Skip to main content
SpringerLink
Log in

Higher order accurate procedures to compare two normal populations

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

A classical statistical problem is the comparison of two normal samples with an emphasis on inference about the difference of their means. When both variances are unknown, this is the notorious Behrens–Fisher problem; however in some applications one sample’s variance can be considered as given. Closed-form confidence intervals for the difference between two means are derived in both cases. These easily computable procedures are based on modern higher-order asymptotic statistical methods and are quite accurate.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. E. Barndorff-Nielsen, “Inference on Full or Partial Parameters Based on the Standardized Signed Log Likelihood Ratio”, Biometrika 73, 307–322 (1986).

    MATH  MathSciNet  Google Scholar 

  2. O. E. Barndorff-Nielsen and D. R. Cox, Inference and Asymptotics (Chapman and Hall, London, 1994).

    Book  MATH  Google Scholar 

  3. M. Bédart, D. A. S. Fraser, and A. C. M. Wong, “Higher Accuracy for Bayesian and Frequentist Inference: Large Sample Theory for Small Sample Likelihood”, Statist. Sci. 22, 301–121 (2007).

    Article  MathSciNet  Google Scholar 

  4. A. Belloni and G. Didier,. “On the Behrens–Fisher Problem: A Globally Convergent Algorithm and a Finite-Sample Study of the Wald, LR, and LM Tests”, Ann. Statist. 36, 2377–2408 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  5. M.-L. G. Bout, S. Hosten, and D. St. P. Richards, “Counting and Locating the Solutions of Polynomials Systems of Maximum Likelihood Equations II: The Behrens–Fisher Problem”, Statist. Sinica 17, 1343–1354 (2007).

    MathSciNet  Google Scholar 

  6. A. R. Brazzale, A. C. Davison, and N. Reid, Applied Asymptotics: Case Studies in Small Sample Statistics (Cambridge University Press, Cambridge, UK, 2007).

    Book  Google Scholar 

  7. R. J. Cook and V. T. Farewell, “Behrens–Fisher Problem”, in Encyclopedia of Biostatistics, Ed. by P. Armitage and T. Colton (Wiley, Chichester, 1998), Vol. 1, pp. 287–288.

    Google Scholar 

  8. M. Drton, “Multiple Solutions to the Likelihood Equations in the Behrens–Fisher Problem”, Statist. and Probab. Lett. 78, 3288–3293 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  9. D. A. S. Fraser, N. Reid, and J. Wu, “A Simple General Formula for Tail Probabilities for Frequentist and Bayesian Inference”, Biometrika 86, 249–264 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  10. D. A. S. Fraser and N. Reid, “StrongMatching of Frequentist and Bayesian Parametric Inference”, J. Statist. Plann. Inf. 103, 263–285 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  11. D. A. S. Fraser, N. Reid, E. Marras, and G. Y. Yi, “Default Priors for Frequentist and Bayesian Inference”, J. R. Statist. Soc. B 72, 631–654 (2010).

    Article  MathSciNet  Google Scholar 

  12. D. A. S. Fraser, A. Wong, and J. Sun, “Three Enigmatic Examples and Inference from Likelihood”, Can. J. Statist. 37, 161–181 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Ghosh and Y.-H. Kim, “The Behrens–Fisher Problem Revisited: A Bayes–Frequentist Synthesis”, Can. J. Statist. 29, 5–17 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  14. J. L. Jensen, “The Modified Signed Likelihood Statistic and Saddle Point Approximations”, Biometrika 79, 693–703 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  15. E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, 3rd ed. (Springer, New York, 2003).

    Google Scholar 

  16. Yu. V. Linnik, Statistical Problems with Nuisance Parameters (Amer Math Soc: Providence, RI, 1967).

    Google Scholar 

  17. G. K. Robinson, “Behrens–Fisher Problem”, in Encyclopedia of Statistical Sciences, Ed. by S. Kotz and N. L. Johnson (Wiley, New York, 1982), Vol. 1, pp. 205–209.

    Google Scholar 

  18. V. I. Pagurova, “On Comparison of Means of Two Normal Samples”, Theory Probab. Appl. 13, 527–534 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  19. A. L. Rukhin, “Assessing Compatibility of Two Laboratories: Formulations as a Statistical Testing Problem”, Metrologia 50, 49–59 (2013).

    Article  Google Scholar 

  20. Y. She, A. Wong, and X. Zhou, “Revisit the Two Sample t-Test With a Known Ratio of Variances”, Open J. Statist. 1, 151–156 (2011).

    Article  MathSciNet  Google Scholar 

  21. I. Skovgaard, “An Explicit Large-Deviation Approximation to One-Parameter Tests”, Bernoulli 2, 145–165 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  22. D. A. Sprott and V. T. Farewell, “The Difference Between Two Normal Means”, Amer. Statist. 47, 126–128 (1993).

    Google Scholar 

  23. A. C. M. Wong and Y. Y. Wu, “Likelihood Analysis for the Difference in Means of Two Independent Normal Distributions with One Variance Known”, J. Statist. Research 42, 17–35 (2008).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Statist. Engng. Division, Inform. Techn. Lab., National Inst. of Standards and Techn., Gaithersburg, Maryland, USA

    A. Rukhin

Authors
  1. A. Rukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Rukhin.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rukhin, A. Higher order accurate procedures to compare two normal populations. Math. Meth. Stat. 24, 292–308 (2015). https://doi.org/10.3103/S1066530715040043

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530715040043

Keywords

2000 Mathematics Subject Classification

Search

Navigation