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On the validity of the likelihood ratio and maximum likelihood methods. (English) Zbl 1022.62048
Summary: When the null or alternative hypothesis of a statistical testing problem is a union of finitely many regions of varying dimensionality, the likelihood ratio test is statistically inappropriate. Its inappropriateness is revealed not by its performance under the Neyman-Pearson criterion but by the fact that it yields incorrect inferences in certain regions of the sample space due to its inability to adapt to the differing dimensions in the composite hypothesis. Maximum likelihood estimators and associated model selection procedures also are inappropriate for such composite models. Tests and estimators based on the $p$-values associated with each of the regions that constitute the composite model are more appropriate for this geometry. Similar issues arise when the boundary of the null hypothesis is a union of finitely many regions of varying dimensionality.

62H15Multivariate hypothesis testing
Full Text: DOI
[1] Barlow, R. E.; Bartholomew, D. J.; Bremner, J. M.; Brunk, H. D.: Statistical inference under order restrictions. (1972) · Zbl 0246.62038
[2] Berger, R. L.: Uniformly more powerful tests for hypotheses concerning linear inequalities and normal means. J. amer. Statist. assoc. 84, 192-199 (1989) · Zbl 0683.62035
[3] Berger, R.; Sinclair, D.: Testing hypotheses concerning unions of linear subspaces. J. amer. Statist. assoc. 79, 158-163 (1984) · Zbl 0553.62053
[4] Bergsma, W. P.; Rudas, T.: Marginal models for categorical data. Ann. statist. 30, 140-159 (2002) · Zbl 1012.62063
[5] Berk, R.; Jones, D.: Relatively optimal combinations of test statistics. Scand. J. Statist. 5, 158-162 (1978) · Zbl 0403.62021
[6] Billera, L., Holmes, S.P., Vogtmann, K., 2000. Geometry of the space of phylogenetic trees. Technical Report, Stanford University, Stanford, CA. http://www-stat.stanford.edu/susan/papers/june1.ps. · Zbl 0995.92035
[7] Brown, L. D.; Hwang, J. T. G.; Munk, A.: An unbiased test for the bioequivalence problem. Ann. statist. 25, 2345-2367 (1997) · Zbl 0905.62107
[8] Cohen, A.; Sackrowitz, H. B.: Directional tests for one-sided alternatives in multivariate models. Ann. statist. 26, 2321-2338 (1998) · Zbl 0927.62056
[9] Cohen, A.; Kemperman, J. H. B.; Sackrowitz, H. B.: Properties of likelihood inference for order restricted models. J. multivariate anal. 72, 50-77 (2000) · Zbl 0978.62046
[10] Efron, B.: Comparing non-nested linear models. J. amer. Statist. assoc. 79, 791-803 (1984) · Zbl 0568.62035
[11] Efron, B.; Tibshirani, R.: The problem of regions. Ann. statist. 26, 1687-1718 (1998) · Zbl 0954.62031
[12] Felsenstein, J.: Evolutionary trees from DNA sequencesa maximum likelihood approach. J. mol. Evol. 17, 246-272 (1981)
[13] Felsenstein, J.: Statistical inference of phylogenies (with discussion). J. roy. Statist. soc. A 146, 368-376 (1983) · Zbl 0528.62090
[14] Geiger, D.; Heckerman, D.; King, H.; Meek, C.: Stratified exponential familiesgraphical models and model selection. Ann. statist. 29, No. 2, 505-529 (2001) · Zbl 1012.62012
[15] Holmes, S.P., 1997. Phylogenetic trees: an overview. In Statistics and Genetics, Institute of Mathematics and its Applications, Minneapolis, MN, pp. 81--118. (Also published by Springer, NY, 1999.) http://www-stat.stanford.edu/susan/papers/ima.ps.
[16] Koziol, J. A.; Perlman, M. D.: Combining independent non-central chi-square tests. J. amer. Statist. assoc. 73, 753-763 (1978) · Zbl 0394.62019
[17] Cam, Le: Maximum likelihoodan introduction. Internat. statist. Rev. 58, 153-171 (1990) · Zbl 0715.62045
[18] Lehmann, E. L.: Testing multiparameter hypotheses. Ann. math. Statist. 23, 541-562 (1952) · Zbl 0048.11702
[19] Liu, H.: Uniformly more powerful, two-sided tests for hypotheses about linear inequalities. Ann. inst. Statist. math. 52, 15-27 (2000) · Zbl 0959.62055
[20] Liu, H.; Berger, R. L.: Uniformly more powerful, one-sided tests for hypotheses about linear inequalities. Ann. statist. 23, 55-72 (1995) · Zbl 0821.62011
[21] Mcdermott, M. P.; Wang, Y.: Construction of uniformly more powerful tests for hypotheses about linear inequalities. J. statist. Plann. inference 107, 207-217 (2002) · Zbl 1016.62070
[22] Menendez, J. A.; Salvador, B.: Anomalies of the likelihood ratio test for testing restricted hypotheses. Ann. statist. 19, 889-898 (1991) · Zbl 0734.62058
[23] Menendez, J. A.; Rueda, C.; Salvador, B.: Dominance of likelihood ratio tests under cone constraints. Ann. statist. 20, 2087-2099 (1992) · Zbl 0774.62057
[24] Mukerjee, H.; Tu, R.: Order-restricted inferences in linear regression. J. amer. Statist. assoc. 90, 717-728 (1995) · Zbl 0826.62050
[25] Munk, A.: A note on unbiased testing for the equivalence problem. Statist. probab. Lett. 41, 401-406 (1999) · Zbl 0932.62116
[26] Perlman, M. D.; Wu, L.: The emperor’s new tests (with discussion). Statist. sci. 14, 355-381 (1999) · Zbl 1059.62515
[27] Perlman, M. D.; Wu, L.: A defense of the likelihood ratio test for one-sided and order-restricted alternatives. J. statist. Plann. inference 107, 173-186 (2002) · Zbl 1095.62502
[28] Resnick, S. I.: Extreme values, regular variation, and point processes. (1987) · Zbl 0633.60001
[29] Robertson, T.; Wright, F. T.; Dykstra, R. L.: Order-restricted statistical inference. (1988) · Zbl 0645.62028
[30] Roy, S. N.: On a heuristic method of test construction and its use in multivariate analysis. Ann. math. Statist. 24, 220-238 (1958) · Zbl 0051.36701
[31] Settimi, R.; Smith, J. Q.: Geometry, moments, and conditional independence trees with hidden variables. Ann. statist. 28, 1179-1205 (2000) · Zbl 1105.62321
[32] Shimodaira, H., 2000a. Approximately unbiased one-sided tests of the maximum of normal means using iterated bootstrap corrections. Technical Report No. 2000-07, Department of Statistics, Stanford University, Stanford, CA.
[33] Shimodaira, H., 2000b. Multiple comparisons of log-likelihoods and combining nonnested models with applications to phylogenetic tree selection. Technical Report, Institute of Statistical Mathematics, Tokyo, Japan. · Zbl 1008.62547
[34] Shimodaira, H.; Hasegawa, M.: Multiple comparisons of log-likelihoods with applications to phylogenetic inference. Mol. biol. Evol. 16, 1114-1116 (1999)
[35] Stein, C.: The admissibility of hotellings’s T2-test. Ann. math. Statist. 27, 616-623 (1956) · Zbl 0073.14301
[36] Tsuda, Y.: On the bioequivalence problem and a testing hypothesis problem for the bivariate normal distribution. J. Japan statist. Soc. 30, 213-236 (2000) · Zbl 0983.62082
[37] Wang, W.: Optimal unbiased tests for equivalence intrasubject variability. J. amer. Statist. assoc. 92, 1163-1170 (1997) · Zbl 1067.62585
[38] Wang, W.; Hwang, J. T. G.; Dasgupta, A.: Statistical tests for multivariate bioequivalence. Biometrika 86, 395-402 (1999) · Zbl 1054.62611
[39] Warrack, G.; Robertson, T.: A likelihood ratio test regarding two nested but oblique order-restricted hypotheses. J. amer. Statist. assoc. 79, 881-886 (1984) · Zbl 0549.62021