Kybernetika 52 no. 3, 348-358, 2016

A versatile scheme for predicting renewal times

Gusztáv Morvai and Benjamin WeissDOI: 10.14736/kyb-2016-3-0348

Abstract:

There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.

Keywords:

stationary processes, nonparametric estimation

Classification:

62G05, 60G25, 60G10

paper.pdf

References:

  1. B. von Bahr and C. G. Esseen: Inequalities for the $r$th Absolute Moment of a Sum of Random Variables, $1\leq r \leq 2$. Annals Math. Statist. 36 (1965), 299-303.   DOI:10.1214/aoms/1177700291
  2. I. Csiszár and P. Shields: The consistency of the BIC Markov order estimator. Annals Statist. 28 (2000), 1601-1619.   DOI:10.1214/aos/1015957472
  3. I. Csiszár: Large-scale typicality of Markov sample paths and consistency of MDL order estimators. IEEE Trans. Inform. Theory 48 (2002), 1616-1628.   DOI:10.1109/tit.2002.1003842
  4. W. Feller: An Introduction to Probability Theory and its Applications Vol. I. Third edition. John Wiley and Sons, New York - London - Sydney 1968.   CrossRef
  5. S. Ghahramani: Fundamentals of Probability with Stochastic Processes. Third edition. Pearson Prentice Hall, Upper Saddle River NJ 2005.   CrossRef
  6. G. Morvai and B. Weiss: Order estimation of Markov chains. IEEE Trans. Inform. Theory 51 (2005), 1496-1497.   DOI:10.1109/tit.2005.844093
  7. G. Morvai and B. Weiss: Estimating the lengths of memory words. IEEE Trans. Inform. Theory 54 (2008), 8, 3804-3807.   DOI:10.1109/tit.2008.926316
  8. G. Morvai and B. Weiss: On universal estimates for binary renewal processes. Ann. Appl. Probab. 18 (2008), 5, 1970-1992.   DOI:10.1214/07-aap512
  9. G. Morvai and B. Weiss: Estimating the residual waiting time for binary stationary time series. In: Proceedings of ITW2009, Volos 2009, pp. 67-70.   DOI:10.1109/itwnit.2009.5158543
  10. G. Morvai and B. Weiss: Universal tests for memory words. IEEE Trans. Inform. Theory 59 (2013), 6873-6879.   DOI:10.1109/tit.2013.2268913
  11. G. Morvai and B. Weiss: Inferring the residual waiting time for binary stationary time series. Kybernetika 50 (2014), 869-882.   DOI:10.14736/kyb-2014-6-0869
  12. B. Ya. Ryabko: Prediction of random sequences and universal coding. Probl. Inform. Transmiss. 24 (1988), 87-96.   CrossRef
  13. P. C. Shields: The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics, American Mathematical Society, Providence 13 1996.   DOI:10.1090/gsm/013