Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
statistical learning; statistical inference; prediction methods; renewal theory
statistical learning; statistical inference; prediction methods; renewal theory
Summary:
A simple renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of $1$'s between successive zeros are independent and identically distributed. After observing ${X_0, X_1, \ldots X_n}$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional distribution of the time to renewal.
References:
[1] Algoet, P.: The strong low of large numbers for sequential decisions under uncertainity. IEEE Trans. Inform. Theory 40 (1994), 609-634. DOI 10.1109/18.335876 | MR 1295308
[2] Bahr, B. von, Esseen, C. G.: Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r \leq 2$. Ann. Math. Statist. 36 (1965), 299-303. DOI 10.1214/aoms/1177700291 | MR 0170407
[3] Denby, L., Vardi, Y.: A short-cut method for estimation in renewal processes. Technometrics 27 (1985), 4, 361-373. DOI 10.1080/00401706.1985.10488075 | MR 0811011
[4] Feller, W.: An Introduction to Probability Theory and its Applications Vol. I. Third edition. John Wiley and Sons, Inc., 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] Györfi, L., Ottucsák, G.: Sequential prediction of unbounded stationary time series. IEEE Trans. Inform. Theory 53 (2007), 1866-1872. DOI 10.1109/tit.2007.894660 | MR 2317147
[7] Khudanpur, S., Narayan, P.: Order estimation for a special class of hidden Markov sources and binary renewal processses. IEEE Trans. Inform. Theory 48 (2002), 1704-1713. DOI 10.1109/tit.2002.1003850 | MR 1909484
[8] Marcinkiewicz, J., Zygmund, A.: Sur les foncions independantes. Fund. Math. 28 (1937), 60-90. MR 0115885
[9] Morvai, G.: Guessing the output of a stationary binary time series. In: Foundations of Statistical Inference (Y. Haitovsky, H. R. Lerche and Y. Ritov, eds.), Physika-Verlag 2003, pp. 207-215. MR 2017826
[10] Morvai, G., Weiss, B.: Forecasting for stationary binary time series. Acta Applic. Math. 79 (2003), 25-34. DOI 10.1023/a:1025862222287 | MR 2021874
[11] Morvai, G., Weiss, B.: Prediction for discrete time series. Probab. Theory Related Fields 132 (2005), 1-12. DOI 10.1007/s00440-004-0386-3 | MR 2136864
[12] Morvai, G., Weiss, B.: On sequential estimation and prediction for discrete time series. Stoch. Dynam. 7 (2007), 4, 417-437. DOI 10.1142/s021949370700213x | MR 2378577 | Zbl 1255.62228
[13] Morvai, G., Weiss, B.: On universal estimates for binary renewal processes. Ann. Appl. Prob. 18 (2008), 5, 1970-1992. DOI 10.1214/07-aap512 | MR 2462556 | Zbl 1158.62053
[14] Morvai, G., Weiss, B.: Estimating the residual waiting time for binary stationary time series. In: ITW 2009, IEEE Information Theory Workshop on Networking and Information Theory, 2009 pp. 67-70. DOI 10.1109/itwnit.2009.5158543 | MR 3301776
[15] Morvai, G., Weiss, B.: Inferring the residual waiting time for binary stationary time series. Kybernetika 50 (2014), 6, 869-882. DOI 10.14736/kyb-2014-6-0869 | MR 3301776 | Zbl 1308.62067
[16] Morvai, G., Weiss, B.: A versatile scheme for predicting renewal times. Kybernetika 52 (2016), 3, 348-358. DOI 10.14736/kyb-2016-3-0348 | MR 3532511
[17] Nobel, A. B.: On optimal sequential prediction for general processes. IEEE Trans. Inform. Theory 49 (2003), 83-98. DOI 10.1109/tit.2002.806141 | MR 1965889
[18] Peña, E. A., Strawderman, R., Hollander, M.: Nonparametric estimation with recurrent event data. J. Amer. Statist. Assoc. 96 (2001), 456, 1299-1315. DOI 10.1198/016214501753381922 | MR 1946578
[19] Petrov, V. V.: Limit Theorems of Probability Theory. Clarendon Press, Oxford 1995. DOI 10.1017/s001309150002335x | MR 1353441
[20] Ryabko, B. Y.: Prediction of random sequences and universal coding. Probl. Inform. Trans. 24 (1988), 87-96. MR 0955983 | Zbl 0666.94009
[21] Shields, P. C.: The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics 13, American Mathematical Society, Providence 1996. DOI 10.1090/gsm/013 | MR 1400225 | Zbl 0879.28031
[22] Shiryayev, A. N.: Probability. Second edition. Springer-Verlag, New York 1996. MR 1368405
[23] Takahashi, H.: Computational limits to nonparametric estimation for ergodic processes. IEEE Trans. Inform. Theory 57 (2011), 10, 6995-6999. DOI 10.1109/tit.2011.2165791 | MR 2882275
[24] Vardi, Y.: Nonparametric estimation in renewal processes. Ann. Statist. (1982), 772-785. DOI 10.1214/aos/1176345870 | MR 0663431
Search
Partner of
