메뉴 건너뛰기




Volumn 40, Issue 3, 1994, Pages 609-633

The Strong Law of Large Numbers for Sequential Decisions Under Uncertainty

Author keywords

asymptotic optimality; ergodic theory; learning from experience; Pattern classification; Sequential decisions; universal decision schemes

Indexed keywords

CLASSIFICATION (OF INFORMATION); INFORMATION THEORY; MATHEMATICAL MODELS; NUMBER THEORY; OPTIMIZATION; PATTERN RECOGNITION; RANDOM PROCESSES;

EID: 0028428283     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.335876     Document Type: Article
Times cited : (59)

References (107)
  • 1
    • 0000551326 scopus 로고
    • Universal schemes for prediction, gambling, and portfolio selection
    • Apr.
    • P. H. Algoet, “Universal schemes for prediction, gambling, and portfolio selection,” Ann. Prob., vol. 20, 901-941, Apr. 1992.
    • (1992) Ann. Prob. , vol.20
    • Algoet, P.H.1
  • 2
    • 0000801141 scopus 로고
    • Asymptotic optimality and asymptotic equipartition properties of log-optimum investment
    • Apr.
    • P. H. Algoet and T. M. Cover, “Asymptotic optimality and asymptotic equipartition properties of log-optimum investment,” Ann. Prob., vol. 16, pp. 876-898, Apr. 1988
    • (1988) Ann. Prob. , vol.16 , pp. 876-898
    • Algoet, P.H.1    Cover, T.M.2
  • 3
    • 0039048754 scopus 로고
    • A sandwich proof of the Shannon-McMillan-Breiman theorem
    • Apr.
    • ——, “A sandwich proof of the Shannon-McMillan-Breiman theorem,” Ann. Prob., vol. 16, pp. 899-909, Apr. 1988.
    • (1988) Ann. Prob. , vol.16 , pp. 899-909
  • 4
    • 0000003501 scopus 로고
    • Almost everywhere convergence of prediction sequences in Lp(1 < p < ∞)
    • T. Ando and I. Amemiya, “Almost everywhere convergence of prediction sequences in Lp(1 < p < ∞),” Z. Wahrscheinlichkeits-theorie verw. Gebiete, vol. 4, pp. 113-120, 1965.
    • (1965) Z. Wahrscheinlichkeits-theorie verw. Gebiete , vol.4 , pp. 113-120
    • Ando, T.1    Amemiya, I.2
  • 5
    • 0000039074 scopus 로고
    • The strong ergodic theorem for densities: generalized Shannon-McMillan-Breiman theorem
    • Nov.
    • A. R. Barron, “The strong ergodic theorem for densities: generalized Shannon-McMillan-Breiman theorem,” Ann. Prob., vol. 13, pp. 1292-1303, Nov. 1985.
    • (1985) Ann. Prob. , vol.13 , pp. 1292-1303
    • Barron, A.R.1
  • 6
    • 0001601975 scopus 로고
    • Game-theoretic optimal portfolios
    • June
    • R. Bell and T. M. Cover, “Game-theoretic optimal portfolios,” Management Sci., vol. 34, pp. 724-733, June 1988.
    • (1988) Management Sci. , vol.34 , pp. 724-733
    • Bell, R.1    Cover, T.M.2
  • 7
  • 9
    • 84972545864 scopus 로고
    • An analog of the minimax theorem for vector payoffs
    • ——, “An analog of the minimax theorem for vector payoffs,” Pacific J. Math., vol. 6, pp. 1-8, 1956.
    • (1956) Pacific J. Math. , vol.6 , pp. 1-8
  • 10
    • 84972508275 scopus 로고
    • A converse to the dominated convergence theorem
    • D. Blackwell and L. E. Dubins, “A converse to the dominated convergence theorem,” Illinois J. Math., vol. 7, 508-514, 1963.
    • (1963) Illinois J. Math. , vol.7
    • Blackwell, D.1    Dubins, L.E.2
  • 11
    • 0001531440 scopus 로고
    • The individual ergodic theorem of information theory
    • L. Breiman, “The individual ergodic theorem of information theory,” Ann. Math. Statist., vol. 28, pp. 809-811, “Correction,” Ann. Math. Statist., vol. 31, pp. 809-810, 1960.
    • (1960) Ann. Math. Statist. , vol.28 , pp. 809-811
    • Breiman, L.1
  • 12
    • 0002522795 scopus 로고
    • Optimal gambling systems for favorable games
    • —,”Optimal gambling systems for favorable games,” in Proc. 4th Berkeley Symp. Math. Statist. Prob. Berkeley, CA: Univ. California Press, 1961, vol. 1, pp. 65-78.
    • (1961) Proc. 4th Berkeley Symp. Math. Statist. Prob. Berkeley , vol.1 , pp. 65-78
  • 13
    • 0001570707 scopus 로고
    • Successive conditional expectations of an integrable function
    • D. L. Burkholder, “Successive conditional expectations of an integrable function,” Ann. Math. Statist., vol. 33, pp. 887-893, 1962.
    • (1962) Ann. Math. Statist. , vol.33 , pp. 887-893
    • Burkholder, D.L.1
  • 14
    • 0010570297 scopus 로고
    • Note on some strong laws of large numbers
    • K. L. Chung, “Note on some strong laws of large numbers,” Amer. J. Math., vol. 69, pp. 189-192, 1947.
    • (1947) Amer. J. Math. , vol.69 , pp. 189-192
    • Chung, K.L.1
  • 15
    • 0000315398 scopus 로고
    • Local convergence of martingales and the law of large numbers
    • Y. S. Chow, “Local convergence of martingales and the law of large numbers,” Ann. Math. Statist., vol. 36, pp. 552-558, 1965.
    • (1965) Ann. Math. Statist. , vol.36 , pp. 552-558
    • Chow, Y.S.1
  • 16
    • 0008954445 scopus 로고
    • On a strong law of large numbers for martingales
    • —, “On a strong law of large numbers for martingales,” Ann. Math. Statist., vol. 38, p. 610, 1967.
    • (1967) Ann. Math. Statist. , vol.38 , pp. 610
  • 17
    • 0004269083 scopus 로고
    • Probability Theory, 2nd ed
    • New York: Springer-Verlag
    • Y. S. Chow and H. Teicher, Probability Theory, 2nd ed. New York: Springer-Verlag, 1988.
    • (1988)
    • Chow, Y.S.1    Teicher, H.2
  • 18
    • 0442294016 scopus 로고
    • Compound decisions and empirical Bayes (with discussion)
    • J. P. Copas, “Compound decisions and empirical Bayes (with discussion),” J. Roy. Statist. Soc., Ser. B, vol. 31, pp. 397-425, 1969.
    • (1969) J. Roy. Statist. Soc. , vol.31 , pp. 397-425
    • Copas, J.P.1
  • 19
    • 0002396094 scopus 로고
    • Behavior of sequential predictors of binary sequences
    • T. M. Cover, “Behavior of sequential predictors of binary sequences,” in Trans 4th Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1965. Prague: Czechoslovak Acad. Sci., 1967, pp. 263-272.
    • (1967) Trans 4th Prague Conf. Inform. Theory , pp. 263-272
    • Cover, T.M.1
  • 20
    • 83455210966 scopus 로고    scopus 로고
    • personal communication
    • —, personal communication.
  • 21
    • 84946245150 scopus 로고
    • Ergodic process selection
    • —, “Ergodic process selection" in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, Ed. New York: Springer-Verlag, 1987, p. 153.
    • (1987) Open Problems in Communication and Computation , pp. 153
  • 22
    • 84986753988 scopus 로고
    • Universal portfolios
    • Jan.
    • —, “Universal portfolios,” Math. Finance, vol. 1, pp. 1-29, Jan. 1991.
    • (1991) Math. Finance , vol.1 , pp. 1-29
  • 23
    • 0008241596 scopus 로고
    • Empirical Bayes stock market portfolios
    • June
    • T. M. Cover and D. H. Gluss, “Empirical Bayes stock market portfolios,” Adv. Appl. Math., vol. 7, pp. 170-181, June 1986.
    • (1986) Adv. Appl. Math. , vol.7 , pp. 170-181
    • Cover, T.M.1    Gluss, D.H.2
  • 24
    • 0017503437 scopus 로고
    • Compound Bayes predictors for sequences with apparent Markov structure
    • June
    • T. M. Cover and A. Shenhar,”Compound Bayes predictors for sequences with apparent Markov structure,” IEEE Trans. Syst., Man, Cybern., vol. SMC-7, pp. 421-424, June 1977.
    • (1977) IEEE Trans. Syst. , vol.SMC-7 , pp. 421-424
    • Cover, T.M.1    Shenhar, A.2
  • 25
    • 0000925020 scopus 로고
    • The strong law of large numbers for k-means and best possible nets of Banach valued random variables
    • J. A. Cuesta and C. Matran, “The strong law of large numbers for k-means and best possible nets of Banach valued random variables,” Prob. Theory Related Fields, vol. 78, pp. 523-534, 1988.
    • (1988) Prob. Theory Related Fields , vol.78 , pp. 523-534
    • Cuesta, J.A.1    Matran, C.2
  • 26
    • 0000582742 scopus 로고
    • Present position and potential developments: Some personal views, statistical theory, the prequential approach
    • A. P. Dawid, “Present position and potential developments: Some personal views, statistical theory, the prequential approach,” J. Roy. Statist. Soc., ser. A, vol. 147, pp. 278-292, 1984.
    • (1984) J. Roy. Statist. Soc. , vol.147 , pp. 278-292
    • Dawid, A.P.1
  • 27
    • 84912451114 scopus 로고
    • Multidimensional information equalities and prediction
    • M. DeGroot and M. M. Rao, “Multidimensional information equalities and prediction,” in Proc. Int. Symp. Multivariate Anal., P. R. Krishnaiah, Ed. New York: Academic, 1966, pp. 287-313.
    • (1966) Proc. Int. Symp. Multivariate Anal. , pp. 287-313
    • DeGroot, M.1    Rao, M.M.2
  • 28
    • 0003954462 scopus 로고
    • Stochastic Processes
    • New York: Wiley
    • J. L. Doob, Stochastic Processes. New York: Wiley, 1953.
    • (1953)
    • Doob, J.L.1
  • 29
    • 84946245667 scopus 로고
    • Continuous random decision processes controlled by experience
    • M. Driml and A. Spacek, “Continuous random decision processes controlled by experience,” in Trans. 1st Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1956. Prague: Czechoslovak Acad. Sci,, 1957, pp. 43-60.
    • (1957) Trans. 1st Prague Conf. Inform. Theory , pp. 43-60
    • Driml, M.1    Spacek, A.2
  • 30
    • 84935756450 scopus 로고
    • On experience theory problems
    • M. Driml and O. Hanš, “On experience theory problems,” in Trans. 2nd Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1959. Prague: Czechoslovak Acad. Sci., 1960, pp. 93-111.
    • (1960) Trans. 2nd Prague Conf. Inform. Theory , pp. 93-111
    • Driml, M.1    Hanš, O.2
  • 31
    • 0001421130 scopus 로고
    • Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems
    • Dec.
    • J. Dupačová and R. Wets, “Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems,” Ann. Statist., vol. 16, pp. 1517-1549, Dec. 1988.
    • (1988) Ann. Statist. , vol.16 , pp. 1517-1549
    • Dupačová, J.1    Wets, R.2
  • 32
    • 51249175084 scopus 로고
    • Asymptotic distribution theory of statistical functionals
    • W. Esty, R. Gillette, M. Hamilton, and D. Taylor, “Asymptotic distribution theory of statistical functionals,” Ann. Inst. Statist. Math., ser. A, vol. 37, pp. 109-129, 1985.
    • (1985) Ann. Inst. Statist. Math. , vol.37 , pp. 109-129
    • Esty, W.1    Gillette, R.2    Hamilton, M.3    Taylor, D.4
  • 33
    • 0642360650 scopus 로고
    • Experience in statistical decision problems
    • V. Fabian and A. Spacek, “Experience in statistical decision problems,” Czechoslovak Math. J., vol. 6, pp. 190-194, 1956, “Correction,” Czechoslovak Math. J., vol. 6, p. 434, 1956.
    • (1956) Czechoslovak Math. J. , vol.6 , pp. 434
    • Fabian, V.1    Spacek, A.2
  • 34
    • 84946244626 scopus 로고
    • von Mises Calculus for Statistical Functionals
    • L. T. Femholtz, von Mises Calculus for Statistical Functionals. New York: Springer-Verlag, 1983.
    • (1983) New York: Springer-Verlag
    • Femholtz, L.T.1
  • 36
    • 0002610193 scopus 로고
    • Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1)
    • R. D. Gill, “Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1),” Scand. J. Statist., vol. 16, pp. 97-128, 1989.
    • (1989) Scand. J. Statist. , vol.16 , pp. 97-128
    • Gill, R.D.1
  • 37
    • 2442584809 scopus 로고
    • Sequential compound estimation
    • D. C. Gilliland, “Sequential compound estimation,” Ann. Math. Statist., vol. 39, pp. 1890-1905, 1968.
    • (1968) Ann. Math. Statist. , vol.39 , pp. 1890-1905
    • Gilliland, D.C.1
  • 38
    • 84946245431 scopus 로고
    • Approximation to Bayes risk in sequences of non-finite games
    • —, “Approximation to Bayes risk in sequences of non-finite games,” Ann. Math. Statist., vol. 40, pp. 467-474, 1969.
    • (1969) Ann. Math. Statist. , vol.40 , pp. 467-474
  • 39
    • 0015399410 scopus 로고
    • Asymptotic risk stability resulting from play against the past in a sequence of decision problems
    • Sept.
    • —, “Asymptotic risk stability resulting from play against the past in a sequence of decision problems,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 614-617, Sept. 1972.
    • (1972) IEEE Trans. Inform. Theory , vol.IT-18 , pp. 614-617
  • 40
    • 2442473449 scopus 로고
    • On an extended compound decision problem
    • D. C. Gilliland and J. F. Hannan, “On an extended compound decision problem,” Ann. Math. Statist., vol. 40, pp. 1536-1541, 1969.
    • (1969) Ann. Math. Statist. , vol.40 , pp. 1536-1541
    • Gilliland, D.C.1    Hannan, J.F.2
  • 41
    • 0017994541 scopus 로고
    • On continuity of the Bayes response
    • July
    • D. C. Gilliland and M. K. Helmers, “On continuity of the Bayes response,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 506-508, July 1978.
    • (1978) IEEE Trans. Inform. Theory , vol.IT-24 , pp. 506-508
    • Gilliland, D.C.1    Helmers, M.K.2
  • 42
    • 0001300935 scopus 로고
    • Pseudo maximum likelihood methods: theory
    • May
    • C. Gouriéroux, A. Montfort, and A. Trognon, “Pseudo maximum likelihood methods: theory,” Econometrica, vol. 52, pp. 681-700, May 1984.
    • (1984) Econometrica , vol.52 , pp. 681-700
    • Gouriéroux, C.1    Montfort, A.2    Trognon, A.3
  • 44
    • 0010774684 scopus 로고
    • Concavity and estimation
    • Dec.
    • S. J. Haberman, “Concavity and estimation,” Ann. Statist., vol. 17, pp. 1631-1661, Dec. 1989.
    • (1989) Ann. Statist. , vol.17 , pp. 1631-1661
    • Haberman, S.J.1
  • 45
    • 84946245658 scopus 로고
    • Ergodic process selection
    • B. Hajek, “Ergodic process selection,” in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, Ed. New York: Springer-Verlag, 1987, pp. 199-203.
    • (1987) Open Problems in Communication and Computation , pp. 199-203
    • Hajek, B.1
  • 46
    • 0003704752 scopus 로고
    • Martingale Limit Theory and Its Applications
    • New York: Academic
    • P. Hall and C. C. Hyde, Martingale Limit Theory and Its Applications. New York: Academic, 1980.
    • (1980)
    • Hall, P.1    Hyde, C.C.2
  • 47
    • 0001976283 scopus 로고
    • Approximation to Bayes risk in repeated play
    • J. F. Hannan, “Approximation to Bayes risk in repeated play,” in Contributions to the Theory of Games III, Ann. Math. Studies. Princeton, NJ: Princeton Univ. Press, 1957, vol. 39, pp. 97-139.
    • (1957) Contributions to the Theory of Games III , vol.39 , pp. 97-139
    • Hannan, J.F.1
  • 48
    • 0003179923 scopus 로고
    • Asymptotic solutions for the compound sequential decision problem for two completely specified distributions
    • J. F. Hannan and H. Robbins, “Asymptotic solutions for the compound sequential decision problem for two completely specified distributions,” Ann. Math. Statist., vol. 26, pp. 37-51, 1955.
    • (1955) Ann. Math. Statist. , vol.26 , pp. 37-51
    • Hannan, J.F.1    Robbins, H.2
  • 49
    • 0038929740 scopus 로고
    • Rate of convergence in the compound decision problem for two completely specified distributions
    • J. F. Hannan and J. R. Van Ryzin, “Rate of convergence in the compound decision problem for two completely specified distributions,” Ann. Math. Statist., vol. 36, pp. 1743-1752, 1965.
    • (1965) Ann. Math. Statist. , vol.36 , pp. 1743-1752
    • Hannan, J.F.1    Van Ryzin, J.R.2
  • 50
    • 84946245405 scopus 로고
    • On conditional Bayes loss
    • O. Hanš, “On conditional Bayes loss,” in Trans 3rd Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1962. Prague: Czechoslovak Acad. Sci., 1964, pp. 207-221.
    • (1964) Trans 3rd Prague Conf. Inform. Theory , pp. 207-221
    • Hanš, O.1
  • 52
    • 0009114174 scopus 로고
    • Best Φ-and NΦ-approaximants in Orlicz spaces of vector valued functions
    • ——, “Best Φ-and NΦ-approaximants in Orlicz spaces of vector valued functions,” Z. Wahrscheinlichkeitstheorie verw. Gebiete, vol. 58, pp. 309-329, 1981.
    • (1981) Z. Wahrscheinlichkeitstheorie verw. Gebiete , vol.58 , pp. 309-329
  • 53
    • 0003157339 scopus 로고
    • Robust estimation of a location parameter
    • P. J. Huber, “Robust estimation of a location parameter,” Ann. Math. Statist., vol. 35, pp. 73-101, 1964.
    • (1964) Ann. Math. Statist. , vol.35 , pp. 73-101
    • Huber, P.J.1
  • 54
    • 0000250624 scopus 로고
    • The behavior of maximum likelihood estimates under standardized conditions
    • ——, “The behavior of maximum likelihood estimates under standardized conditions,” in Proc. 5th Berkeley Symp. Math. Statist. Prob. Berkeley, CA: Univ. California Press, 1967, vol. I, pp. 221-233.
    • (1967) Proc. 5th Berkeley Symp. Math. Statist. Prob. Berkeley , vol.1 , pp. 221-233
  • 55
    • 0004262735 scopus 로고
    • New York: Wiley
    • ——, Robust Statistics. New York: Wiley, 1981.
    • (1981) Robust Statistics
  • 56
    • 84946244259 scopus 로고
    • On the experience problem
    • S. Jilovec, “On the experience problem,” in Trans. 6th Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1971. Prague: Czechoslovak Acad. Sci., 1973, pp. 207-221.
    • (1973) Trans. 6th Prague Conf. Inform. Theory , pp. 207-221
    • Jilovec, S.1
  • 57
    • 84946244115 scopus 로고
    • Repetitive play of a game against nature
    • S. Jilovec and B. Subert, “Repetitive play of a game against nature,” Aplikace Matematiky, vol. 12, pp. 383-397, 1967.
    • (1967) Aplikace Matematiky , vol.12 , pp. 383-397
    • Jilovec, S.1    Subert, B.2
  • 59
    • 0000733254 scopus 로고
    • A new interpretation of information rate
    • J. L. Kelly, Jr., “A new interpretation of information rate,” Bell Syst. Tech. J., vol. 35, pp. 917-926, 1956.
    • (1956) Bell Syst. Tech. J. , vol.35 , pp. 917-926
    • Kelly, J.L.1
  • 60
    • 84968465513 scopus 로고
    • An ergodic theorem for constrained sequences of functions
    • Oct.
    • J. C. Kieffer, “An ergodic theorem for constrained sequences of functions,” Bull. Amer. Math. Soc., vol. 21, pp. 249-254, Oct. 1989.
    • (1989) Bull. Amer. Math. Soc. , vol.21 , pp. 249-254
    • Kieffer, J.C.1
  • 61
    • 0002706524 scopus 로고
    • An almost sure convergence theorem for sequences of random variables selected from log-convex sets
    • ——, “An almost sure convergence theorem for sequences of random variables selected from log-convex sets,” in Almost Sure Convergence II, A. Bellow and R. L. Jones, Ed. San Diego, CA: Academic, 1991, pp. 151-166.
    • (1991) Almost Sure Convergence II , pp. 151-166
  • 62
    • 84946244046 scopus 로고
    • Best approximants in Musielak-Orlicz spaces
    • S. J. Kilmer, “Best approximants in Musielak-Orlicz spaces,” in Approximation Theory, G. A. Anastassiou, Ed., Lecture Notes on Pure and Appl. Math. New York: Marcel Dekker, 1992, vol. 138, pp. 363-376.
    • (1992) Approximation Theory , vol.138 , pp. 363-376
    • Kilmer, S.J.1
  • 63
    • 84946245924 scopus 로고
    • Information bounds, certainty equivalence and learning in asymptotically efficient adaptive control of time-invariant control of time-invariant stochastic systems
    • T. L. Lai, “Information bounds, certainty equivalence and learning in asymptotically efficient adaptive control of time-invariant control of time-invariant stochastic systems,” Tech. Rep., Dep. Statist., Stanford Univ., 1991.
    • (1991) Tech. Rep.
    • Lai, T.L.1
  • 65
    • 84946244432 scopus 로고
    • Theorie de l'Addition des Variables Aléatories, 2nd Ed
    • P. Levy, Theorie de l'Addition des Variables Aléatories, 2nd Ed. Paris: Gauthier-Villars, 1954.
    • (1954) Paris: Gauthier-Villars
    • Levy, P.1
  • 67
    • 0001457509 scopus 로고
    • Some methods for classification and analysis of multivariate observations
    • J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proc. 5th Berkeley Symp. Math. Statist. Prob. Berkeley, CA: Univ. California Press, 1967, vol. I, 281-297.
    • (1967) Proc. 5th Berkeley Symp. Math. Statist. Prob. Berkeley , vol.1
    • MacQueen, J.1
  • 68
    • 0027599448 scopus 로고
    • Some properties of sequential predictors of binary Markov sources
    • May
    • N. Merhav, M. Feder, and M. Gutman, “Some properties of sequential predictors of binary Markov sources,” in IEEE Trans. Inform. Theory, May 1993, vol. 39, pp. 887-892.
    • (1993) IEEE Trans. Inform. Theory , vol.39 , pp. 887-892
    • Merhav, N.1    Feder, M.2    Gutman, M.3
  • 69
    • 0642268720 scopus 로고
    • Two breakthroughs in the theory of statistical decision making
    • J. Neyman, “Two breakthroughs in the theory of statistical decision making,” Rev. Inst. Int. Statist., vol. 30, pp. 11-27, 1962.
    • (1962) Rev. Inst. Int. Statist. , vol.30 , pp. 11-27
    • Neyman, J.1
  • 70
    • 21144480023 scopus 로고
    • Asymptotics for M-estimators defined by convex minimization
    • Sept.
    • W. Niemiro, “Asymptotics for M-estimators defined by convex minimization,” Ann. Statist., vol. 20, pp. 1514-1533, Sept. 1992.
    • (1992) Ann. Statist. , vol.20 , pp. 1514-1533
    • Niemiro, W.1
  • 71
    • 3142591626 scopus 로고
    • On the Shannon-Pérez-Moy theorem
    • S. Orey, “On the Shannon-Pérez-Moy theorem,” Contemporary Math., vol. 41, pp. 319-327, 1985.
    • (1985) Contemporary Math. , vol.41 , pp. 319-327
    • Orey, S.1
  • 72
    • 51249183423 scopus 로고
    • Guessing the next output of a stationary process
    • D. Ornstein, “Guessing the next output of a stationary process,” Israel J. Math., vol. 30, pp. 292-296, 1978.
    • (1978) Israel J. Math. , vol.30 , pp. 292-296
    • Ornstein, D.1
  • 73
    • 84946244508 scopus 로고
    • L'expérience et Pinformation puisée dans elle a l'aide des lois limites de la theorie des probability
    • A. Perez, “L'expérience et Pinformation puisée dans elle a l'aide des lois limites de la theorie des probability,” in Trans. 2nd Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1959. Prague: Czechoslovak Acad. Sci., 1960, pp. 499-519.
    • (1960) Trans. 2nd Prague Conf. Inform. Theory , pp. 499-519
    • Perez, A.1
  • 74
    • 0003831421 scopus 로고
    • Ergodic Theory
    • Cambridge: Cambridge Univ. Press
    • K. E. Petersen, Ergodic Theory. Cambridge: Cambridge Univ. Press, 1983.
    • (1983)
    • Petersen, K.E.1
  • 75
    • 0000963889 scopus 로고
    • Strong consistency of k-means clustering
    • Jan.
    • D. Pollard, “Strong consistency of k-means clustering,” Ann. Statist., vol. 9, pp. 135-140, Jan. 1981.
    • (1981) Ann. Statist. , vol.9 , pp. 135-140
    • Pollard, D.1
  • 76
    • 0020101636 scopus 로고
    • Quantization and the method of k-means
    • Mar.
    • —, “Quantization and the method of k-means,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 199-205, Mar. 1982.
    • (1982) IEEE Trans. Inform. Theory , vol.IT-28 , pp. 199-205
  • 77
    • 0000963885 scopus 로고
    • A central limit theorem for k-means clustering
    • Nov.
    • —, “A central limit theorem for k-means clustering,” Ann. Prob., vol. 10, pp. 919-926, Nov. 1982.
    • (1982) Ann. Prob. , vol.10 , pp. 919-926
  • 79
    • 49649154211 scopus 로고
    • Abstract nonlinear prediction and operator martingales
    • —, “Abstract nonlinear prediction and operator martingales,” Multivariate Anal., vol. 1, pp. 129-159, 1971. “Correction,” Multivariate Anal., vol. 9, p. 615, 1979.
    • (1979) Multivariate Anal. , vol.1 , pp. 129-159
  • 80
    • 0000318553 scopus 로고
    • Stochastic complexity and modeling
    • Sept.
    • J. Rissanen, “Stochastic complexity and modeling,” Ann Statist., vol. 14, pp. 1080-1100, Sept. 1986.
    • (1986) Ann Statist. , vol.14 , pp. 1080-1100
    • Rissanen, J.1
  • 81
    • 0011026447 scopus 로고
    • Asymptotically subminimax solutions of compound statistical decision problems
    • H. Robbins, “Asymptotically subminimax solutions of compound statistical decision problems,” in Proc. 2nd Berkeley Symp. Math. Statist. Prob. Berkeley, CA: Univ. California Press, 1950, pp. 131-148.
    • (1950) Proc. 2nd Berkeley Symp. Math. Statist. Prob. Berkeley , pp. 131-148
    • Robbins, H.1
  • 82
    • 0000033354 scopus 로고
    • An empirical Bayes approach to statistics
    • —, “An empirical Bayes approach to statistics,” in Proc. 3rd Berkeley Symp. Math. Statist. Prob. Berkeley, CA: Univ. California Press, 1955, vol. I, pp. 157-164.
    • (1955) Proc. 3rd Berkeley Symp. Math. Statist. Prob. Berkeley , vol.1 , pp. 157-164
  • 83
    • 0002533801 scopus 로고
    • The empirical Bayes approach to statistical decision problems
    • —, “The empirical Bayes approach to statistical decision problems,” Ann. Math. Statist., vol. 35, pp. 1-20, 1964.
    • (1964) Ann. Math. Statist. , vol.35 , pp. 1-20
  • 84
    • 0003389747 scopus 로고
    • Asymptotic solutions of the sequential compound decision problem
    • E. Samuel, “Asymptotic solutions of the sequential compound decision problem,” Ann. Math. Statist., vol. 34, pp. 1079-1094, 1963.
    • (1963) Ann. Math. Statist. , vol.34 , pp. 1079-1094
    • Samuel, E.1
  • 85
    • 84946245278 scopus 로고
    • Convergence of the losses of certain decision rules for the sequential compounded decision problem
    • —, “Convergence of the losses of certain decision rules for the sequential compounded decision problem,” Ann. Math. Statist., vol. 35, 1606-1621, 1964.
    • (1964) Ann. Math. Statist. , vol.35
  • 86
    • 84946244279 scopus 로고
    • Sequential compounded estimators
    • —, “Sequential compounded estimators,” Ann. Math. Statist., vol. 36, 879-889, 1965.
    • (1965) Ann. Math. Statist. , vol.36
  • 87
    • 84919677556 scopus 로고
    • The compound statistical decision problem
    • —, “The compound statistical decision problem,” Sankhya: Indian J. Statist., ser. A, vol. 29, pp. 123-140, 1967.
    • (1967) Sankhya: Indian J. Statist. , vol.29 , pp. 123-140
  • 88
    • 0027553276 scopus 로고
    • Universal redundancy rates do not exist
    • Mar.
    • P. S. Shields, “Universal redundancy rates do not exist,” IEEE Trans. Inform. Theory, vol. 39, pp. 520-524, Mar. 1993.
    • (1993) IEEE Trans. Inform. Theory , vol.39 , pp. 520-524
    • Shields, P.S.1
  • 91
    • 84946244823 scopus 로고
    • Processus aleatoires de decision statistique conditionnee. Le calcul des probability et ses applications
    • —, “Processus aleatoires de decision statistique conditionnee. Le calcul des probability et ses applications,” in Colloques Inter-national: du Centre National de la Recherche Scientifique LXXXVII, Paris, 1959, pp. 157-164.
    • (1959) Colloques Inter-national: du Centre National de la Recherche Scientifique LXXXVII , pp. 157-164
  • 93
    • 84946245199 scopus 로고
    • An asymptotically optimal decision procedure
    • B. Subert, “An asymptotically optimal decision procedure,” in Trans. 4th Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1965. Prague: Czechoslovak Acad. Sci., 1967, 573-587.
    • (1967) Trans. 4th Prague Conf. Inform. Theory
    • Subert, B.1
  • 94
    • 84937355605 scopus 로고
    • Bayesian model of decision-making as a result of learning from experience
    • —, “Bayesian model of decision-making as a result of learning from experience,” Ann. Math. Statist., vol. 40, pp. 2127-2142, 1969.
    • (1969) Ann. Math. Statist. , vol.40 , pp. 2127-2142
  • 95
    • 0012721984 scopus 로고
    • Empirical Bayes Theory
    • V. Susarla, “Empirical Bayes Theory,” Encyclopedia of Statistical Sciences, Vol. 2, S. Kotz and N. L. Johnson, Ed. Wiley-Interscience, 1982, pp. 490-503.
    • (1982) Encyclopedia of Statistical Sciences , vol.2 , pp. 490-503
    • Susarla, V.1
  • 96
    • 0001439513 scopus 로고
    • Strong law of large numbers for measures of central tendency and dispersion of random variables in compact metric spaces
    • Jan.
    • H. Sverdrup-Thygeson, “Strong law of large numbers for measures of central tendency and dispersion of random variables in compact metric spaces,” Ann. Statist., vol. 9, pp. 141-145, Jan. 1981.
    • (1981) Ann. Statist. , vol.9 , pp. 141-145
    • Sverdrup-Thygeson, H.1
  • 97
    • 0042548685 scopus 로고
    • Bounds and rates of convergence for the extended compound estimation problems
    • D. D. Swain, “Bounds and rates of convergence for the extended compound estimation problems,” Tech. Rep. Dep. Statist., Stanford Univ., 1965.
    • (1965) Tech. Rep. Dep. Statist.
    • Swain, D.D.1
  • 98
    • 51649149993 scopus 로고
    • Asymptotic distribution theory for general statistical functionals
    • D. C. Taylor, “Asymptotic distribution theory for general statistical functionals,” Ann. Inst. Statist. Math., ser. A, vol, 37, pp. 131-138, 1985.
    • (1985) Ann. Inst. Statist. Math. , pp. 131-138
    • Taylor, D.C.1
  • 99
    • 0642330006 scopus 로고
    • The compound decision problem with m × n finite loss matrix
    • J. R. Van Ryzin, “The compound decision problem with m × n finite loss matrix,” Ann. Math. Statist., vol. 37, pp. 412-424, 1966.
    • (1966) Ann. Math. Statist. , vol.37 , pp. 412-424
    • Van Ryzin, J.R.1
  • 100
    • 0001402719 scopus 로고
    • The sequential compound decision problem with m X n finite loss matrix
    • —, “The sequential compound decision problem with m X n finite loss matrix,” Ann. Math. Statist., vol. 37, pp. 954-975, 1966.
    • (1966) Ann. Math. Statist. , vol.37 , pp. 954-975
  • 101
    • 2442553157 scopus 로고
    • Repetitive play in finite statistical games with unknown distributions
    • —, “Repetitive play in finite statistical games with unknown distributions,” Ann. Math. Statist., vol. 37, pp. 976-994, 1966.
    • (1966) Ann. Math. Statist. , vol.37 , pp. 976-994
  • 102
    • 84938023621 scopus 로고
    • Non-parametric Bayesian decision procedures for (pattern) classification with stochastic learning
    • —, “Non-parametric Bayesian decision procedures for (pattern) classification with stochastic learning,” in Trans. 4th Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1965. Prague: Czechoslovak Acad. Sci., 1967, pp. 479-494.
    • (1967) Trans. 4th Prague Conf. Inform. Theory , pp. 479-494
  • 103
    • 0000167944 scopus 로고
    • A note on the consistency of the maximum likelihood estimate
    • A. Wald, “A note on the consistency of the maximum likelihood estimate,” Ann. Math. Statist., vol. 60, pp. 595-603, 1943.
    • (1943) Ann. Math. Statist. , vol.60 , pp. 595-603
    • Wald, A.1
  • 105
    • 0002644952 scopus 로고
    • Maximum likelihood estimation of misspecified models
    • Jan.
    • H. White, “Maximum likelihood estimation of misspecified models,” Econometrica, vol. 50, pp. 1-25, Jan. 1982.
    • (1982) Econometrica , vol.50 , pp. 1-25
    • White, H.1
  • 106
    • 84967702091 scopus 로고
    • The ergodic theorem
    • N. Wiener, “The ergodic theorem,” Duke Math. J., vol. 5, pp. 1-18, 1978.
    • (1978) Duke Math. J. , vol.5 , pp. 1-18
    • Wiener, N.1
  • 107
    • 84946245153 scopus 로고
    • Experience in games of strategy and in statistical decision
    • K. Winkelbauer, “Experience in games of strategy and in statistical decision,” in Trans. 1st Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes, Prague, 1956. Prague: Czechoslovak Acad. Sci., 1957, pp. 297-354.
    • (1957) Trans. 1st Prague Conf. Inform. Theory , pp. 297-354
    • Winkelbauer, K.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.