Abstract
SenGupta and Pal (1991,J. Statist. Plann. Inference,29, 145–155) have recently obtained the locally optimal test for zero intraclass correlation coefficient in symmetric multivariate normal mixtures, with known mixing proportion, for the case when the common mean,m, and the common variance, σ2, are known. Here, we establish that even under the general situation, when some or none ofm and σ2 are known, simple optimal tests can be derived, which are locally most powerful similar, whose exact cut-off points are already available and which retain all the previous optimality properties, e.g. unbiasedness, monotonicity and consistency. Some power tables are presented to demonstrate the favorable performances of these tests.
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References
Bai, Z. D., Chen, X. R., Miao, B. Q. and Rao, C. R. (1990). Asymptotic theory of least distances estimate in multivariate linear models,Statistics,21, 503–519.
Bartels, R. (1984). Estimation in a bidirectional mixture of von Mises distributions,Biometrics,40, 777–784.
Behboodian, J. (1972). On the distribution of a symmetric statistic from a mixed population,Technometrics,14, 919–923.
Brewer, K. R. W. and Tam, S. M. (1990). Is the assumption of uniform intra-class correlation ever needed?,Austral. J. Statist.,32, 411–423.
Dugundji, J. (1975).Topology, Prentice Hall, New Delhi.
Durairajan, T. M. and Kale, B. K. (1982). Locally most powerful similar test for mixing proportion,Sankhyā Ser. A,44, 153–161.
Everitt, B. S. (1985). Mixture distributions,Encyclopedia of Statistical Sciences (eds. N. L. Johnson and S. Kotz),5, 559–569.
Ferguson, T. S. (1967).Mathematical Statistics: A Decision Theoretic Approach, Academic Press, New York.
Gokhale, D. V. and SenGupta, A. (1986). Optimal tests for the correlation coefficient in a symmetric multivariate normal population,J. Statist. Plann. Inference,14, 263–268.
Henze, N. and Zirkler, B. (1990). A class of invariant consistent tests for multivariate normality,Comm. Statist. Theory Methods,19, 3595–3617.
Johnson, R. A. and Wichern, D. W. (1982).Applied Multivariate Statistical Analysis, Prentice Hall, New Jersey.
Koch, G. G. (1983). Intraclass correlation coefficient,Encyclopedia of Statistical Sciences (eds. N. L. Johnson and S. Kotz),4, 212–217.
Koch, G. G., Klashoff, J. D. and Amar, I. A. (1988). Repeated measurements-design and analysis,Encyclopedia of Statistical Sciences (eds. N. L. Johnson and S. Kotz),8, 46–73.
Kocherlakota, K. and Kocherlakota, S. (1981). On the distribution of r in samples from the mixtures of bivariate normal populations,Comm. Statist. Theory Methods,10, 1943–1966.
Rachev, S. T. and SenGupta, A. (1992). Geometric stable distributions and Laplace-Weibull mixtures,Statist. Decisions,10, 251–271.
Rao, C. R. (1973).Linear Statistical Inference and Its Applications, 2 ed., Wiley, New York.
SenGupta, A. (1982). On tests for equicorrelation coefficient and the generalized variance of a standard symmetric multivariate normal distribution, Tech. Report, # 54 Department of Statistics, Stanford University, California.
SenGupta, A. (1987). On tests for equicorrelation coefficient of a standard symmetric multivariate normal distribution,Austral. J. Statist.,29, 49–59.
SenGupta, A. (1991). A review of optimality of multivariate tests, Special issue on multivariate optimality and related topics (guest co-editors: S. R. Jammalamadaka and A. SenGupta),Statist. Probab. Lett.,12, 527–535.
SenGupta, A. and Pal, C. (1991). Locally optimal tests for no contamination in standard symmetric multivariate normal mixtures, Special issue on reliability theory,J. Statist. Plann. Inference,29, 145–155.
SenGupta, A. and Pal, C. (1993). Optimal tests in some applied mixture models with non-regularity problems,Recent Developments in Probability and Statistics: Calcutta Symposium (to appear).
SenGupta, A. and Vermeire, L. (1986). Locally optimal tests for multiparameter hypotheses,J. Amer. Statist. Assoc.,81, 819–825.
Spjøtvoll, E. (1968). Most powerful test for some non-exponential families,Ann. Math. Statist.,39, 772–784.
Srivastava, M. S. and Lee, G. C. (1984). On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: robustness and the effect of outliers,Canad. J. Statist.,12, 119–133.
Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985).Statistical Analysis of Finite Mixture Distributions, Wiley, New York.
Verrill, S., Axelford, M. and Durst, M. (1990). We've got the positive correlation BLUEs,The American Statistician,44, 171–173.
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Sengupta, A., Pal, C. Optimal tests for no contamination in symmetric multivariate normal mixtures. Ann Inst Stat Math 45, 137–146 (1993). https://doi.org/10.1007/BF00773674
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DOI: https://doi.org/10.1007/BF00773674