Skip to main content

Metrics, Information Theory, Convergence, and Poisson Approximations

  • Chapter
Asymptotic Theory of Statistics and Probability

Part of the book series: Springer Texts in Statistics ((STS))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approximation and the Chen-Stein method, Stat. Sci., 5(4), 403–434.

    MATH  MathSciNet  Google Scholar 

  • Barbour, A., Chen, L., and Loh, W-L. (1992). Compound Poisson approximation for non-negative random variables via Stein’s method, Ann. Prob., 20, 1843–1866.

    Article  MATH  MathSciNet  Google Scholar 

  • Barbour, A., Holst, L., and Janson, S. (1992). Poisson Approximation, Clarendon Press, New York.

    Google Scholar 

  • Barron, A. (1986). Entropy and the central limit theorem, Ann. Prob., 14(1), 336–342.

    Article  MATH  MathSciNet  Google Scholar 

  • Boos, D. (1985). A converse to Scheffe’s theorem, Ann. Stat., 1, 423–427.

    Article  MathSciNet  Google Scholar 

  • Brown, L. (1982). A proof of the central limit theorem motivated by the Cramér-Rao inequality, G. Kallianpur, P.R. Krishnaiah, and J.K. Ghosh Statistics and Probability, Essays in Honor of C.R. Rao, North-Holland, Amsterdam, 141–148.

    Google Scholar 

  • Brown, L., DasGupta, A., Haff, L.R., and Strawderman, W.E. (2006). The heat equation and Stein’s identity: Connections, applications, J. Stat. Planning Infer, Special Issue in Memory of Shanti Gupta, 136, 2254–2278.

    MATH  MathSciNet  Google Scholar 

  • Chen, L.H.Y. (1975). Poisson approximation for dependent trials, Ann. Prob., 3, 534–545.

    Article  MATH  Google Scholar 

  • DasGupta, A. (1999). The matching problem and the Poisson approximation, Technical Report, Purdue University.

    Google Scholar 

  • DasGupta, A. (2005). The matching, birthday, and the strong birthday problems: A contemporary review, J. Stat. Planning Infer, Special Issue in Honor of Herman Chernoff, 130, 377–389.

    MATH  Google Scholar 

  • Dembo, A. and Rinott, Y. (1996). Some examples of normal approximations by Stein’s method, in Random Discrete Structures, D. Aldous and Pemantle R. IMA Volumes in Mathematics and Its Applications, Vol. 76, Springer, New York, 25–44.

    Google Scholar 

  • Diaconis, P. and Holmes, S. (2004). Stein’s Method: Expository Lectures and Applications, IMS Lecture Notes Monograph Series, vol. 46, Institute of Mathematical Statistics, Beachwood, OH.

    Google Scholar 

  • Dudley, R. (1989). Real Analysis and Probability, Wadsworth, Pacific Grove, CA.

    MATH  Google Scholar 

  • Galambos, J. and Simonelli, I. (1996). Bonferroni-Type Inequalities with Applications, Springer, New York.

    MATH  Google Scholar 

  • Hwang, J.T. (1982). Improving upon standard estimators in discrete exponential families with applications to Poisson and negative binomial cases, Ann. Stat., 10(3), 857–867.

    Article  MATH  Google Scholar 

  • Johnson, O. (2004). Information Theory and the Central Limit Theorem, Imperial College Press, Yale University London.

    Google Scholar 

  • Johnson, O. and Barron, A. (2003). Fisher information inequalities and the central limit theorem, Technical Report.

    Google Scholar 

  • Kolchin, V., Sevas’tyanov, B., and Chistyakov, V. (1978). Random Allocations,V.H. Winston & Sons, Washington, distributed by Halsted Press, New York.

    Google Scholar 

  • LeCam, L. (1960). An approximation theorem for the Poisson binomial distribution, Pac. J. Math., 10, 1181–1197.

    MathSciNet  Google Scholar 

  • Linnik, Y. (1959). An information theoretic proof of the central limit theorem, Theory Prob. Appl., 4, 288–299.

    MathSciNet  Google Scholar 

  • Rachev, S. (1991). Probability Metrics and the Stability of Stochastic Models, JohnWiley, Chichester.

    MATH  Google Scholar 

  • Reiss, R. (1989). Approximate Distributions of Order Statistics, with Applications to Nonparametric Statistics, Springer-Verlag, New York.

    MATH  Google Scholar 

  • Sevas’tyanov, B.A. (1972). A limiting Poisson law in a scheme of sums of dependent random variables, Teor. Veroyatni. Primen., 17, 733–738.

    Google Scholar 

  • Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, L. Le Cam, J. Neyman, and E. Scott in Proceedings of the Sixth Berkeley Symposium, Vol. 2, University of California Press, Berkeley, 583–602.

    Google Scholar 

  • Stein, C. (1981). Estimation of the mean of a multivariate normal distribution, Ann. Stat., 9, 1135–1151.

    Article  MATH  Google Scholar 

  • Stein, C. (1986). Approximate Computations of Expectations, Institute of Mathematical Statistics, Hayward, CA.

    Google Scholar 

  • Sweeting, T. (1986). On a converse to Scheffe’s theorem, Ann. Stat., 3, 1252–1256.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

DasGupta, A. (2008). Metrics, Information Theory, Convergence, and Poisson Approximations. In: Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75971-5_2

Download citation

Publish with us

Policies and ethics