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Keywords:
nonparametric estimation; stationary processes
Summary:
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.
References:
[1] Algoet, P.: Universal schemes for learning the best nonlinear predictor given the infinite past and side information. IEEE Trans. Inform. Theory 45 (1999), 1165-1185. DOI 10.1109/18.761258 | MR 1686250 | Zbl 0959.62078
[2] Bailey, D. H.: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph. D. Thesis, Stanford University 1976. MR 2626644
[3] Bunea, F., Nobel, A.: Sequential procedures for aggregating arbitrary estimators of a conditional mean. IEEE Trans. Inform. Theory 54 (2008), 4, 1725-1735. DOI 10.1109/TIT.2008.917657 | MR 2450298
[4] Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II. Second edition. John Wiley and Sons, New York - London - Sydney 1971. MR 0270403
[5] Györfi, L., Morvai, G., Yakowitz, S.: Limits to consistent on-line forecasting for ergodic time series. IEEE Trans. Inform. Theory 44 (1998), 886-892. DOI 10.1109/18.661540 | MR 1607704 | Zbl 0899.62122
[6] Morvai, G., Weiss, B.: Inferring the conditional mean. Theory Stoch. Process. 11 (2005), 112-120. MR 2327452 | Zbl 1164.62382
[7] 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
[8] Morvai, G., Weiss, B.: On sequential estimation and prediction for discrete time series. Stoch. Dyn. 7 (2007), 4, 417-437. DOI 10.1142/S021949370700213X | MR 2378577 | Zbl 1255.62228
[9] 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
[10] Morvai, G., Weiss, B.: Estimating the residual waiting time for binary stationary time series. In: Proc. ITW2009, Volos 2009, pp. 67-70.
[11] Morvai, G., Weiss, B.: A note on prediction for discrete time series. Kybernetika 48 (2012), 4, 809-823. MR 3013400
[12] Ryabko, B. Ya.: Prediction of random sequences and universal coding. Probl. Inf. Trans. 24 (1988), 87-96. MR 0955983 | Zbl 0666.94009
[13] Ryabko, D., Ryabko, B.: Nonparametric statistical inference for ergodic processes. IEEE Trans. Inform. Theory 56 (2010), 3, 1430-1435. DOI 10.1109/TIT.2009.2039169 | MR 2723689
[14] Shiryayev, A. N.: Probability. Springer-Verlag, New York 1984. MR 0737192
[15] 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
[16] Zhou, Z., Xu, Z., Wu, W. B.: Long-term prediction intervals of time series. IEEE Trans. Inform. Theory 56 (2010), 3, 1436-1446. DOI 10.1109/TIT.2009.2039158 | MR 2723690

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