Abstract
A general class of conditional U-statistics was introduced by W. Stute as a generalization of the Nadaraya–Watson estimates of a regression function. It was shown that such statistics are universally consistent. Also, universal consistencies of the window and k n -nearest neighbor estimators (as two special cases of the conditional U-statistics) were proved. Later, (Harel and Puri, Ann Inst Stat Math 56(4):819–832, 2004) extended his results from the i.i.d. case to the absolute regular case. In this paper, we extend these results from the stationary case to the nonstationary case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bennett G (1962) Probability inequalities for the sum on independent random variables. J Am Stat Assoc 19:33–45
Devroye LP, Wagner TJ (1980) Distribution-free consistency results in nonparametric discrimination and regression function estimate. Ann Stat 8:231–239.
Harel M, Puri LM (1996) Conditional U-statistics for dependent random varaibles. J Multivar Anal 57:84–100
Harel M, Puri LM (2004) Universally consistent conditional U-statistics for absolutely regular processes and its applications for hidden markov models. Ann Inst Stat Math 56(4):819–832
Speigelman G, Sacks J (1980) Consistent window estimation inonparametric regression. Ann Stat 8:240–246
Stone G (1977) Consistent nonparametric regression, Ann Stat 5:595–620
Stute W (1991) Conditional U-statistics. Ann Probab 19:812–825
Stute W (1994a) Universally consistent conditional U-statistics. Ann Stat 22:460–473
Stute W (1994b) L p convergence of conditional U-statistics. J Multivar Anal 51:71–82
Yoshihara KL (1976) Limiting behavior of U-statistics for stationary absolutely regular processes. Z Wahrsch Verw Gebiete 35:237–252
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Elharfaoui, E., Harel, M., Puri, M.L. (2011). Asymptotic Behavior of the Universally Consistent Conditional U-Statistics for Nonstationary and Absolutely Regular Processes. In: Wells, M., SenGupta, A. (eds) Advances in Directional and Linear Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2628-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-7908-2628-9_2
Published:
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-2627-2
Online ISBN: 978-3-7908-2628-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)