Uniform in Bandwidth Consistency of Local Polynomial Regression Function Estimators
DOI:
https://doi.org/10.17713/ajs.v35i2&3.359Abstract
We generalize a method for proving uniform in bandwidth consistency results for kernel type estimators developed by the two last named authors. Such results are shown to be useful in establishing consistency of local polynomial estimators of the regression function.References
Blondin, D., Massiani, A., and Ribereau, P. (2005). Vitesses de convergence uniforme presque sûre d’estimateurs non-paramétriques de la régression. C.R. Acad. Sci. Paris, Ser., I 340, 525-528.
Deheuvels, P. (2000). Uniform limit laws for kernel density estimators on possibly unbounded intervals. In N. Limnios and M. Nikulin (Eds.), Recent Advances in Reliability Theory: Methodology, Practice and Inference (p. 477-492). Basel: Birkhäuser.
Einmahl, U., and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab., 13, 1-37.
Einmahl, U., and Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat., 33, 1380-1403.
Fan, J., and Gijbels, I. (1996). Local Polynomial Modelling and its Applications (Monographs on Statistics and Applied Probability, 66 ed.). London: Chapman & Hall.
Giné, E., and Guillou, A. (2001). On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals. Ann. Inst. H. Poincaré, 37, 503-522.
Giné, E., and Guillou, A. (2002). Rates of strong consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré, 38, 907-921.
Giné, E., and Koltchinskii, V. (2005). Concentration inequalities and asymptotic results for ratio type empirical processes. Ann. Probab., to appear.
Nolan, D., and Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist., 15, 780-799.
Pollard, D. (1984). Convergence of Stochastic Processes. New York: Springer-Verlag.
Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton, New Jersey: Princeton University Press.
Stute, W. (1982a). The oscillation behavior of empirical processes. Ann. Probab, 10, 86-107.
Stute, W. (1982b). The law of the iterated logarithm for kernel density estimators. Ann. Probab, 10, 414-422.
Stute, W. (1984). The oscillation behavior of empirical processes: the multivariate case. Ann. Probab, 12, 361-379.
Stute, W. (1986). On almost sure convergence of conditional empirical distribution functions. Ann. Probab, 14, 891-901.
Talagrand, M. (1994). Sharper bounds for gaussian and empirical processes. Ann. Probab, 22, 28-76.
Tsybakov, A. B. (2004). Introduction à l’estimation non–paramétrique. In Mathematics & applications. Paris: Springer.
van der Vaart, A. W., and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer-Verlag.
Downloads
Published
How to Cite
Issue
Section
License
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.