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Computational Statistics

, Volume 31, Issue 3, pp 973–988 | Cite as

Least squares generalized inferences in unbalanced two-component normal mixed linear model

  • Xuhua LiuEmail author
  • Xingzhong Xu
  • Jan Hannig
Original Paper
  • 253 Downloads

Abstract

In this paper, we make use of least squares idea to construct new fiducial generalized pivotal quantities of variance components in two-component normal mixed linear model, then obtain generalized confidence intervals for two variance components and the ratio of the two variance components. The simulation results demonstrate that the new method performs very well in terms of both empirical coverage probability and average interval length. The newly proposed method also is illustrated by a real data example.

Keywords

Variance component Least squares Fiducial generalized pivotal quantity Fiducial generalized confidence interval 

Notes

Acknowledgments

Xuhua Liu’s work was supported by the National Natural Science Foundation of China under Grant No. 11201478 and 11471030. Xingzhong Xu’s work was supported by the National Natural Science Foundation of China under Grant No. 11471035. Jan Hannig’s research was supported in part by the National Science Foundation under Grant No. 1016441. The authors are very grateful to the two reviewers for their valuable comments and suggestions on the earlier versions of this paper.

Supplementary material

180_2015_604_MOESM1_ESM.pdf (67 kb)
Supplementary material 1 (pdf 66 KB)

References

  1. Ahrens H, Pincus R (1981) On two measures of unbalancedness in a one-way model and their relation to efficiency. Biom J 23:227–235MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arendacká B (2005) Generalized confidence intervals on the variance component in mixed linear models with two variance components. Statistics 39:275–286MathSciNetCrossRefzbMATHGoogle Scholar
  3. Burch BD (1996) Confidence intervals and prediction intervals in a mixed linear model. Unpublished doctoral thesis, Colorado State University, Department of StatisticsGoogle Scholar
  4. Burch BD, Iyer HK (1997) Exact confidence intervals for a variance ratio (or heritability) in a mixed linear model. Biometrics 53:1318–1333MathSciNetCrossRefzbMATHGoogle Scholar
  5. Burdick RK, Eickman J (1986) Confidence intervals on the among group variance component in the unbalanced one-fold nested design. J Stat Comput Simul 26:205–219CrossRefzbMATHGoogle Scholar
  6. Burdick RK, Graybill FA (1984) Confidence intervals on linear combinations of variance components in the unbalanced one-way classification. Technometrics 26:131–136CrossRefGoogle Scholar
  7. Fenech AP, Harville DA (1991) Exact confidence sets for variance components in unbalanced mixed linear models. Ann Stat 19:1771–1785MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hannig J, Iyer HK, Patterson P (2006) Fiducial generalized confidence intervals. J Am Stat Assoc 101:254–269MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hannig J (2009) On generalized fiducial intervals. Stat Sin 19:491–544MathSciNetzbMATHGoogle Scholar
  10. Hartung J, Knapp G (2000) Confidence intervals for the between-group variance in the unbalanced one-way random effects model of analysis of variance. J Stat Comput Simul 65:311–323MathSciNetCrossRefzbMATHGoogle Scholar
  11. Harville DA, Fenech AP (1985) Confidence interval for a variance ratio, or for heritability, in an unbalanced mixed linear model. Biometrics 41:137–152MathSciNetCrossRefzbMATHGoogle Scholar
  12. Henderson CR (1984) Application of linear models in animal breeding. University of Guelph Press, OntarioGoogle Scholar
  13. Krishnamoorthy K, Mathew T (2003) Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals. J Stat Plan Inference 115:103–121MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li X, Li G, Xu X (2005) Fiducial intervals of restricted parameters and their applications. Sci China (Ser A Math) 48:1567–1583MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lidong E, Hannig J, Iyer HK (2008) Fiducial intervals for variance components in an unbalanced two-component normal mixed linear model. J Am Stat Assoc 103:854–865MathSciNetCrossRefzbMATHGoogle Scholar
  16. Li X, Li G (2007) Comparison of confidence intervals on the among group variance in the unbalanced variance component model. J Stat Comput Simul 77:477–486MathSciNetCrossRefzbMATHGoogle Scholar
  17. Liu X, Xu X (2010) A new generalized p-value approach for testing homogeneity of variances. Stat Probab Lett 80:1486–1491MathSciNetCrossRefzbMATHGoogle Scholar
  18. Montgomery DC (1997) Design and analysis of experiments, 4th edn. Wiley, New YorkzbMATHGoogle Scholar
  19. Olsen A, Seely J, Birkes D (1976) Invariant quadratic unbiased estimation for two variance components. Ann Stat 4:878–890MathSciNetCrossRefzbMATHGoogle Scholar
  20. Roy A, Mathew T (2005) A generalized confidence limit for the reliability function of a two-parameter exponential distribution. J Stat Plan Inference 128:509–517MathSciNetCrossRefzbMATHGoogle Scholar
  21. Thomas JD, Hultquist RA (1978) Interval estimation for the unbalanced case of the one-way random effects model. Ann Stat 6:582–587MathSciNetCrossRefzbMATHGoogle Scholar
  22. Tian L (2006) Testing equality of inverse Gaussian means based on the generalized test variable. Comput Stat Data Anal 51:1156–1162MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tsui KW, Weerahandi S (1989) Generalized p-Values in significance testing of hypotheses in the presence of nuisance parameters. J Am Stat Assoc 84:602–607MathSciNetGoogle Scholar
  24. Weerahandi S (1993) Generalized confidence intervals. J Am Stat Assoc 88:899–905MathSciNetCrossRefzbMATHGoogle Scholar
  25. Weerahandi S (1995) Exact statistical methods for data analysis. Springer, New YorkCrossRefzbMATHGoogle Scholar
  26. Weerahandi S (2004) Generalized inference in repeated measures. Wiley, New YorkzbMATHGoogle Scholar
  27. Xu X, Li G (2006) Fiducial inference in the pivotal family of distributions. Sci China (Ser A Math) 49:410–432MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ye RD, Wang SG (2009) Inferences on the intraclass correlation coefficients in the unbalanced two-way random effects model with interaction. J Stat Plan Inference 139:396–410MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zhou L, Mathew T (1994) Some tests for variance components using generalized p-values. Technometrics 36:394–402MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsChina Agricultural UniversityBeijingChina
  2. 2.College of MathematicsBeijing Institute of TechnologyBeijingChina
  3. 3.Department of Statistics and Operations ResearchThe University of North Carolina at Chapel HillChapel HillUSA

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