Advertisement

Statistical Inference for Stochastic Processes

, Volume 10, Issue 3, pp 209–221 | Cite as

Nonparametric density estimation for nonmixing approximable Stochastic Processes

  • Salim LardjaneEmail author
Article

Abstract

The author deals with nonparametric density estimation for stochastic processes which satisfy the L -approximability property. He considers a Parzen–Rosenblatt estimator of the density for general stationary L -approximable processes. He states conditions under which it is consistent and investigates its rate of convergence. Finally, he applies his results to general nonmixing linear processes and nonmixing nonlinear autoregressive processes.

Keywords

Nonparametric density estimation Nonmixing processes Near Epoch Dependence Linear processes Autoregressive processes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews DWK (1984) Non-strong mixing autoregressive processes. J Appl Probab 21:930–934zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bartlett M (1990) Chance or chaos?. J R Statist Soc A 153(3):321–347CrossRefGoogle Scholar
  3. Bosq D (1995) Optimal asymptotic quadratic error of density estimators for strong mixing or chaotic data. Statist Probab Lett 22:339–347zbMATHCrossRefMathSciNetGoogle Scholar
  4. Bosq D (1998) Nonparametric statistics for stochastic processes: estimation and prediction. Number 110 in lecture notes in statistics, 2nd edn. Springer, BerlinGoogle Scholar
  5. Chan K-S, Tong H (2001) Chaos: a statistical perspective. Springer, BerlinzbMATHGoogle Scholar
  6. Chernick MR (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann Probab 9(1):145–149zbMATHMathSciNetGoogle Scholar
  7. Chernick MR (1982) Extremes in autoregressive processes with uniform marginal distributions. Statist Probab Lett 1:85–88zbMATHCrossRefMathSciNetGoogle Scholar
  8. Davidson J (1994) Stochastic limit theory. Advanced texts in econometrics. Oxford University Press, OxfordGoogle Scholar
  9. Doukhan P (1994) Mixing: properties and examples. Springer, BerlinzbMATHGoogle Scholar
  10. Gallant AR, White H (1988) A unified theory of estimation and inference for nonlinear dynamic models. Basil Blackwell, OxfordGoogle Scholar
  11. Habutsu T, Nishio Y, Sasase Y, Mori S (1990) A secret key cryptosystem by iterating a chaotic map. Trans Inst Elec Inf Comm Eng Jpn E 73:1041–1044Google Scholar
  12. Ibragimov IA (1962) Some limit theorems for stationary processes. Theor Probab Appl 7:349–382CrossRefGoogle Scholar
  13. Jacod J, Shiryaev AN (1987) Limit theorems for stochastic processes. Springer-Verlag, BerlinzbMATHGoogle Scholar
  14. Lawrance AJ (1992) Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators. J Appl Probab 29:896–903zbMATHCrossRefMathSciNetGoogle Scholar
  15. Nadaraja E (1965) On nonparametric estimation of density function and regression. Theor Probab Appl 10:186–190CrossRefMathSciNetGoogle Scholar
  16. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Statist 33:1065–1076MathSciNetGoogle Scholar
  17. Pötscher B, Prucha I (1991) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: consistency and approximation concepts. Economet Rev 10:125–126zbMATHCrossRefGoogle Scholar
  18. Robinson P (1983) Nonparametric estimators for time series. J Time Ser Anal 4:185–297zbMATHMathSciNetGoogle Scholar
  19. Rosenblatt M (1956) Remarks on some non parametric estimates of a density function. Ann Math Statist 27:832–837MathSciNetGoogle Scholar
  20. Rosenblatt M (1985) Stationary sequences and random fields. Birkhäuser, BaselzbMATHGoogle Scholar
  21. Roussas G (1988) Nonparametric estimation in mixing sequences of random variables. J Statist Plan Inf 18:135–149zbMATHCrossRefMathSciNetGoogle Scholar
  22. Taniguchi M, Kazikawa Y (2000) Asymptotic theory of statistical inference for time series. Springer, BerlinzbMATHGoogle Scholar
  23. Tiago de Oliveira J (1963) Estatistica de densidades. Resultados assintoticos. Rev Fac Sci Univ Lisboa Ser A 9:111–206Google Scholar
  24. Tong H (1992) A note on one-dimensional chaotic maps under time reversal. Adv Appl Prob 24:219–220zbMATHCrossRefGoogle Scholar
  25. Tran L (1989) The L 1 convergence of kernel density estimates under dependence. Can J Statist 17(2):197–208zbMATHCrossRefMathSciNetGoogle Scholar
  26. Vieu P (1991) Quadratic errors for nonparametric estimators under dependence. J. Multivar Anal 39:324–347zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Laboratoire de Statistique d’EnquêtesCREST-ENSAIBruzFrance

Personalised recommendations