Skip to main content
SpringerLink
Log in

Contributions of Rabi Bhattacharya to the Central Limit Theory and Normal Approximation

  • Chapter
  • First Online:
Rabi N. Bhattacharya

Part of the book series: Contemporary Mathematicians ((CM))

Abstract

Rabi Bhattacharya has made signal contributions to central limit theory and normal approximation, particularly for sums of independent random vectors. His monograph in the area (Bhattacharya and Ranga Rao 1976) has become a classic, its importance being so great that it has had significant influence on mathematical statistics as well as probability. The methods developed in that monograph led to Bhattacharya and Ghosh’s (1978) seminal account of general Edgeworth expansions under the smooth function model, as it is now commonly called. That article had a profound influence on the development of bootstrap methods, not least because it provided a foundation for influential research that enabled different bootstrap methods to be compared. At a vital time in the evolution of bootstrap methods, it led to an authoritative and enduring assessment of many of the bootstrap’s important contributions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bergström, H. (1969/70). On the central limit theorem in R k. The remainder term for special Borel sets. Z. Wahrscheinlichkeitstheorie verw. Gebiete 14, 113–126.

    Google Scholar 

  2. Berry, A.C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49, 122–136.

    Google Scholar 

  3. Bhattacharya, R.N. (1968). Berry-Esseen bounds for the multi-dimensional central limit theorem. Bull. Amer. Math. Soc. 74, 285–287.

    Google Scholar 

  4. Bhattacharya, R.N. (1970). Rates of weak convergence for the multidimensional central limit theorem. Teor. Verojatnost. i Primenen 15, 69–85.

    Google Scholar 

  5. Bhattacharya, R.N. (1971). Rates of weak convergence and asymptotic expansions for classical central limit theorems. Ann. Math. Statist. 42, 241–259.

    Google Scholar 

  6. Bhattacharya, R.N. (1972). Recent results on refinements of the central limit theorem. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol. II: Probability theory, pp. 453–484. Univ. California Press, Berkeley, Calif.

    Google Scholar 

  7. Bhattacharya, R.N. (1972/3). Speed of convergence of the n-fold convolution of a probability measure on a compact group. Z. Wahrscheinlichkeitstheorie verw. Gebiete 25, 1–10.

    Google Scholar 

  8. Bhattacharya, R.N. (1975). On errors of normal approximation. Ann. Probability 3, 815–828.

    Google Scholar 

  9. Bhattacharya, R.N. (1977). Refinements of the multidimensional central limit theorem and applications. Ann. Probability 5, 1–28.

    Google Scholar 

  10. Bhattacharya, R.N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 185–201.

    Google Scholar 

  11. Bhattacharya, R.N. (1985). Some recent results on Cramér-Edgeworth expansions with applications. In Multivariate Analysis VI, Ed. P.R. Krishnaiah, pp. 57–75. North-Holland, Amsterdam.

    Google Scholar 

  12. Bhattacharya, R.N. (1987). Some aspects of Edgeworth expansions in statistics and probability. In New Perspectives in Theoretical and Applied Statistics, Eds. M.L. Puri, J.-P. Vilaplana and W. Wertz, pp. 157–170. Wiley, New York.

    Google Scholar 

  13. Bhattacharya, R.N. and Chan, N.H. (1996). Comparisons of chisquare, Edgeworth expansions and bootstrap approximations to the distribution of the frequency chisquare. Sankhyā Ser. A 58, 57–68.

    MathSciNet  MATH  Google Scholar 

  14. Bhattacharya, R.N. and Ghosh, J.K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6, 434–451. Correction ibid. 8 (1980), 1399.

    Google Scholar 

  15. Bhattacharya, R.N. and Denker, M (1990). Asymptotic Statistics, Birkhäuser Verlag, Basel.

    Book  MATH  Google Scholar 

  16. Bhattacharya, R.N. and Ghosh, J.K. (1988). On moment conditions for valid formal Edgeworth expansions. J. Multivariate Anal. 27, 68–79.

    Google Scholar 

  17. Bhattacharya, R.N. and Ghosh, J.K. (1992). A class of U-statistics and asymptotic normality of the number of k-clusters. J. Multivariate Anal. 43, 300–330.

    Google Scholar 

  18. Bhattacharya, R. and Qumsiyeh, M. (1989). Second order and L p -comparisons between the bootstrap and empirical Edgeworth expansion methodologies. Ann. Statist. 17, 160–169.

    Google Scholar 

  19. Bhattacharya, R.N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.

    MATH  Google Scholar 

  20. Bickel, P.J. (1974). Edgeworth expansions in nonparametric statistics. Ann. Statist. 2, 1–20.

    Google Scholar 

  21. Bickel, P.J. and Freedman, D.A. (1980). On Edgeworth expansions and the bootstrap. Unpublished manuscript.

    Google Scholar 

  22. Chandra, T.K. and Ghosh, J.K. (1979). Valid asymptotic expansions for the likelihood ratio statistic and other perturbed chi-square variables. Sankhyā Ser. A 41, 22–47.

    MathSciNet  MATH  Google Scholar 

  23. Chibisov, D.M. (1972). An asymptotic expansion for the distribution of a statistic admitting an asymptotic expansion. Theor. Probab. Appl. 17, 620–630.

    Google Scholar 

  24. Chibisov, D.M. (1973a). An asymptotic expansion for a class of estimators containing maximum likelihood estimators. Theor. Probab. Appl. 18, 295–303.

    Google Scholar 

  25. Chibisov, D.M. (1973b). An asymptotic expansion for the distribution of sums of a special form with an application to minimum-contrast estimates. Theor. Probab. Appl. 18, 649–661.

    Google Scholar 

  26. Contaldi, C.R., Ferreira, P.G., Magueijo, J. and Górski, K.M. (2000). A Bayesian estimate of the skewness of the cosmic microwave background. Astrophysical J. 534, 25–28.

    Google Scholar 

  27. Cramér, H. (1928). On the composition of elementary errors. Aktuarietidskr. 11, 13–74, 141–180.

    Google Scholar 

  28. Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.

    MATH  Google Scholar 

  29. Daniels, H.E. and Young, G.A. (1991). Saddlepoint approximation for the Studentized mean, with an application to the bootstrap. Biometrika 78, 169–179.

    Google Scholar 

  30. Davison, A.C. and Hinkley, D.V. (1988). Saddlepoint approximations in resampling methods. Biometrika 75, 417–431.

    Google Scholar 

  31. DiCiccio, T.J., Martin, M.A. and Young, G.A. (1994). Analytical approximations to bootstrap distribution functions using saddlepoint methods. Statist. Sinica 4, 281–295.

    Google Scholar 

  32. Efron, B. (1979a). Bootstrap methods: another look at the jackknife. Ann. Statist. 7, 1–26.

    Google Scholar 

  33. Efron, B. (1979b). Computers and the theory of statistics: thinking the unthinkable. SIAM Rev. 21, 460–480.

    Google Scholar 

  34. Esseen, C.-G. (1945). Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77, 1–125.

    Google Scholar 

  35. Fisher, R.A. (1935). The Design of Experiments. Oliver and Boyd, Edinburgh.

    Google Scholar 

  36. Gayen, A.K. (1949). The distribution of “Student’s” t in random samples of any size drawn from non-normal universes. Biometrika 36, 353–369.

    Google Scholar 

  37. Geary, R.C. (1936). The distribution of “Student’s” t ratio for non-normal samples. J. Roy. Statist. Soc. Supp. 3, 178–184.

    Google Scholar 

  38. Gonçalvesa, S. and Meddahia, N. (2008). Edgeworth corrections for realized volatility. Econometric Reviews 27, 139–162.

    Google Scholar 

  39. Götze, F. and Hipp, C. (1978). Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 42, 67–87.

    Google Scholar 

  40. Hall, P. (1983a). Chi squared approximations to the distribution of a sum of independent random variables. Ann. Probability 11 1028–1036.

    Google Scholar 

  41. Hall, P. (1983b). Inverting an Edgeworth expansion. Ann. Statist. 11, 569–576.

    Google Scholar 

  42. Hall, P. (1990). On the relative performance of bootstrap and Edgeworth approximations of a distribution function. J. Multivariate Anal. 35, 108–129.

    Google Scholar 

  43. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.

    Book  MATH  Google Scholar 

  44. Hall, P. (2003). A short prehistory of the bootstrap. Statist. Sci. 18, 158–167.

    Google Scholar 

  45. Hsu, P.L. (1945). The approximate distributions of the mean and variance of a sample of independent variables. Ann. Math. Statist. 16, 1–29.

    Google Scholar 

  46. Hu, Z., Shao, Q.-M. and Wang, Q. (2009). Cramér type moderate deviations for the maximum of self-normalized sums. Electron. J. Probab. 14, 1181–1197.

    Google Scholar 

  47. Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2008). Towards a universal self-normalized moderate deviation. Trans. Amer. Math. Soc. 360, 4263–4285.

    Google Scholar 

  48. Liu, W. and Shao, Q.-M. (2013). A Cramér moderate deviation theorem for Hotelling’s T 2 statistic with applications to global tests. Ann. Statist. 41, 296–322.

    Google Scholar 

  49. Pfaff, Th. and Pfanzagl, J. (1980). On the numerical accuracy of approximations based on Edgeworth expansions. J. Statist. Comput. Simulation 11, 223–239.

    Google Scholar 

  50. Pfanzagl, J. (1973). Asymptotic expansions related to minimum contrast estimators. Ann. Statist. 1, 993–1026. Correction ibid. 2, 1357–1358.

    Google Scholar 

  51. Sadikova, S. M. (1968). The multidimensional central limit theorem. Teor. Verojatnost. i Primenen. 13, 164–170.

    Google Scholar 

  52. Sargan, J.D. (1975). Gram-Charlier approximations applied to t ratios of k-class estimators. Econometrica 43, 327–346.

    Google Scholar 

  53. Sargan, J.D. (1976). Econometric estimators and the Edgeworth approximation. Econometrica 44, 421–448.

    Google Scholar 

  54. Sazonov, V.V. (1968). On the multidimensional central limit theorem. Sankhyā Ser. A 30, 181–204.

    MathSciNet  Google Scholar 

  55. Simon, J. (1969). Basic Research Methods in Social Science. The Art of Empirical Investigation. Random House, New York.

    Google Scholar 

  56. Singh, K. (1981). On the asymptotic accuracy of Efron’s bootstrap. Ann. Statist. 9, 1187–1195.

    Google Scholar 

  57. von Bahr, B. (1987a). On the central limit theorem in R k. Ark. Mat. 7, 61–69.

    Google Scholar 

  58. von Bahr, B. (1987b). Multi-dimensional integral limit theorems for large deviations. Ark. Mat. 7, 89–99.

    Google Scholar 

  59. Wallace, D.L. (1958). Asymptotic approximations to distributions. Ann. Math. Statist. 29, 635–654.

    Google Scholar 

  60. Zolotarev, V.M. (1967). A sharpening of the inequality of Berry-Esseen. Z. Wahrscheinlichkeitstheorie verw. Gebiete 8, 32–42.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sydney, Australia

    Peter Hall

Authors
  1. Peter Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics, The Pennsylvania State University, State College, Pennsylvania, USA

    Manfred Denker

  2. Department of Mathematics, Oregon State University, Corvallis, Oregon, USA

    Edward C. Waymire

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hall, P. (2016). Contributions of Rabi Bhattacharya to the Central Limit Theory and Normal Approximation. In: Denker, M., Waymire, E. (eds) Rabi N. Bhattacharya. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30190-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation