Advertisement

A Normal Approximation for the Multivariate Likelihood Ratio Statistics

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

Summary

For many multivariate hypotheses, under the normality assumptions, the likelihood ratio tests are optimal in the sense of having maximal exact slopes. The exact distributions needed for implementing these tests are complex and their tabulation is limited in scope and accessibility. In this paper, a method of constructing normal approximations to these distributions is described, and illustrated using the problems of testing sphericity and independence between two sets of variates. The normal approximations are compared with well-known competing approximations and are seen to fare well.

Key Words

Sphericity independence between two sets of variates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T.W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York.zbMATHGoogle Scholar
  2. Bahadur, R.R. (1967). An optimal property of the likelihood ratio statistics. Proceedings of Fifth Berkeley Symposium Mathematical Statistics and Probability. University of California Press. Pages 13–26.Google Scholar
  3. Bartlett, M.S. (1937). Properties of sufficiency and statistical tests. Proceedings of Royal Statistical Society, Series A, 160, 268.CrossRefGoogle Scholar
  4. Bishop, D.J. (1939). On the comprehensive test of the homogenectiy of variances and covariances of multivariate problems. Biometrika, 31, 31–55.MathSciNetzbMATHGoogle Scholar
  5. Box, G.E.P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317–346.MathSciNetGoogle Scholar
  6. Consul, P.C. (1967a). On the exact distribution of likelihood ratio criteria for testing independence of sets of variates under the null hypothesis. Annals of Mathematical Statistics, 38, 1160–1169.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Consul, P.C. (1967b). On the exact distribution of the criterion W for testing sphericity in a p-variate normal distribution. Annals of Mathematical Statistics, 38, 1170–1174.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Engelman, L., Frane, J.W., Jennrich, R.I. (1977). Biomedical Computer Programs P-Series. University of California Press, Berkeley, California.Google Scholar
  9. Hsieh, H.K. (1979). On asymptotic optimality of likelihood ratio tests for multivariate normal distributions. Annals of Statistics, 7, 592–598.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Krishnaiah, P.R. (1979). Some Recent Developments on The Real Multivariate Distributions. North Holland Co., New York.Google Scholar
  11. Mann, H.B., Wald. A. (1943). On stochastic limit and order relationships. Annals of Mathematical Statistics, 14, 217–226.zbMATHGoogle Scholar
  12. Mauchly, J.W. (1940). Significance test for sphericity of a normal n-variate distribution. Annals of Mathematical Statistics, 11, 204–209.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Nagarsanker, B.N., Pillai, K.C.S. (1973). Distribution of the sphericity test criterion. Journal of Multivariate Analysis, 3, 226–235.MathSciNetCrossRefGoogle Scholar
  14. Nayer, P.P.N. (1936). An investigation into the application of Neyman and Pearson’s L1 test, with tables of percentage limits. Statistical Research Memorandum, 1, 38. University of London.Google Scholar
  15. Neyman, J., Pearson, E.S. (1931). On the problem of k samples. Bulletin International Academy Cracovie, A, 6, 460.Google Scholar
  16. Pearson, E.S., Hartley, H.O. (1972). Biometrika Tables for Statisticians, 2. Cambridge University Press, New York.zbMATHGoogle Scholar
  17. Rao, C.R. (1948). Tests of significance in multivariate analysis. Biometrika, 35, 58–79.MathSciNetzbMATHGoogle Scholar
  18. Srivastava, M.S., Khatri, C.G. (1979). An Introduction to Multivariate Statistics. North Holland Co., New York.zbMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Govind S. Mudholkar
    • 1
  • Madhusudan C. Trivedi
    • 2
  1. 1.Department of StatisticsUniversity of RochesterRochesterUSA
  2. 2.Pennwalt Corporation Pharmaceutical DivisionRochesterUSA

Personalised recommendations