1984, Comput. Sci., Physica-Verlag, Vienna, 1984, pp. 84-89.10.1007/978-3-642-51883-6_10
Search in Google Scholar
[10] , Optimal designs for nonparametric kernel regression, Statist. Probab. Lett. 2 (1984), 285-290.10.1016/0167-7152(84)90066-X
Search in Google Scholar
[11] , Kernel estimators of zeros and of location and size of extrema or regression functions, Scand. J. Stat. 12 (1985), 221-232.
Search in Google Scholar
[12] M¨ULLER, W. G.: Optimal design for local fitting, J. Statist. Plann. Inference 55 (1996), 389-397.10.1016/S0378-3758(95)00197-2
Search in Google Scholar
[13] NOˇZIˇCKA, F.: Calculus of variations (5th ed.), in: Pˇrehled uˇzit´e matematiky (K. Rektorys, ed.), SNTL, Prague, 1998, pp. 841-858. (In Czech)
Search in Google Scholar
[14] PELIGRAD, M.: On the asymptotic normality of sequences of weak dependent random variables, J. Theoret. Probab. 9 (1996), 703-715.10.1007/BF02214083
Search in Google Scholar
[15] R DEVELOPMENT CORE TEAM: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2011.
Search in Google Scholar
[16] ROUSSAS, G. G.: Consistent regression estimation with fixed design points under depen- dence conditions, Statist. Probab. Lett. 8 (1989), 41-50.10.1016/0167-7152(89)90081-3
Search in Google Scholar
[17] , Exponential probability inequalities with some applications, in: Statistics, Probability and Game Theory, IMS Lecture Notes Monogr. Ser., Vol. 30, Institute of Mathematical Statistics, Hayward, CA, 1996, pp. 303-319.10.1214/lnms/1215453579
Search in Google Scholar
[18] ROUSSAS, G. G.-TRAN, L. T.-IOANNIDES, D. A.: Fixed design regression for time series: asymptotic normality, J. Multivariate Anal. 40 (1992), 262-291.10.1016/0047-259X(92)90026-C
Search in Google Scholar
[19] SMIRNOW, W. I.: Lehrgang der H¨oheren Mathematik, Teil IV. VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.
Search in Google Scholar
[20] TITTERINGTON, D. M.: Optimal design in flexible models, including feed-forward net- works and nonparametric regression, in: Optimum Design 2000 (A. Atkinson, B. Bogacka and A. Zhigljavsky, eds.), Nonconvex Optim. Appl., Vol. 51, Kluwer, Dordrecht, 2001, pp. 261-272.10.1007/978-1-4757-3419-5_23
Search in Google Scholar
[21] WIENS, D. P.: Minimax robust designs and weights for approximately specified regression models with heteroscedastic errors, J. Amer. Statist. Assoc. 93 (1998), 1440-1450.10.1080/01621459.1998.10473804
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10127-012-0006-z | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 55 - 65

Published on: Nov 13, 2012

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

empirical zero,

location of maximum,

nonparametric regression,

Related subjects:

© 2012 Zdeněk Hlávka, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons License.

Volume 51 (2012): Issue 1 (April 2012)