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To how many simultaneous hypothesis tests can normal student's t or bootstrap calibrations 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.

Suggested Citation

  • Fan, Jianqing & Hall, Peter & Yao, Qiwei, 2007. "To how many simultaneous hypothesis tests can normal student's t or bootstrap calibrations be applied," LSE Research Online Documents on Economics 5399, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:5399
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. Hidalgo, Javier & Schafgans, Marcia, 2017. "Inference and testing breaks in large dynamic panels with strong cross sectional dependence," Journal of Econometrics, Elsevier, vol. 196(2), pages 259-274.
    2. 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.
    3. 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.
    4. Shi, Zhentao, 2016. "Econometric estimation with high-dimensional moment equalities," Journal of Econometrics, Elsevier, vol. 195(1), pages 104-119.
    5. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    6. 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.
    7. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers CWP45/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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    JEL classification:

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

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