Summary
In the present article we are interested in presenting some properties of testing homogeneity of multivariate normal mean vectors against an order restriction for two cases, the covariance matrices are known, and the case that they have an unknown scale factor. This problem of testing with these two different cases was considered by Sasabuchi et al. (1983) and Kulatunga and Sasabuchi (1984). They only derived the test statistic and studied its null distribution. In this article, we obtain the critical values for the proposed test statistic by Kulatunga and Sasabuchi (1984) for the first case, at different significance levels for some of the two and three dimensional normal distributions. The power and p-value are computed using Monte Carlo simulation. We consider the case that covariance matrices have an unknown scale factor. In this case the specific conditions are given which under those the estimator of the unknown scale factor does not exist. Also we derive the unique test statistic. Some properties of this test, for instance, the critical values and power are computed by simulation study.
Similar content being viewed by others
References
Anderson, T.W. (1984) An Introduction to Multivariate Statistical Analysis, 2nd edition. Wley, New York.
Barlow, R. E., Bartholomew, D. J., Bremner, J.M. and Brunk, H. D. (1972) Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression, John Wiley, New York.
Bartholomew, D.J. (1959a) A test of homogeneity for ordered alternatives, Biometrika, 46,36–48.
Bartholomew, D.J. (1959b) A test of homogeneity for ordered alternatives II, Biometrika, 46, 328–335.
Bartholomew, D.J. (1961) Ordered tests in the analysis of variance, Biometrika, 48, 325–332.
Chacko, V.J. (1963) Testing homogeneity against ordered alternatives, The Annals of Mathematical Statistics, 34, 945–956.
Inada, K. (1978) Some bivariate tests of composite hypotheses with restricted alternatives, Reports of the Faculty of Science, Kagoshima University (Mathematics, Physics and Chemistry), 11, 25–31.
Kudo, A. (1963) A multivariate analogue of the one-sided test, Biometrika, 50,403–418.
Kudo, A. and Choi, J. R. (1975) A generalized multivariate analogue of the one-sided test, Memoirs of the Faculty of Science, Kyushu University, 29(2), 303–328.
Kudo, A. and Yao, J. S. (1982) Tables for testing ordered alternatives in an analysis of variance without replications, Biometrika, 69(1), 237–238.
Kulatunga, D.D.S. (1984) Convolutions of the probabilities P(l, k) used in order restricted inference, Memoirs of the Faculty of Science, Kyushu University, series A, Mathematics, 38, 9–15.
Kulatunga, D.D.S. and Sasabuchi, S. (1984) A test of homogeneity of mean vectors against multivariate isotonic alternatives, Memoirs of the Faculty of Science, Kyushu University, series A, Mathematics, 38, 151–161.
Kulatunga, D. D. S., Inutsuka, M. and Sasabuchi, S. (1990) A Simulation Study of Some Test Procedures for Testing Homogeneity of Mean Vectors against Multivariate Isotonic Alternatives, Technical Report, Tokuyama University, Japan.
Perlman, M.D. (1969) One-sided testing problems in multivariate analysis, The Annals of Mathematical Statistics, 40(2), 549–567.
Robertson, T. and Wegman, E. T. (1978) Likelihood ratio tests for order restrictions in exponential families, The Annals of Statistics, 6(3), 485–505.
Robertson, T., Wright, F.T. and Dykstra, R. L. (1988) Order Restricted Statistical Inference, John Wiley, New York.
Sarka, S. K., Snapinn, S. and Wang, W. (1995) On improving the min test for the analysis of combination drug trails (Corr: 1998V60 pl80-181), Journal of Statistical Computation and Simulation, 51, 197–213.
Sasabuchi, S. (1980) A test of a multivariate normal mean with composite hypotheses determined by linear inequalities, Biometrika, 67,429–439.
Sasabuchi, S. (1988a) A multivariate test with composite hypotheses determined by linear inequalities when the covariance matrix is completely unknown, Memoirs of the Faculty of Science, Kyushu University, series A, Mathematics, 42(1), 37–46.
Sasabuchi, S. (1988b) A multivariate test with composite hypotheses determined by linear inequalities when the covariance matrix has an unknown scale factor, Memoirs of the Faculty of Science, Kyushu University, series A, Mathematics, 42(1), 9–19.
Sasabuchi, S. (2007) More powerful tests for homogeneity of multivariate normal mean vectors under an order restriction, Sankhya, 69(4), 700–716.
Sasabuchi, S. and Kulatunga, D. D. (1985) Some approximations for the null distribution of the $$⩈erline E2$$ statistic used in order restricted inference, Biometrika, 72, 476–480.
Sasabuchi, S., Inutsuka, M. and Kulatunga, D. D. S. (1983) A multivariate version of isotonic regression, Biometrika, 70, 465–472.
Sasabuchi, S., Inutsuka, M. and Kulatunga, D. D. S. (1992) An algorithm for computing multivariate isotonic regression, Hiroshima Mathematical Journal, 22, 551–560.
Sasabuchi, S. and Kulatunga, D. D. S. and Saito, H. (1998) Comparison of powers some tests for testing homogeneity under order restrictions in multivariate normal means, American Journal of Mathematical and Management Sciences, 18, 131–158.
Sasabuchi, S., Tanaka, K. and Takeshi, T. (2003) Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown, The Annals of Mathematical Statistics, 31(5), 1517–1536.
Shorack, G.R. (1967) Testing against ordered alternative in model I analysis of variance: normal theory and nonparametric, The Annals of Mathematical Statistics, 38,1740–1752.
Silvapulle, M.J. and Sen, P. K. (2005) Constrained Statistical Inference: Inequality Order and Shape Restrictions, John Wiley, New York.
Tang, D.-I., Gnecco, C. and Geller, N. (1989) An approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials, Biometrika, 76, 577–583.
van Eeden, C. (1958) Testing and Estimating Ordered Parameters of Probability Distributions., Ph.D. Dissertation, University of Amsterdam, Amsterdam, the Netherlands.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bazyari, A. On the computation of some properties of testing homogeneity of multivariate normal mean vectors against an order Restriction. METRON 70, 71–88 (2012). https://doi.org/10.1007/BF03263572
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03263572