On choosing and bounding probability metrics

International Statistical Review

Volumn 70, Issue 3, 2002, Pages 419-435

  Gibbs, Alison L ^a   Su, Francis Edward ^b  

^a YORK UNIVERSITY   (Canada)

^b HARVEY MUDD COLLEGE   (United States)

Author keywords

Discrepancy; Hellinger distance; Probability metrics; Prokhorov metric; Rates of convergence; Relative entropy; Wasserstein distance

Indexed keywords

EID: 0036896227     PISSN: 03067734     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1751-5823.2002.tb00178.x     Document Type: Article

References (53)

1. Aldous, D. & Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. in Appl. Math., 8, 69-97.
2. Barron, A. (1986). Entropy and central limit theorem. Ann. Probab., 14, 336-342.
3. Bernardo, J.M. & Smith, A.F.M. (1994). Bayesian Theory. Chichester, UK: John Wiley & Sons.
4. Borovkov, A.A. (1998). Mathematical Statistics. Amsterdam: Gordon and Breach Science Publishers. Translated from the Russian by A. Moullagaliev and revised by the author.
5. Chung, F., Diaconis, P. & Graham, R.L. (1987). Random walks arising in random number generation. Ann, Probab., 15, 1148-1165.
6. Cover, T.M. & Thomas, J.A. (1991). Elements of Information Theory. New York: John Wiley & Sons Inc. A Wiley-Interscience Publication.
7. Csiszar, I. (1967). Information type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar., 2, 299-318.
8. Diaconis, P. (1988). Group Representations in Probability and Statistics. Hayward, CA: Institute of Mathematical Statistics.
9. Diaconis, P. & Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab., 6, 695-750.
10. Diaconis, P. & Zabell, S.L. (1982). Updating subjective probability. J. Amer. Statist. Assoc., 77, 822-830.
11. Dudley, R.M. (1968). Distances of probability measures and random variables. Ann. Math. Statist., 39, 1563-1572.
12. Dudley, R.M. (1989). Real Analysis and Probability. Belmont, CA: Wadsworth & Brooks/Cole.
13. Gibbs, A.L. (2000). Convergence of Markov Chain Monte Carlo Algorithms with Applications to Image Restoration, PhD thesis, University of Toronto.
14. Gilks, W.R., Richardson, S. & Spiegelhalter, D.J. (Eds.) (1996). Markov Chain Monte Carlo in Practice. London: Chapman and Hall.
15. Hartigan, J.A. (1998). The maximum likelihood prior. Ann. Statist., 26, 2083-2103.
16. Hellinger, E. (1907). Die Orthogonalinvarianten quadratischer Formen von unendlich vielen Variablen, Dissertation, Göttingen.
17. Huber, P.J. (1981). Robust Statistics. New York: John Wiley & Sons.
18. Ibragimov, I.A. & Has'minskii, R.Z. (1981). Statistical Estimation: Asymptotic Theory. New York: Springer-Verlag.
19. Jacod, J. & Shiryaev, A.N. (1987). Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
20. Kakutani, S. (1948). On equivalence of infinite product measures. Ann. of Math., 49, 214-224.
21. Kolmogorov, A.N. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari, 4, 83-91.
22. Kuipers, L. & Niederreiter, H. (1974). Uniform Distribution of Sequences. Pure and Applied Mathematics. New York: Wiley-Interscience [John Wiley & Sons].
23. Kullback, S. (1967). A lower bound for discrimination in terms of variation. IEEE Trans. Inform. Theory, 4, 126-127.
24. Kullback, S. & Leibler, R.A. (1951). On information and sufficiency. Ann. Math. Statistics, 22, 79-86.
25. Le Cam, L. & Yang, G.L. (1990). Asymptotics in Statistics: Some Basic Concepts. New York: Springer-Verlag.
26. LeCam, L.M. (1969). Théorie Asymptotique de la Décision Statistique. Montréal: Les Presses de l'Université de Montréal.
27. LeCam, L.M. (1986). Asymptotic Methods in Statistical Decision Theory. New York: Springer-Verlag.
28. Lehmann, E.L. (1994). Testing Statistical Hypotheses, 2nd edn. New York: Chapman & Hall.
29. Lévy, P. (1925). Calcul des Probabilitiés. Paris: Gauthiers-Villars.
30. Liese, F. & Vajda, I. (1987). Convex Statistical Distances. Leipzig: B.G. Teubner Verlagsgesellschaft.
31. Lindsay, B.G. (1994). Efficiency versus robustness: The case for minimum Hellinger distance and related methods. Ann. Statist., 22, 1081-1114.
32. Lindvall, T. (1992). Lectures on the Coupling Method. New York: John Wiley & Sons.
33. Linnik, Y.V. (1959). An information-theoretic proof of the central limit theorem with the Lindeberg condition. Theory Probab. Appl., 4, 288-299.
34. Lukacs, E. (1975). Stochastic Convergence, 2nd edn, Probability and Mathematical Statistics; Vol. 30, New York: Academic Press [Harcourt Brace Jovanovich Publishers].
35. Mathai, A.M. & Rathie, P.N. (1975). Basic Concepts in Information Theory. New Delhi: Wiley Eastern.
36. Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge: Cambridge University Press.
37. Orey, S. (1971). Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Mathematical Studies, No. 34. London: Van Nostrand Reinhold Co.
38. Petrov, V.V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. New York: The Clarendon Press Oxford University Press. Oxford Science Publications.
39. Prokhorov, Y.V. (1956). Convergence of random processes and limit theorems in probability theory. Teor. Veroyatnost. i Primenen., 1, 177-238. English translation: Theory Probab. Appl. (1956) 1, 157-214.
40. Rachev, S.T. (1991). Probability Metrics and the Stability of Stochastic Models. Chichester/New York: John Wiley & Sons.
41. Reiss, R.-D. (1989). Approximate Distributions of Order Statistics. New York: Springer-Verlag.
42. Rosenthal, J.S. (1995). Convergence rates of Markov chains. SIAM Rev., 37, 387-405.
43. Rüschendorf, L., Schweizer, B. & Taylor, M.D. (Eds.) (1996). Distributions with Fixed Marginals and Related Topics. Lecture Notes-Monograph Series, 28. Hayward, CA: Institute of Mathematical Statistics.
44. Shannon, C.E. (1948). A mathematical theory of communication. Bell System Tech. J., 27, 379-423, 623-656.
45. Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist., 36, 423-439.
46. Su, F.E. (1995). Methods for Quantifying Rates of Convergence for Random Walks on Groups, PhD thesis, Harvard University.
47. Su, F.E. (1998). Convergence of random walks on the circle generated by an irrational rotation. Trans. Amer. Math. Soc., 350, 3717-3741.
48. Su, F.E. (2001). Discrepancy convergence for the drunkard'S walk on the sphere. Electron. J. Probab., 6(2), 20 pp. (electronic).
49. Szulga, A. (1982). On minimal metrics in the space of random variables. Theory Probab. Appl., 27, 424-430.
50. Tierney, L. (1994). Markov chains for exploring posterior distributions, with discussion. Ann. Statist., 22, 1701-1762.
51. Tierney, L. (1996). Introduction to general state-space Markov chain theory. In Markov Chain Monte Carlo in Practice, Eds. W.R. Gilks, S. Richardson and D.J. Spiegelhalter, pp. 59-74. London: Chapman and Hall.
52. Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge University Press.
53. Zolotarev, V.M. (1983). Probability metrics. Theory Probab. Appl., 28, 278-302.