Bernoulli

Large deviations of the kernel density estimator in L1(Rd) for reversible Markov processes

Liangzhen Lei

Full-text: Open access

Abstract

We consider a reversible Rd-valued Markov process {Xi; i≥0} with the unique invariant measure μ(dx)=f(x)dx, where the density f is unknown. The large-deviation principles for the nonparametric kernel density estimator fn* in L1(Rd,dx) and for {||fn*-f||}1 are established. This generalizes the known results in the independent and identically distributed case. Furthermore, we show that fn* is asymptotically efficient in the Bahadur sense for estimating the unknown density f.

Article information

Source
Bernoulli, Volume 12, Number 1 (2006), 65-83.

Dates
First available in Project Euclid: 28 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1141136649

Mathematical Reviews number (MathSciNet)
MR2202321

Keywords
Bahadur efficiency kernel density estimator large deviations reversible Markov processes uniformly integrable operators

Citation

Lei, Liangzhen. Large deviations of the kernel density estimator in L1(Rd) for reversible Markov processes. Bernoulli 12 (2006), no. 1, 65--83. https://projecteuclid.org/euclid.bj/1141136649


Export citation

References

  • [1] Bosq, D., Merlevède, F. and Peligrad, M. (1999) Asymptotic normality for density kernel estimators in discrete and continuous time. J. Multivariate Anal., 68, 78-95.
  • [2] Csörgo?, M. and Horváth, L. (1988) Central limit theorems for Lp-norms of density estimators. Z. Wahrscheinlichkeitstheorie Verw. Geb., 80, 269-291.
  • [3] Devroye, L. (1983) The equivalence of weak, strong and complete convergence in L1 for kernel density estimates. Ann. Statist., 11, 896-904.
  • [4] Donsker, M.D. and Varadhan, S.R.S. (1975a) Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math., 28, 1-47.
  • [5] Donsker, M.D. and Varadhan, S.R.S. (1975b) Asymptotic evaluation of certain Markov process expectations for large time, II. Commun. Pure Appl. Math., 28, 279-301.
  • [6] Donsker, M.D. and Varadhan, S.R.S. (1976) Asymptotic evaluation of certain Markov process expectations for large time, III. Commun. Pure Appl. Math., 29, 389-461.
  • [7] Donsker, M.D. and Varadhan, S.R.S. (1983) Asymptotic evaluation of certain Markov process expectations for large time, IV. Commun. Pure Appl. Math., 36, 183-212.
  • [8] Giné, E., Mason, D.M. and Zaitsev, A.Yu. (2003) The L1-norm density estimator process. Ann. Probab., 31, 719-768.
  • [9] Gao, F.Q. (2003) Moderate deviations and the law of iterated logarithm for kernel density estimators in L1. Preprint.
  • [10] Kato, T. (1984) Perturbation Theory for Linear Operators, 2nd corr. printing of the 2nd edn. Berlin: Springer-Verlag.
  • [11] Lei, L.Z. and Wu, L.M. (2005) Large deviations of kernel density estimator in L1(Rd) for uniformly ergodic Markov processes. Stochastic Process. Appl., 115, 275-298.
  • [12] Lei, L.Z., Wu, L.M. and Xie, B. (2003) Large deviations and deviation inequality for kernel density estimator in L1(Rd)-distance. In H. Zhang and J. Huang (eds), Development of Modern Statistics and Related Topics, pp. 89-97. River Edge, NJ: World Scientific.
  • [13] Louani, D. (2000) Large deviations for the L1-distance in kernel density estimation. J. Statist. Plann. Inference, 177-182.
  • [14] Meyn, S.P. and Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. London: Springer- Verlag.
  • [15] Peligrad, M. (1992) Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumption. Stochastics Stochastics Rep., 40, 147-168.
  • [16] Wu, L.M. (1995) Moderate deviations of dependent random variables related to CLT. Ann. Probab., 23, 420-445.
  • [17] Wu, L.M. (1997) An introduction to large deviations. In: J.A. Yan, S.G. Peng, S.Z. Fang and L.M. Wu (eds), Several Topics in Stochastic Analysis, pp. 225-336. Beijing: Academic Press of China (in Chinese).
  • [18] Wu, L.M. (2000a) Uniformly integrable operator and large deviations for Markov processes. J. Funct. Anal., 172, 301-376.
  • [19] Wu, L.M. (2000b) A deviation inequality for non-reversible Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 36, 435-445.
  • [20] Wu, L.M. (2002) Large deviations for reversible Markov chains. Preprint.