Skip to main content
SpringerLink
Log in

Inferences on Linear Combinations of Normal Means with Unknown and Unequal Variances

  • Published:
Sankhya A Aims and scope Submit manuscript

Abstract

This paper considers the problem of making inferences on a linear combination of means from independent normal distributions with unknown variances. While standard inference procedures require the assumption that the variances of the normal distributions are all equal or have known ratios, in this paper procedures are developed which allow the variances to take any unknown values. Bounds are developed for the confidence levels of confidence intervals, and for the sizes and p-values of hypothesis tests, which are valid for any values of the variances. These bounds are sharp in the sense that they are attained for certain limiting values of the variances. Applications to the one-way linear model are illustrated, and the Behrens-Fisher problem, which is a special case with just two means and for which the popular approximate Welch method can be employed, is also discussed. Some examples and simulations of the implementation of the procedures are provided, and comparisons are made with other procedures.

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.

Similar content being viewed by others

References

  • Banerjee, S.K. (1960). Approximate confidence interval for linear functions of means of k-populations when the populations are not equal. Sankhya, 22, 357–258.

    MathSciNet  Google Scholar 

  • Behrens, W.V. (1929). Ein beitrag zur Fehlerberechnung bei wenigen Beobachtungen (translation: A contribution to error estimation with few observations). Landwirtschaftliche Jahrbcher, 68, 807–837.

    Google Scholar 

  • Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978). Statistics for experimenters. Wiley.

  • Chang, C.H. & Pal, N. (2008). A revisit to the Behrens-Fisher problem: Comparison of five test methods. Comm. Statist. Simulation Comput., 37, 1064–1085.

    Article  MathSciNet  MATH  Google Scholar 

  • Dalal, S.R. (1978). Simultaneous confidence procedures for univariate and multivariate Behrens-Fisher type problems. Biometrika, 65, 221–224.

    MathSciNet  MATH  Google Scholar 

  • Dobson, A.J. (1990). An introduction to generalized linear models. Chapman & Hall.

  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, R.A. (1935). The fiducial argument in statistical inference. Ann. Eugenics, 8, 391–398.

    Article  Google Scholar 

  • Hajek, J. (1962). Inequalities for the generalized Student’s distribution. Sel. Transl. Math. Statist. Prob., 2, 63–74.

    MATH  Google Scholar 

  • Hsu, P.L. (1938). Contributions to the theory of “Student’s” t-test as applied to the problem of two samples. Statistical Research Memoirs, Vol. II. Department of Statistics, University College, London, pp. 1–24.

  • Huang, P., Tilley, B.C., Woolson, R.F. and Lipsitz, S. (2005). Adjusting O’Brien’s test to control type I error for the generalized nonparametric Behrens-Fisher problem. Biometrics, 61, 532–539.

    Article  MathSciNet  MATH  Google Scholar 

  • Jensen, J.L.W.V. (1906). Sur les fonctions convexes et les inegalities entre les valeurs moyennes. Acta Math., 30, 175–193.

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, R.H. (1987). Serial correlation in unbalanced mixed models. Bull. Int. Stat. Inst., 52, 105–122.

    Google Scholar 

  • Kabaila, P. and Tuck, J. (2008). Confidence intervals utilizing prior information in the Behrens-Fisher problem. Aust. N. Z. J. Stat., 50, 309–328.

    Article  MathSciNet  Google Scholar 

  • Lawton, W.H. (1965). Some inequalities for central and non-central distributions. Ann. Math. Statist., 36, 1521–1524.

    Article  MathSciNet  MATH  Google Scholar 

  • Lord, E. (1947). The use of range in place of the standard deviation in the t-test. Biometrika, 34, 41–67.

    Article  MathSciNet  MATH  Google Scholar 

  • Lord, E. (1950). Power of the modified t-test (u-test) based on range. Biometrika, 37, 64–77.

    Article  MathSciNet  MATH  Google Scholar 

  • Mickey, M.R. and Brown, M.B. (1966). Bounds on the distribution functions of the Behrens-Fisher statistic. Ann. Math. Stat., 37, 639–642.

    Article  MathSciNet  MATH  Google Scholar 

  • Nel, D.G. and van der Merwe, C.A. (1986). A solution to the multivariate Behrens-Fisher problem. Comm. Statist. Theory Methods, 15, 3719–3735.

    Article  MATH  Google Scholar 

  • Neubert, K. and Brunner, E. (2007). A studentized permutation test for the non-parametric Behrens-Fisher problem. Comput. Statist. Data Anal., 51, 5192–5204.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramos, P.D. and Ferreir, D.F. (2010). A Bayesian solution to the multivariate Behrens-Fisher problem. Comput. Statist. Data Anal., 54, 1622–1633.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsey, P.H. and Ramsey, P.P. (2008). Power of pairwise comparisons in the equal variance and unequal sample size case. Br. J. Math. Stat. Psychol., 61, 115–131.

    Article  MathSciNet  Google Scholar 

  • Ramsey, P.H., Barrera, K., Hachimine-Semprebom, P. and Liu, C. (2011). Pairwise comparisons of means under realistic nonnormality, unequal variances, outliers and equal sample sizes. J. Stat. Comput. Simul., 81, 125–135.

    Article  MathSciNet  MATH  Google Scholar 

  • Satterthwaite, F.E. (1946). An approximate distribution of estimates of variance components. Biometrics Bull., 2, 110–114.

    Article  Google Scholar 

  • Spjotvoll, E. (1972). Joint confidence intervals for all linear functions of means in ANOVA with unknown variances. Biometrika, 59, 684–685.

    Article  MathSciNet  Google Scholar 

  • Tamhane, A.C. (1979). A comparison of procedures for multiple comparisons of means with unequal variances. J. Amer. Statist. Assoc., 74, 366, 471–480.

    MathSciNet  Google Scholar 

  • Welch, B.L. (1947). The generalization of “Student’s” problem when several different population variances are involved. Biometrika, 34, 28–35.

    MathSciNet  MATH  Google Scholar 

  • Welch, B.L. (1949). Further note on Mrs. Aspin’s tables and on certain approximations to the tabled function. Biometrika, 36, 293–296.

    MathSciNet  Google Scholar 

  • Wilks, S.S. (1962) Mathematical statistics. Wiley.

Download references

Author information

Authors and Affiliations

  1. Department of Business Information and Analytics, University of Denver, Denver, USA

    Anthony J. Hayter

Authors
  1. Anthony J. Hayter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anthony J. Hayter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hayter, A.J. Inferences on Linear Combinations of Normal Means with Unknown and Unequal Variances. Sankhya A 76, 257–279 (2014). https://doi.org/10.1007/s13171-013-0041-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-013-0041-0

Keywords and phrases

AMS (2000) subject classification

Search

Navigation