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To How Many Simultaneous Hypothesis Tests Can Normal, Student's t or Bootstrap Calibration Be Applied?

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  • Fan, Jianqing
  • Hall, Peter
  • Yao, Qiwei

Abstract

In the analysis of microarray data, and in some other contemporary statistical problems, it is not uncommon to apply hypothesis tests in a highly simultaneous way. The number, N say, of tests used can be much larger than the sample sizes, n, to which the tests are applied, yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate. Often the sampling distribution is quite different for each test, so there may not be an opportunity to combine data across samples. In this setting, how large can N be, as a function of n, before level accuracy becomes poor? Here we answer this question in cases where the statistic under test is of Student's t type. We show that if either the normal or Student t distribution is used for calibration, then the level of the simultaneous test is accurate provided that log N increases at a strictly slower rate than n1/3 as n diverges. On the other hand, if bootstrap methods are used for calibration, then we may choose log N almost as large as n1/2 and still achieve asymptotic-level accuracy. The implications of these results are explored both theoretically and numerically.
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  • Fan, Jianqing & Hall, Peter & Yao, Qiwei, 2007. "To How Many Simultaneous Hypothesis Tests Can Normal, Student's t or Bootstrap Calibration Be Applied?," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1282-1288, December.
    • Handle: RePEc:bes:jnlasa:v:102:y:2007:m:december:p:1282-1288
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    1. Fan, Jianqing & Peng, Heng & Huang, Tao, 2005. "Semilinear High-Dimensional Model for Normalization of Microarray Data: A Theoretical Analysis and Partial Consistency," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 781-796, September.
    2. Efron, Bradley, 2004. "Large-Scale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 96-104, January.
    3. Huang, Jian & Wang, Deli & Zhang, Cun-Hui, 2005. "A Two-Way Semilinear Model for Normalization and Analysis of cDNA Microarray Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 814-829, September.
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    3. Wang, Siyang & Cui, Hengjian, 2013. "Generalized F test for high dimensional linear regression coefficients," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 134-149.
    4. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    5. Javier Hidalgo & Marcia M Schafgans, 2015. "Inference and Testing Breaks in Large Dynamic Panels with Strong Cross Sectional Dependence," STICERD - Econometrics Paper Series /2015/583, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    6. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers 45/12, Institute for Fiscal Studies.
    7. Santu Ghosh & Alan M. Polansky, 2022. "Large-Scale Simultaneous Testing Using Kernel Density Estimation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 808-843, August.
    8. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    9. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    10. Shi, Zhentao, 2016. "Econometric estimation with high-dimensional moment equalities," Journal of Econometrics, Elsevier, vol. 195(1), pages 104-119.
    11. Yang, Jun & Zou, Ran & Cheng, Jixin & Geng, Zhifei & Li, Qi, 2023. "Environmental technical efficiency and its dynamic evolution in China's industry: A resource endowment perspective," Resources Policy, Elsevier, vol. 82(C).
    12. Hidalgo, Javier & Schafgans, Marcia, 2017. "Inference and testing breaks in large dynamic panels with strong cross sectional dependence," LSE Research Online Documents on Economics 68839, London School of Economics and Political Science, LSE Library.
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    15. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2022. "High-dimensional Data Bootstrap," Papers 2205.09691, arXiv.org.

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    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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