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The limiting bound of Efron’s W-formula for hypothesis testing when a nuisance parameter is present only under the alternative. (English) Zbl 1189.62033

Summary: When testing a hypothesis with a nuisance parameter present only under the alternative, the maximum of a test statistic over the nuisance parameter space has been proposed. Different upper bounds for the one-sided tail probabilities of the maximum tests were provided. R. B. Davies [Biometrika 64, 247–254 (1977; Zbl 0362.62026)] studied the problem when the parameter space is an interval, while B. Efron [ibid. 84, No. 1, 143–157 (1997; Zbl 0892.62048)] considered the problem with some finite points of the parameter space and obtained a W-formula.
We study the limiting bound of Efron’s W-formula when the number of points in the parameter space goes to infinity. The conditions under which the limiting bound of the W-formula is identical to that of Davies are given. The results are also extended to two-sided tests. Examples are used to illustrate the conditions, including case-control genetic association studies. Efficient calculations of upper bounds for the tail probability with finite points in the parameter space are described.

MSC:

62F05 Asymptotic properties of parametric tests
62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] Azaïs, J. M.; Cierco-Ayrolles, C., An asymptotic test for quantitative gene detection, Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques, 38, 1087-1092 (2002) · Zbl 1011.62113
[2] Davies, R. B., Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika, 64, 247-254 (1977) · Zbl 0362.62026
[3] Davies, R. B., Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika, 74, 33-43 (1987) · Zbl 0612.62023
[4] Davies, R. B., Hypothesis testing when a nuisance parameter is present only under the alternative: linear mode case, Biometrika, 89, 484-489 (2002) · Zbl 1023.62017
[5] Efron, B., The length heuristic for simultaneous hypothesis test, Biometrika, 84, 143-157 (1997) · Zbl 0892.62048
[6] Gastwirth, J. L., The use of maximin efficiency robust tests in combining contingency tables and survival analysis, Journal of the American Statistical Association, 80, 380-384 (1985) · Zbl 0573.62042
[7] Li, Q.; Zheng, G.; Li, Z.; Yu, K., Efficient approximation of P-value of the maximum of correlated tests, with applications to genome-wide association studies, Annals of Human Genetics, 72, 397-406 (2008)
[8] Li, Q.; Zheng, G.; Liang, X.; Yu, K., Robust tests for single-marker analysis in case-control genetic association studies, Annals of Human Genetics, 73, 245-252 (2009)
[9] Shoukri, M. M.; Lathrop, G. M., Statistical testing of genetic linkage under heterogeneity, Biometrics, 49, 151-161 (1993) · Zbl 0775.62312
[10] Stein, M. L., Interpolation of Spatial Data: Some Theory for Kriging (1999), Springer: Springer New York · Zbl 0924.62100
[11] Yamada, R.; Okada, Y., An optimal dose-effect mode trend test for SNP genotype tables, Genetic Epidemiology, 33, 114-127 (2009)
[12] Zhang, H. P.; Feng, R.; Zhu, H., A latent variable model of segregation analysis for ordinal traits, Journal of the American Statistical Association, 98, 1023-1034 (2003) · Zbl 1045.62115
[13] Zheng, G.; Chen, Z., Comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative, Biometrics, 61, 254-258 (2005)
[14] Zheng, G.; Joo, J.; Yang, Y. N., Pearson’s test, trend test and MAX are all trend tests with different types of scores, Annals of Human Genetics, 73, 133-140 (2009)
[15] Zhu, H.; Zhang, H. P., Generalized score test of homogeneity for mixed effects models, Annals of Statistics, 34, 1545-1569 (2006) · Zbl 1113.62018
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