Dependency and false discovery rate: Asymptotics

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.Comment: Published in at http://dx.doi.org/10.1214/009053607000000046 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the false discovery rate and an asymptotically optimal rejection curve

In this paper we introduce and investigate a new rejection curve for asymptotic control of the false discovery rate (FDR) in multiple hypotheses testing problems. We first give a heuristic motivation for this new curve and propose some procedures related to it. Then we introduce a set of possible assumptions and give a unifying short proof of FDR control for procedures based on Simes' critical values, whereby certain types of dependency are allowed. This methodology of proof is then applied to other fixed rejection curves including the proposed new curve. Among others, we investigate the problem of finding least favorable parameter configurations such that the FDR becomes largest. We then derive a series of results concerning asymptotic FDR control for procedures based on the new curve and discuss several example procedures in more detail. A main result will be an asymptotic optimality statement for various procedures based on the new curve in the class of fixed rejection curves. Finally, we briefly discuss strict FDR control for a finite number of hypotheses.Comment: Published in at http://dx.doi.org/10.1214/07-AOS569 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Edgeworth expansions and rates of convergence for normalized sums: Chung's 1946 method revisited

In this work we revisit, correct and extend Chung's 1946 method for deriving higher order Edgeworth expansions with respect to t-statistics and generalized self-normalized sums. Thereby we provide a set of formulas which allows the computation of the approximation of any order and specify the first four polynomials in the Edgeworth expansion, the first two of which are well known. It turns out that knowledge of the first four polynomials is necessary and sufficient for characterizing the rate of convergence of the Edgeworth expansion in terms of moments and the norming sequence appearing in generalized self-normalized sums. It will be shown that depending on the moments and the norming sequence, the rate of convergence can be O(n-i/2), i=1,...,4. Finally, we study expansions and rates of convergence if the normal distribution is replaced by the t-distribution.Generalized self-normalized sum Hermite polynomials Normal approximation Statistical algorithms Student's t