Bernoulli

  • Bernoulli
  • Volume 16, Number 2 (2010), 418-434.

Strong approximations of level exceedences related to multiple hypothesis testing

Peter Hall and Qiying Wang

Full-text: Open access

Abstract

Particularly in genomics, but also in other fields, it has become commonplace to undertake highly multiple Student’s t-tests based on relatively small sample sizes. The literature on this topic is continually expanding, but the main approaches used to control the family-wise error rate and false discovery rate are still based on the assumption that the tests are independent. The independence condition is known to be false at the level of the joint distributions of the test statistics, but that does not necessarily mean, for the small significance levels involved in highly multiple hypothesis testing, that the assumption leads to major errors. In this paper, we give conditions under which the assumption of independence is valid. Specifically, we derive a strong approximation that closely links the level exceedences of a dependent “studentized process” to those of a process of independent random variables. Via this connection, it can be seen that in high-dimensional, low sample-size cases, provided the sample size diverges faster than the logarithm of the number of tests, the assumption of independent t-tests is often justified.

Article information

Source
Bernoulli, Volume 16, Number 2 (2010), 418-434.

Dates
First available in Project Euclid: 25 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1274821077

Digital Object Identifier
doi:10.3150/09-BEJ220

Mathematical Reviews number (MathSciNet)
MR2668908

Zentralblatt MATH identifier
1323.62049

Keywords
false discovery rate family-wise error rate genomic data large deviation probability moving average Poisson approximation Student’s t-statistic upper tail dependence upper tail independence

Citation

Hall, Peter; Wang, Qiying. Strong approximations of level exceedences related to multiple hypothesis testing. Bernoulli 16 (2010), no. 2, 418--434. doi:10.3150/09-BEJ220. https://projecteuclid.org/euclid.bj/1274821077


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