Kybernetika 50 no. 6, 869-882, 2014

Inferring the residual waiting time for binary stationary time series

Gusztáv Morvai and Benjamin WeissDOI: 10.14736/kyb-2014-6-0869

Abstract:

For a binary stationary time series define $\sigma_n$ to be the number of consecutive ones up to the first zero encountered after time $n$, and consider the problem of estimating the conditional distribution and conditional expectation of $\sigma_n$ after one has observed the first $n$ outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state the stopping times along which we estimate have density one.

Keywords:

stationary processes, nonparametric estimation

Classification:

62G05, 60G25, 60G10

paper.pdf

References:

  1. P. Algoet: Universal schemes for learning the best nonlinear predictor given the infinite past and side information. IEEE Trans. Inform. Theory 45 (1999), 1165-1185.   CrossRef
  2. D. H. Bailey: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph. D. Thesis, Stanford University 1976.   CrossRef
  3. F. Bunea and A. Nobel: Sequential procedures for aggregating arbitrary estimators of a conditional mean. IEEE Trans. Inform. Theory 54 (2008), 4, 1725-1735.   CrossRef
  4. W. Feller: An Introduction to Probability Theory and its Applications. Vol. II. Second edition. John Wiley and Sons, New York - London - Sydney 1971.   CrossRef
  5. L. Györfi, G. Morvai and S. Yakowitz: Limits to consistent on-line forecasting for ergodic time series. IEEE Trans. Inform. Theory 44 (1998), 886-892.   CrossRef
  6. G. Morvai and B. Weiss: Inferring the conditional mean. Theory Stoch. Process. 11 (2005), 112-120.   CrossRef
  7. G. Morvai and B. Weiss: Order estimation of Markov chains. IEEE Trans. Inform. Theory 51 (2005), 1496-1497.   CrossRef
  8. G. Morvai and B. Weiss: On sequential estimation and prediction for discrete time series. Stoch. Dyn. 7 (2007), 4, 417-437.   CrossRef
  9. G. Morvai and B. Weiss: On Universal estimates for binary renewal processes. Ann. Appl. Probab. 18 (2008), 5, 1970-1992.   CrossRef
  10. G. Morvai and B. Weiss: Estimating the residual waiting time for binary stationary time series. In: Proc. ITW2009, Volos 2009, pp. 67-70.   CrossRef
  11. G. Morvai and B. Weiss: A note on prediction for discrete time series. Kybernetika 48 (2012), 4, 809-823.   CrossRef
  12. B. Ya. Ryabko: Prediction of random sequences and universal coding. Probl. Inf. Trans. 24 (1988), 87-96.   CrossRef
  13. D. Ryabko and B. Ryabko: Nonparametric statistical inference for ergodic processes. IEEE Trans. Inform. Theory 56 (2010), 3, 1430-1435.   CrossRef
  14. A. N. Shiryayev: Probability. Springer-Verlag, New York 1984.   CrossRef
  15. H. Takahashi: Computational limits to nonparametric estimation for ergodic processes. IEEE Trans. Inform. Theory 57 (2011), 10, 6995-6999.   CrossRef
  16. Z. Zhou, Z. Xu and W. B. Wu: Long-term prediction intervals of time series. IEEE Trans. Inform. Theory 56 (2010), 3, 1436-1446.   CrossRef