The Annals of Statistics

Dependency and false discovery rate: Asymptotics

Helmut Finner, Thorsten Dickhaus, and Markus Roters

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Abstract

Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes’ rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1432-1455.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405617

Digital Object Identifier
doi:10.1214/009053607000000046

Mathematical Reviews number (MathSciNet)
MR2351092

Zentralblatt MATH identifier
1125.62076

Subjects
Primary: 62J15: Paired and multiple comparisons 62F05: Asymptotic properties of tests
Secondary: 62F03: Hypothesis testing 60F99: None of the above, but in this section

Keywords
exchangeable test statistics expected error rate false discovery rate Glivenko–Cantelli theorem largest crossing point least favorable configurations multiple comparisons multiple test procedure multivariate total positivity of order 2 positive regression dependency Simes’ test

Citation

Finner, Helmut; Dickhaus, Thorsten; Roters, Markus. Dependency and false discovery rate: Asymptotics. Ann. Statist. 35 (2007), no. 4, 1432--1455. doi:10.1214/009053607000000046. https://projecteuclid.org/euclid.aos/1188405617


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