Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
nonparametric estimation; stationary processes
nonparametric estimation; stationary processes
Summary:
Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$ and that $f(X)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional expectation $E(f(X_{\lambda_n+1})|X_0,\dots,X_{\lambda_n} )$ from the observations $(X_0,\dots,X_{\lambda_n})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} \frac{n}{\lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upper bounded by a polynomial, eventually almost surely.
References:
[1] D. H. Bailey: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph.D. Thesis, Stanford University 1976. MR 2626644
[2] A. Berlinet, I. Vajda, E. C. van der Meulen: About the asymptotic accuracy of Barron density estimates. IEEE Trans. Inform. Theory 44 (1998), 3, 999-1009. DOI 10.1109/18.669143 | MR 1616679 | Zbl 0952.62029
[3] K. L. Chung: A note on the ergodic theorem of information theory. Ann. Math. Statist. 32 (1961), 612-614. DOI 10.1214/aoms/1177705069 | MR 0131782 | Zbl 0115.35503
[4] T. M. Cover, J. Thomas: Elements of Information Theory. Wiley, 1991. MR 1122806 | Zbl 1140.94001
[5] I. Csiszár, P. Shields: The consistency of the BIC Markov order estimator. Ann. Statist. 28 (2000), 1601-1619. DOI 10.1214/aos/1015957472 | MR 1835033 | Zbl 1105.62311
[6] 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 | MR 1909476 | Zbl 1060.62092
[7] G. A. Darbellay, I. Vajda: Estimation of the information by an adaptive partitioning of the observation space. {IEEE Trans. Inform. Theory 45 (1999), 4, 1315-1321.} DOI 10.1109/18.761290 | MR 1686274 | Zbl 0957.94006
[8] J. Feistauerová, I. Vajda: Testing system entropy and prediction error probability. IEEE Trans. Systems Man Cybernet. 23 (1993), 5 1352-1358. DOI 10.1109/21.260666
[9] L. Györfi, G. Morvai, S. Yakowitz: 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
[10] L. Györfi, G. Morvai, I. Vajda: Information-theoretic methods in testing the goodness of fit. {In: Proc. 2000 IEEE Internat. Symposium on Information Theory}, ISIT 2000, New York and Sorrento, p. 28.
[11] W. Hoeffding: Probability inequalities for sums of bounded random variables. {J. Amer. Statist. Assoc.} 58 (1963), 13-30. DOI 10.1080/01621459.1963.10500830 | MR 0144363 | Zbl 0127.10602
[12] S. Kalikow: Random Markov processes and uniform martingales. {Israel J. Math.} 71 (1990), 33-54. DOI 10.1007/BF02807249 | MR 1074503 | Zbl 0711.60041
[13] M. Keane: Strongly mixing g-measures. {Invent. Math. } 16 (1972), 309-324. DOI 10.1007/BF01425715 | MR 0310193 | Zbl 0241.28014
[14] H. Luschgy, L. A. Rukhin, I. Vajda: Adaptive tests for stochastic processes in the ergodic case. {Stochastic Process. Appl.} 45 (1993), 1, 45-59. MR 1204860 | Zbl 0770.62071
[15] G. Morvai, I. Vajda: A survay on log-optimum portfolio selection. In: Second European Congress on Systems Science, Afcet, Paris 1993, pp. 936-944.
[16] G. Morvai, B. Weiss: Prediction for discrete time series. {Probab. Theory Related Fields} 132 (2005), 1-12. DOI 10.1007/s00440-004-0386-3 | MR 2136864
[17] G. Morvai, B. Weiss: Estimating the memory for finitarily Markovian processes. { Ann. Inst. H. Poincaré Probab. Statist.} 43 (2007), 15-30. DOI 10.1016/j.anihpb.2005.11.001 | MR 2288267 | Zbl 1106.62094
[18] B. Ya. Ryabko: Prediction of random sequences and universal coding. {Problems Inform. Transmission} 24 (1988), 87-96. MR 0955983 | Zbl 0666.94009
[19] B. Ryabko: Compression-based methods for nonparametric prediction and estimation of some characteristics of time series. {IEEE Trans. Inform. Theory} 55 (2009), 9, 4309-4315. DOI 10.1109/TIT.2009.2025546 | MR 2582884
[20] I. Vajda, F. Österreicher: Existence, uniqueness and evaluation of log-optimal investment portfolio. {Kybernetika} 29 (1993), 2, 105-120. MR 1227745 | Zbl 0799.90013
[21] I. Vajda, P. Harremoës: On the Bahadur-efficient testing of uniformity by means of entropy. IEEE Trans. Inform. Theory 54 (2008), 321-331. DOI 10.1109/TIT.2007.911155 | MR 2446756
Search
Partner of
