Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
nonparametric estimation; stationary processes
nonparametric estimation; stationary processes
Summary:
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.
References:
[1] Bahr, B. von, Esseen, C. G.: 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 | MR 0170407
[2] Csiszár, I., Shields, P.: The consistency of the BIC Markov order estimator. Annals Statist. 28 (2000), 1601-1619. DOI 10.1214/aos/1015957472 | MR 1835033 | Zbl 1105.62311
[3] Csiszár, I.: 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 | MR 1909476 | Zbl 1060.62092
[4] Feller, W.: An Introduction to Probability Theory and its Applications Vol. I. Third edition. John Wiley and Sons, New York - London - Sydney 1968. MR 0228020
[5] Ghahramani, S.: Fundamentals of Probability with Stochastic Processes. Third edition. Pearson Prentice Hall, Upper Saddle River NJ 2005.
[6] Morvai, G., Weiss, B.: Order estimation of Markov chains. IEEE Trans. Inform. Theory 51 (2005), 1496-1497. DOI 10.1109/tit.2005.844093 | MR 2241507
[7] Morvai, G., Weiss, B.: Estimating the lengths of memory words. IEEE Trans. Inform. Theory 54 (2008), 8, 3804-3807. DOI 10.1109/tit.2008.926316 | MR 2451043 | Zbl 1329.60095
[8] Morvai, G., Weiss, B.: On universal estimates for binary renewal processes. Ann. Appl. Probab. 18 (2008), 5, 1970-1992. DOI 10.1214/07-aap512 | MR 2462556 | Zbl 1158.62053
[9] Morvai, G., Weiss, B.: 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] Morvai, G., Weiss, B.: Universal tests for memory words. IEEE Trans. Inform. Theory 59 (2013), 6873-6879. DOI 10.1109/tit.2013.2268913 | MR 3106870
[11] Morvai, G., Weiss, B.: Inferring the residual waiting time for binary stationary time series. Kybernetika 50 (2014), 869-882. DOI 10.14736/kyb-2014-6-0869 | MR 3301776 | Zbl 1308.62067
[12] Ryabko, B. Ya.: Prediction of random sequences and universal coding. Probl. Inform. Transmiss. 24 (1988), 87-96. MR 0955983 | Zbl 0666.94009
[13] Shields, P. C.: The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics, American Mathematical Society, Providence 13 1996. DOI 10.1090/gsm/013 | MR 1400225 | Zbl 0879.28031
Search
Partner of
