The limiting bound of Efron's W-formula for hypothesis testing when a nuisance parameter is present only under the alternative |
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| Authors: | Qizhai Li Gang Zheng Aiyi Liu Shifeng Xiong Zhaohai Li Kai Yu |
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| Institution: | 1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;2. Office of Biostatistics Research, Division of Prevention and Population Sciences, National Heart, Lung and Blood Institute, Bethesda, MD 20892, USA;3. Biometry and Mathematical Statistics Branch, National Institute of Child Health and Human Development, Bethesda, MD 20892, USA;4. Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda MD 20892, USA;5. Department of Statistics, George Washington University, Washington, DC 20052, USA |
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| Abstract: | 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. Davies (1977. Biometrika 64, 247–254) studied the problem when the parameter space is an interval, while Efron (1997. Biometrika 84, 143–157) 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. |
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| Keywords: | Hypothesis testing Limiting bound MAX Nuisance parameter Robustness W-formula |
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