Advertisement

Complete Independence in the Multivariate Normal Distribution

  • Govind S. Mudholkar
  • Perla Subbaiah
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)

Summary

Testing complete independence is one of the simplest problems concerning the covariance structure of a set of measurements. A stepwise procedure proposed by Roy and Bargmann (1958) and a trace criterion due to Nagao (1973) are two well-known competitors of the likelihood ratio test of the hypothesis derived assuming the multivariate normality. We consider some modifications of the Roy-Bargmann procedure based on combinations of independent tests and find them to be asymptotically equivalent to the likelihood ratio test, which is optimal in terms of the exact slopes. The operating characteristics of various tests with samples of moderate size are examined empirically.

Key Words

Combination of tests exact slopes stepdown procedure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bahadur, R.R. (1971). Some Limit Theorems in Statistics. Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.Google Scholar
  2. Berk, R.H., Cohen, A. (1979). Asymptotically optimal method of combining tests. Journal of American Statistical Association, 74, 821–814.MathSciNetCrossRefGoogle Scholar
  3. George, E.O. (1977). Combining independent one-sided and two- sided statistical tests - some theory and applications. Ph.D. dissertation, University of Rochester.Google Scholar
  4. Hsieh, H.K. (1979). On asymptotic optimality of likelihood ratio tests for multivariate normal distributions. Annals of Statistics, 7, 592–598.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Liptak, T. (1958). On the combination of independent tests. Magyar Tudomangos Akademia. Matematikai Kutato Intezetenek, 3, 171–197.Google Scholar
  6. Littell, R.C., Folks, J.L. (1971). Asymptotic optimality of Fisher’s method of combining independent tests. Journal of American Statistical Association, 66, 802–806.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Mathai, A.M., Katiyar, R.S. (1979). Exact percentage points for testing independence, Biometrika, 66, 353–356.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Mudholkar, G.S., George, E.O. (1979). The logit statistic for combining probabilities - an overview. In Optimizing Methods in Statistics, J.S. Rustagi, ed.Google Scholar
  9. Mudholkar, G.S., Trivedi, M.C., Lin, C.C. (1980). An approximation to the distribution of the likelihood ratio for complete independence. To appear in Technometrics.Google Scholar
  10. Nagao, H. (1973). On some test criteria for covariance matrix. Annals of Statistics, 1, 700–709.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Oosterhoff, J. (1969). Combination of One-sided Statistical Tests, Mathematical Centre Tracts, 28, The Mathematical Centre, Amsterdam.Google Scholar
  12. Roy, S.N., Bargmann, R.E. (1958). Tests of multiple independence and the associated confidence bounds. Annals of Mathematical Statistics, 29, 491–503.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Wilks, S.S. (1935). On the independence of k sets of normally distributed statistical variables. Econometrica, 3, 309–326.zbMATHCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Govind S. Mudholkar
    • 1
  • Perla Subbaiah
    • 2
  1. 1.University of RochesterUSA
  2. 2.Oakland UniversityUSA

Personalised recommendations