Electronic Journal of Statistics

Asymptotic theorems for kernel U-quantiles

Stefan Ralescu

Full-text: Open access

Abstract

For a locally smooth statistical model, we investigate kernel U-quantiles estimators. Under suitable assumptions, we establish a strong Bahadur representation theorem, an invariance principle, and the asymptotic normality for randomly indexed sequences of observations.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 664-671.

Dates
First available in Project Euclid: 18 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1334754010

Digital Object Identifier
doi:10.1214/12-EJS687

Mathematical Reviews number (MathSciNet)
MR2988423

Zentralblatt MATH identifier
1276.62028

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Bahadur representation asymptotic normality U-statistics kernel estimator

Citation

Ralescu, Stefan. Asymptotic theorems for kernel U-quantiles. Electron. J. Statist. 6 (2012), 664--671. doi:10.1214/12-EJS687. https://projecteuclid.org/euclid.ejs/1334754010


Export citation

References

  • [1] Arcones, M. (1996). The Bahadur-Kiefer representation for, U-quantiles. Ann. Statist. 24, 1400–1422.
  • [2] Choudhury, J. and R. Serfling (1988). Generalized order statistics, Bahadur representations and sequential nonparametric fixed-width confidence intervals., J. Statist. Plann. Inference 19, 269–282.
  • [3] Dehling, H., M. Denker and W. Philipp (1987). The almost sure invariance principle for the empirical process of, U-statistic structure. Annales de l’I.H.P. 23, 121–134.
  • [4] Falk, M. (1984). Relative deficiency of kernel type estimators of quantiles., Ann. Statistics. 12, 261–268.
  • [5] Falk, M. (1985). Asymptotic normality of kernel quantile estimators., Ann. Statistics. 13, 428–433.
  • [6] Miller, R.G.,Jr. and P.K.Sen (1972). Weak convergence of U-statistics and von Mises’ differentiable statistical functions., Ann. Math. Statist. 43, 31–41.
  • [7] Parzen, E. (1979). Nonparametric statistical data modeling., J. Amer. Statist. Assoc. 74, 105–131.
  • [8] Parzen, E. (1991). Unification of statistical methods for continuous and discrete data., Proceedings of Computer Science and Statistics: Interface’90 (C. Page and R. La Page eds.), Springer Verlag, N. Y.
  • [9] Ralescu, S. S. (1995). Strong approximation theorems for integrated kernel quantiles., Mathematical Methods of Statistics. 4, 201–215.
  • [10] Serfling, R. (1984). Generalized L-, M- and R- estimates., Ann. Statist. 12, 76–86.
  • [11] Sheather, S. J. and J. S. Marron (1990). Kernel quantile estimators., J. Amer. Statist. Assoc. 85, 410–416.
  • [12] Veraverbeke, N. (1987). A Kernel-type estimator for generalized quantiles., Statist. and Probab. Lett. 5, 175–180.
  • [13] Wendler, M. (2011). Bahadur representation for, U-quantiles of dependent data. J. Multivariate Anal. 102, 1064–1079.
  • [14] Yang, S. S. (1985). A smooth nonparametric estimator of a quantile function., J. Amer. Statist. Assoc. 80, 1004–1011.