Abstract
Let Π i be an i-tb population with a probability density function f(· | θ i ) with one dimensional unknown parameter θ i = 1, 2, ... , k. Let n i sample be drawn from each Π i . The likelihood ratio criteria λ j|(j−1) for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j − 1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, ... , k. The paper shows the asymptotic independence of λ j|(j−1)'s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples.
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References
Barndorff-Nielsen, O. E. and Cox, D. R. (1994). Inference and Asymptotics, Chapman and Hall, London.
Bartlett, M. S. (1937). Properties of sufficiency and statistical tests, Proc. Royal Soc. London Ser. A, 160, 268–282.
Bickel, P. J. and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction—A Bayesian argument, Ann. Statist., 18, 1070–1090.
Cordeiro, G. M. (1987). On the corrections to the likelihood ratio statistics, Biometrika, 74, 265–274.
Harris, P. (1986). A note on Bartlett adjustments to likelihood ratio tests, Biometrika, 73, 735–737.
Hayakawa, T. (1977). The likelihood ratio criterion and the asymptotic expansion of its distribution, Ann. Inst. Statist. Math., 29, 359–378 (Correction: ibid. (1987). 39, P. 681).
Hayakawa, T. (1993). Test of homogeneity of parameters, Statistical Science and Data Analysis (eds. K. Matusita, M. L. Puri and T. Hayakawa), 337–343, VSP Zeist.
Hayakawa, T. (1994). Test of homogeneity of multiple parameters, J. Statist. Plann. Inference, 38, 351–357.
Hayakawa, T. (2001). Rao's statistic for homogeneity of multiple parameters, J. Statist. Plann. Inference, Special Issue for Rao's Score Statistic (eds. A. K. Bera and R. Mukerjee), 97, 101–111.
Hayakawa, T. and Doi, M. (1999). Modified Wald statistic for homogeneity of multiple parameters, Comm. Statist. Theory. Methods, 28, 755–771.
Lawley, D. N. (1956). A general method of approximating to the distribution of likelihood ratio criteria, Biometrika, 43, 295–303.
Ogawa, J. (1949). On the independence of bilinear form and quadratic forms of a random sample from a normal population, Ann. Inst. Statist. Math., 1, 83–108.
Takemura, A. and Kuriki, S. (1996). A proof of independent Bartlett correctability of nested likelihood ratio tests, Ann. Inst. Statist. Math., 48, 603–620.
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Hayakawa, T. Independence of Likelihood Ratio Criteria for Homogeneity of Several Populations. Annals of the Institute of Statistical Mathematics 54, 918–933 (2002). https://doi.org/10.1023/A:1022431906333
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DOI: https://doi.org/10.1023/A:1022431906333