False Discovery Rate Control under Archimedean Copula
FFalse Discovery Rate Control underArchimedean Copula
Taras Bodnar
Department of MathematicsHumboldt-University BerlinUnter den Linden 6D-10099 BerlinGermanye-mail: [email protected] andThorsten Dickhaus
Department of MathematicsHumboldt-University BerlinUnter den Linden 6D-10099 BerlinGermanye-mail: [email protected]
Abstract:
We are considered with the false discovery rate (FDR) of thelinear step-up test ϕ LSU considered by Benjamini and Hochberg (1995).It is well known that ϕ LSU controls the FDR at level m q/m if the jointdistribution of p -values is multivariate totally positive of order 2. In this, m denotes the total number of hypotheses, m the number of true nullhypotheses, and q the nominal FDR level. Under the assumption of an Ar-chimedean p -value copula with completely monotone generator, we derivea sharper upper bound for the FDR of ϕ LSU as well as a non-trivial lowerbound. Application of the sharper upper bound to parametric subclasses ofArchimedean p -value copulae allows us to increase the power of ϕ LSU bypre-estimating the copula parameter and adjusting q . Based on the lowerbound, a sufficient condition is obtained under which the FDR of ϕ LSU is exactly equal to m q/m , as in the case of stochastically independent p -values. Finally, we deal with high-dimensional multiple test problems withexchangeable test statistics by drawing a connection between infinite se-quences of exchangeable p -values and Archimedean copulae with completelymonotone generators. Our theoretical results are applied to important cop-ula families, including Clayton copulae and Gumbel copulae. AMS 2000 subject classifications:
Primary 62J15, 62F05; secondary62F03.
Keywords and phrases:
Clayton copula, exchangeability, Gumbel cop-ula, linear step-up test, multiple hypotheses testing, p -values. Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a r X i v : . [ m a t h . S T ] N ov odnar and Dickhaus/Copula-based FDR Control Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
Control of the false discovery rate (FDR) has become a standard type I er-ror criterion in large-scale multiple hypotheses testing. When the number m of hypotheses to be tested simultaneously is of order 10 − , as it is preva-lent in many modern applications from the life sciences like genetic associationanalyses, gene expression studies, functional magnetic resonance imaging, orbrain-computer interfacing, it is typically infeasible to model or to estimate thefull joint distribution of the data. Hence, one is interested in generic proceduresthat control the FDR under no or only qualitative assumptions regarding thisjoint distribution. The still by far most popular multiple test for FDR control,the linear step-up test ϕ LSU (say) considered in the seminal work by Benjaminiand Hochberg (1995), operates on marginal p -values p , . . . , p m . As shown byBenjamini and Yekutieli (2001) and Sarkar (2002), ϕ LSU is generically FDR-controlling over the class of models that lead to positive dependency amongthe random p -values P , . . . , P m in the sense of positive regression dependencyon subsets (PRDS), including p -value distributions which are multivariate to-tally positive of order 2 (MTP ). Under the PRDS assumption, the FDR of ϕ LSU is upper-bounded by m q/m , where m denotes the number of true nullhypotheses and q the nominal FDR level.In this work, we extend these findings by deriving a sharper upper bound forthe FDR of ϕ LSU in the case that the dependency structure among P , . . . , P m can be expressed by an Archimedean copula. Our respective contributions arethreefold. First, we quantify the magnitude of conservativity (non-exhaustionof the FDR level q ) of ϕ LSU in various copula models as a function of the cop-ula parameter η . This allows for a gain in power in practice by pre-estimating η and adjusting the nominal value of q . Second, we demonstrate by computersimulations that the proposed upper bound leads to a robust procedure in thesense that the variance of this bound over repeated Monte Carlo simulations ismuch smaller than the corresponding variance of the false discovery proportion(FDP) of ϕ LSU . This makes the utilization of our upper bound an attractivechoice in practice, addressing the issue that the FDP is typically not well con-centrated around its mean, the FDR, if p -values are dependent. As a by-product,we directly obtain that the FDR of ϕ LSU is bounded by m q/m under the as-sumption of an Archimedean p -value copula, without explicitly relying on the odnar and Dickhaus/Copula-based FDR Control MTP property (which is fulfilled in the class of Archimedean p -value copulaewith completely monotone generator functions, cf. M¨uller and Scarsini (2005)).Let us point out already here that the FDR criterion is only suitable if thenumber m of tests is large. In this case, the restriction to completely monotonegenerators is essentially void, because every copula generator is necessarily m -monotone. Third, in an asymptotic setting ( m → ∞ ), we show that the classof Archimedean p -value copulae with completely monotone generators includescertain models with p -values or test statistics, respectively, which are exchange-able under null hypotheses, H -exchangeable for short. Such H -exchangeabletest statistics occur naturally in many multiple test problems, for instance inmany-to-one comparisons or if test statistics are given by jointly Studentizedmeans (cf. Finner, Dickhaus and Roters (2007)).In addition, we also derive and discuss a lower FDR bound for ϕ LSU interms of the generator of an Archimedean p -value copula. Application of thislower bound leads to sufficient conditions under which the FDR of ϕ LSU isexactly equal to m q/m , at least asymptotically as m tends to infinity and m /m converges to a fixed value. Hence, if the latter conditions are fulfilled, theFDR behaviour of ϕ LSU is under dependency the same as in the case of jointlystochastically independent p -values.The paper is organized as follows. In Section 2, we set up the necessary nota-tion, define our class of statistical models for P , . . . , P m , and recall propertiesand results around the FDR. Our main contributions are presented in Section3, dealing with FDR control of ϕ LSU under the assumption of an Archimedeancopula. Special parametric copula families are studied in Section 4, where wequantify the realized FDR of ϕ LSU as a function of η . Section 5 outlines meth-ods for pre-estimation of η . We conclude with a discussion in Section 6. Lengthyproofs are deferred to Section 7.
2. Notation and preliminaries
All multiple test procedures considered in this work depend on the data only via(realized) marginal p -values p , . . . , p m and their ordered values p (1) ≤ p (2) ≤ . . . ≤ p ( m ) . Hence, it suffices to model the distribution of the random vector P = ( P , . . . , P m ) (cid:62) of p -values and we consider statistical models of the form([0 , m , B ([0 , m ) , ( P ϑ,η : ϑ ∈ Θ , η ∈ Ξ)). In this, we assume that ϑ is the (main)parameter of statistical interest and we identify the null hypotheses H i : 1 ≤ i ≤ m with non-empty subsets of Θ, with corresponding alternatives K i = Θ \ H i .The null hypothesis H i is called true if ϑ ∈ H i and false otherwise. We let I ≡ I ( ϑ ) = { ≤ i ≤ m : ϑ ∈ H i } denote the index set of true hypothesesand m ≡ m ( ϑ ) = | I | the number of true nulls. Without loss of generality, wewill assume I ( ϑ ) = { , . . . , m } throughout the work. Analogously, we define I = { , . . . , m } , I ≡ I ( ϑ ) = I \ I and m ≡ m ( ϑ ) = | I | = m − m . Theintersection hypothesis H = (cid:84) mi =1 H i will be referred to as the global (null)hypothesis.The parameter η is the copula parameter of the joint distribution of P , thus odnar and Dickhaus/Copula-based FDR Control representing the dependency structure among P , . . . , P m . Its parameter spaceΞ may be of infinite dimension. In particular, in Section 3 we will consider theclass of all Archimedean copulas which can be indexed by the generator function ψ . However, we sometimes restrict our attention to parametric subclasses, forinstance the class of Clayton copulae which can be indexed by a one-dimensionalcopula parameter η ∈ R . In any case, we will assume that η is a nuisanceparameter in the sense that it does not depend on ϑ and that the marginaldistribution of each P i is invariant with respect to η . Therefore, to simplifynotation, we will write P ϑ instead of P ϑ,η if marginal p -value distributions areconcerned. Throughout the work, the p -values P , . . . , P m are assumed to bevalid in the sense that ∀ ≤ i ≤ m : ∀ ϑ ∈ H i : ∀ t ∈ [0 ,
1] : P ϑ ( P i ≤ t ) ≤ t. A (non-randomized) multiple test operating on p -values is a measurable map-ping ϕ = ( ϕ i : 1 ≤ i ≤ m ) : [0 , m → { , } m the components of which have theusual interpretation of a statistical test for H i versus K i , 1 ≤ i ≤ m . For fixed ϕ ,we let V m ≡ V m ( ϑ ) = |{ i ∈ I ( ϑ ) : ϕ i = 1 }| denote the (random) number of falserejections (type I errors) of ϕ and R m ≡ R m ( ϑ ) = |{ i ∈ { , . . . , m } : ϕ i = 1 }| the total number of rejections. The FDR under ( ϑ, η ) of ϕ is then defined byFDR ϑ,η ( ϕ ) = E ϑ,η (cid:20)(cid:18) V m R m ∨ (cid:19)(cid:21) , (1)and ϕ is said to control the FDR at level q ∈ (0 ,
1) if sup ϑ ∈ Θ ,η ∈ Ξ FDR ϑ,η ( ϕ ) ≤ q . The random variable V m / max( R m ,
1) is referred to as the false discoveryproportion of ϕ , FDP ϑ,η ( ϕ ) for short. Notice that, although the trueness ofthe null hypotheses is determined by ϑ alone, the FDR depends on ϑ and η ,because the dependency structure among the p -values typically influences thedistribution of ϕ when regarded as a statistic with values in { , } m .The linear step-up test ϕ LSU , also referred to as Benjamini-Hochberg test or the
FDR procedure in the literature, rejects exactly hypotheses H (1) , . . . , H ( k ) ,where the bracketed indices correspond to the order of the p -values and k =max { ≤ i ≤ m : p ( i ) ≤ q i } for linearly increasing critical values q i = iq/m . If k does not exist, no hypothesis is rejected. The sharpest characterization of FDRcontrol of ϕ LSU that we are aware of so far is given in the following theorem.
Theorem 2.1 (Finner, Dickhaus and Roters (2009)) . Consider the following assumptions. (D1) ∀ ( ϑ, η ) ∈ Θ × Ξ : ∀ j ∈ I : ∀ i ∈ I ( ϑ ) : P ϑ,η ( R m ≥ j | P i ≤ t ) is non-increasing in t ∈ (0 , q j ] . (D2) ∀ ϑ ∈ Θ : ∀ i ∈ I ( ϑ ) : P i ∼ UNI ([0 , . (I1) ∀ ( ϑ, η ) ∈ Θ × Ξ : The p -values ( P i : i ∈ I ( ϑ )) are independent and identi-cally distributed (iid). (I2) ∀ ( ϑ, η ) ∈ Θ × Ξ : The random vectors ( P i : i ∈ I ( ϑ )) and ( P i : i ∈ I ( ϑ )) are stochastically independent. odnar and Dickhaus/Copula-based FDR Control Then, the following two assertions hold true.Under (D1), ∀ ( ϑ, η ) ∈ Θ × Ξ :
FDR ϑ,η ( ϕ LSU ) ≤ m ( ϑ ) m q. (2) Under (D2)-(I2), ∀ ( ϑ, η ) ∈ Θ × Ξ :
FDR ϑ,η ( ϕ LSU ) = m ( ϑ ) m q. (3)The crucial assumption (D1) is fulfilled for multivariate distributions of P which are positively regression dependent on the subset I (PRDS on I ) in thesense of Benjamini and Yekutieli (2001). In particular, if the joint distributionof P is MTP , then (D1) holds true.To mention also a negative result, Guo and Rao (2008) have shown that thereexists a multivariate distribution of P such that the FDR of ϕ LSU is equal to m q/m (cid:80) mj =1 j − , showing that ϕ LSU is not generically FDR-controlling over allpossible joint distributions of P . The main purpose of the present work (Section3) is to derive a sharper upper bound on the right-hand side of (2), assumingthat Ξ is the space of completely monotone generator functions of Archimedeancopulae.
3. FDR control under Archimedean Copula
In this section, it is assumed that the joint distribution of P is given by anArchimedean copula such that F P ( p , ..., p m ) = P ϑ,ψ ( P ≤ p , ..., P m ≤ p m ) = ψ (cid:32) m (cid:88) i =1 ψ − ( F P i ( p i )) (cid:33) , (4)where the function ψ ( · ) is the so-called copula generator and takes the role of η in our general setup. In (4) and throughout the work, F ξ denotes the cumulativedistribution function (cdf) of the variate ξ . The generator ψ fully determinesthe type of the Archimedean copula; see, e.g. Nelsen (2006). A necessary andsufficient condition under which a function ψ : R + → [0 ,
1] with ψ (0) = 1 andlim x →∞ ψ ( x ) = 0 can be used as a copula generator is that ψ ( · ) is an m -alteringfunction, that is, ( − d ψ ( d ) ( · ) ≥ d ∈ { , , ..., m } , cf. M¨uller and Scarsini(2005). Throughout the present work, we impose a slightly stronger assumptionon ψ . Namely, we assume that ψ is completely monotone, i. e. ( − d ψ ( d ) ( · ) ≥ d ∈ N . If m is large as it is usual in applications of the FDR criterion, thedistinction between the class of completely monotone functions and the class of m -altering functions becomes negligible.A very useful property of an Archimedean copula with completely monotonegenerator ψ is the stochastic representation of P . Namely, there exists a sequenceof jointly independent and identically UNI[0 , Y , . . . , Y m such that (cf. Marshall and Olkin (1988), Section 5) P = ( P i : 1 ≤ i ≤ m ) d = (cid:16) F − P i (cid:16) ψ (cid:16) log (cid:16) Y − /Zi (cid:17)(cid:17)(cid:17) : 1 ≤ i ≤ m (cid:17) , (5) odnar and Dickhaus/Copula-based FDR Control where the symbol d = denotes equality in distribution. The random variable Z with Laplace transform t (cid:55)→ ψ ( t ) = E [(exp( − tZ ))] is independent of Y , . . . , Y m ,and its distribution is determined by ψ only. Throughout the remainder, P and E refer to the distribution of Z , for ease of presentation. The stochasticrepresentation (5) shows that the type of the Archimedean copula can equiva-lently be expressed in terms of the random variable Z . Moreover, the p -values( P i : 1 ≤ i ≤ m ) are conditionally independent given Z = z . This second prop-erty allows us to establish the following sharper upper bound for the FDR of ϕ LSU . Theorem 3.1 (Upper FDR bound) . Let Z be as in (5) and let P ( i ) consist ofthe ( m − remaining p-values obtained by dropping P i from P so that P ( i )(1) ≤ P ( i )(2) ≤ ... ≤ P ( i )( m − . The random set D ( i ) k is then given by D ( i ) k = { q k +1 ≤ P ( i )( k ) , . . . , q m ≤ P ( i )( m − } . (6) For a given value Z = z we define the function T : [0 , m → [0 , m by T ( p ) = ( T ( p ) , ..., T m ( p m )) T with T j ( p j ) = exp (cid:0) − zψ − (cid:0) F P j ( p j ) (cid:1)(cid:1) for p =( p , ..., p m ) T ∈ [0 , m . This function transforms, for fixed Z = z , realizationsof P into realizations of Y = ( Y , . . . , Y m ) (cid:62) given in (5) . Let D ( i,z ) Y ; k denote theimage of the set D ( i ) k under T for given Z = z and let G ik ( z ) = P ϑ,ψ (cid:16) D ( i,z ) Y ; k (cid:17) .Then it holds ∀ ϑ ∈ Θ :
FDR ϑ,ψ ( ϕ LSU ) ≤ m ( ϑ ) m q − b ( m, ϑ, ψ ) , where b ( m, ϑ, ψ ) = qm m (cid:88) i =1 m − (cid:88) k =1 (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z )= qm m (cid:88) i =1 m − (cid:88) k =1 E (cid:34)(cid:32) exp (cid:0) − Zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − Zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( Z ) − G ik ( z ∗ k )) [ z ∗ k , ∞ ) ( Z ) (cid:105) (7) with z ∗ k = log q k +1 − log q k ψ − ( q k ) − ψ − ( q k +1 ) = log (1 + 1 /k ) ψ − ( kq/m ) − ψ − (( k + 1) q/m ) (8) and A denoting the indicator function of the set A . Noticing that b ( m, ϑ, ψ ) is always non-negative, we obtain the following resultas a straightforward corollary of Theorem 3.1. odnar and Dickhaus/Copula-based FDR Control Corollary 3.1.
Let the copula of P = ( P , ..., P m ) (cid:62) be an Archimedean copula,where P i is continuously distributed on [0 , for ≤ i ≤ m . Then it holds that ∀ ϑ ∈ Θ : ∀ ψ ∈ Ξ :
FDR ϑ,ψ ( ϕ LSU ) ≤ m ( ϑ ) m q, (9) where Ξ denotes the set of all completely monotone generator functions of Ar-chimedean copulae. The result of Corollary 3.1 is in line with the findings obtained by Benjaminiand Yekutieli (2001) and Sarkar (2002) that we have recalled in Section 1.Namely, M¨uller and Scarsini (2005) pointed out that an Archimedean copulapossesses the MTP property if the copula generator ψ is completely monotoneand, hence, the FDR is controlled by ϕ LSU in this case.From the practical point of view, it is problematic that b ( m, ϑ, ψ ) depends onthe (main) parameter ϑ of statistical interest. In practice, one will therefore oftenonly be able to work with sup ϑ ∈ Θ { m ( ϑ ) q/m − b ( m, ϑ, ψ ) } . Since b ( m, ϑ, ψ ) ≥ ϑ ∈ Θ, the latter ϑ -free upper bound will typically still yield an improve-ment over the ”classical” upper bound. The issue of minimization of b ( m, · , ψ )over ϑ ∈ Θ is closely related to the challenging task of determining least fa-vorable parameter configurations (LFCs) for the FDR. So-called Dirac-uniformconfigurations are least favorable (provide upper FDR bounds) for ϕ LSU underindependence assumptions and are assumed to be generally least favorable for ϕ LSU also in models with dependent p -values, at least for large values of m (cf.,e. g., Finner, Dickhaus and Roters (2007), Blanchard et al. (2013)). Troendle(2000) motivated the consideration of Dirac-uniform configurations from thepoint of view of consistency of marginal tests with respect to the sample size.Furthermore, the expectations in (7) can in general not be calculated analyti-cally. However, they can easily be approximated by means of computer simula-tions. Namely, the approximation is performed by generating random numberswhich behave like independent realizations of Z , which completely specifies thetype of the Archimedean copula, evaluating the functions G ik at the generatedvalues and replacing the theoretical expectation of Z by the arithmetic mean ofthe resulting values of the integrand in (7). Under Dirac-uniform configurations,evaluation of G ik can efficiently be performed by means of recursive formulas forthe joint cdf of the order statistics of Y . We discuss these points in detail inSection 4.Next, we discuss a lower bound for the FDR of ϕ LSU under the assumptionof an Archimedean copula.
Theorem 3.2 (Lower FDR bound) . Let the copula of P = ( P , ..., P m ) T bean Archimedean copula with generator function ψ , where P i is continuouslydistributed on [0 , for i = 1 , . . . , m . Then it holds that ∀ ϑ ∈ Θ :
FDR ϑ,ψ ( ϕ LSU ) ≥ m qm γ min , (10) odnar and Dickhaus/Copula-based FDR Control where γ min ≡ γ min ( ψ ) = (cid:90) min k ∈{ ,...,m } (cid:40) exp (cid:0) − zψ − ( kq/m ) (cid:1) kq/m (cid:41) dF Z ( z ) . (11)From the assertion of Theorem 3.2 we conclude that the lower bound forthe FDR of ϕ LSU under the assumption of an Archimedean copula cruciallydepends on the extreme points of the function g ( ·| z ), given by g ( x | z ) = exp ( − zx ) ψ ( x ) (12)for x ∈ { ψ − ( q/m ) , ψ − (2 q/m ) , . . . , ψ − ( q ) } . If for all z > g ( x | z ) is always attained for the same index k ∗ (say), then γ min = 1 andtogether with Theorem 3.1 we get FDR ϑ,ψ ( ϕ LSU ) = m ( ϑ ) q/m . This followsdirectly from the identity (cid:90) exp (cid:0) − zψ − ( k ∗ q/m ) (cid:1) k ∗ q/m dF Z ( z ) = ψ (cid:0) ψ − ( k ∗ q/m ) (cid:1) k ∗ q/m = 1 . However, the latter holds true only in some specific cases. To obtain a moreexplicit constant γ min ( ψ ) in the general case, we notice that, due to the analyticproperties of ψ , there exists a point z ∗ such that g (cid:0) ψ − ( q ) | z (cid:1) < g (cid:0) ψ − ( q/m ) | z (cid:1) for z < z ∗ and g (cid:0) ψ − ( q ) | z (cid:1) > g (cid:0) ψ − ( q/m ) | z (cid:1) for z > z ∗ . The point z ∗ isobtained as the solution of0 = g (cid:0) ψ − ( q ) | z (cid:1) − g (cid:0) ψ − ( q/m ) | z (cid:1) = exp (cid:0) − zψ − ( q ) (cid:1) q − exp (cid:0) − zψ − ( q/m ) (cid:1) q/m = 1 q (cid:16) exp (cid:0) − zψ − ( q ) (cid:1) − exp (cid:16) log m − zψ − (cid:16) qm (cid:17)(cid:17)(cid:17) = exp (cid:0) − zψ − ( q ) (cid:1) q (cid:16) − exp (cid:16) log m + z (cid:16) ψ − ( q ) − ψ − (cid:16) qm (cid:17)(cid:17)(cid:17)(cid:17) , which leads to z ∗ = log mψ − ( q/m ) − ψ − ( q ) . (13)Next, we analyze the function x (cid:55)→ g ( x | z ) for given z . For its derivative withrespect to x , it holds that g (cid:48) ( x | z ) = − exp ( − zx )( ψ ( x )) ( zψ ( x ) + ψ (cid:48) ( x )) . Setting this expression to zero, we get that any extreme point of g ( ·| z ) satisfies zψ ( x ) + ψ (cid:48) ( x ) = 0 . (14) odnar and Dickhaus/Copula-based FDR Control Let x z be a solution of (14). Then, the second derivative of g ( ·| z ) at x z isgiven by g (cid:48)(cid:48) ( x z | z ) = − exp ( − zx z )( ψ ( x z )) ( zψ (cid:48) ( x z ) + ψ (cid:48)(cid:48) ( x z )) . (15)Substituting (14) with x = x z in (15), we obtain g (cid:48)(cid:48) ( x z | z ) = − exp ( − zx z )( ψ ( x z )) (cid:18) − ( ψ (cid:48) ( x z )) ψ ( x z ) + ψ (cid:48)(cid:48) ( x z ) (cid:19) = − exp ( − zx z )( ψ ( x z )) (cid:0) ψ ( x z ) ψ (cid:48)(cid:48) ( x z ) − ( ψ (cid:48) ( x z )) (cid:1) and application of the Cauchy-Schwarz inequality leads to ψ ( x z ) ψ (cid:48)(cid:48) ( x z ) = (cid:90) exp ( − zx z ) dF Z ( z ) (cid:90) z exp ( − zx z ) dF Z ( z ) ≥ (cid:18)(cid:90) z exp ( − zx z ) dF Z ( z ) (cid:19) = ( ψ (cid:48) ( x z )) . This proves that g (cid:48)(cid:48) ( x z | z ) ≤ x z is an extreme point of g ( x z | z ). Thus, any such x z is a maximum and the minimum in (11) is attained at ψ − ( q ) for z ≤ z ∗ aswell as at ψ − ( q/m ) for z ≥ z ∗ . This allows for a more explicit characterizationof the lower bound. Lemma 3.1.
The quantity γ min ≡ γ min ( ψ ) from (11) can equivalently be ex-pressed as γ min = 1 − (cid:90) z ∗ (cid:0) g (cid:0) ψ − ( q/m ) | z (cid:1) − g (cid:0) ψ − ( q ) | z (cid:1)(cid:1) dF Z ( z ) (16)= 1 − E (cid:2) ( g (cid:0) ψ − ( q/m ) | Z (cid:1) − g (cid:0) ψ − ( q ) | Z (cid:1) ) [0 ,z ∗ ] ( Z ) (cid:3) , (17) where g ( ·| z ) and z ∗ are defined in (12) and (13) , respectively. If the integral in (16) cannot be calculated analytically, then it can easily beapproximated via a Monte Carlo simulation by using the expression on the right-hand side of (17) and replacing the theoretical expectation by its pseudo-sampleanalogue.Lemma 3.1 possesses several interesting applications. We consider the quan-tity γ min itself. It holds that 1 ≥ γ min ≥ γ min , where γ min = 1 − min (cid:40)(cid:90) z ∗ sup z ∈ [0 ,z ∗ ] h ( z ) dF Z ( z ) , (cid:90) ∞ z ∗ sup z ∈ [ z ∗ , ∞ ] ( − h ( z )) dF Z ( z ) (cid:41) (18)with h ( z ) = g (cid:0) ψ − ( q/m ) | z (cid:1) − g (cid:0) ψ − ( q ) | z (cid:1) = exp (cid:0) − zψ − (cid:0) qm (cid:1)(cid:1) q/m − exp (cid:0) − zψ − ( q ) (cid:1) q , odnar and Dickhaus/Copula-based FDR Control because (cid:90) z ∗ h ( z ) dF Z ( z ) = − (cid:90) ∞ z ∗ h ( z ) dF Z ( z ) . (19)However, both of the integrals in (19) can be bounded by different values. Tosee this, we study the behavior of the function z (cid:55)→ h ( z ). It holds that h (cid:48) ( z ) = − ψ − (cid:16) qm (cid:17) exp (cid:0) − zψ − (cid:0) qm (cid:1)(cid:1) q/m + ψ − ( q ) exp (cid:0) − zψ − ( q ) (cid:1) q = ψ − ( q ) exp (cid:0) − zψ − ( q ) (cid:1) q × (cid:32) − exp (cid:32) log m + log ψ − (cid:0) qm (cid:1) ψ − ( q ) + z (cid:16) ψ − ( q ) − ψ − (cid:16) qm (cid:17)(cid:17)(cid:33)(cid:33) . Since ψ − is a non-increasing function, we get that there exists a unique mini-mum of h ( z ) at z ∗ = log m + log ψ − ( q/m ) − log ψ − ( q ) ψ − ( q/m ) − ψ − ( q ) ≥ z ∗ . Consequently, we get (cid:90) z ∗ sup z ∈ [0 ,z ∗ ] h ( z ) dF Z ( z ) = (cid:90) z ∗ h (0) dF Z ( z ) = h (0) F Z ( z ∗ )= m − q F Z ( z ∗ ) , (cid:90) ∞ z ∗ sup z ∈ [ z ∗ , ∞ ] ( − h ( z )) dF Z ( z ) = (cid:90) ∞ z ∗ h ( z ∗ ) dF Z ( z ) = h ( z ∗ )(1 − F Z ( z ∗ ))= exp (cid:0) − z ∗ ψ − ( q ) (cid:1) q (cid:18) − ψ − ( q ) ψ − ( q/m ) (cid:19) × (1 − F Z ( z ∗ )) . Corollary 3.2.
Under the assumptions of Theorem 3.2, the following two as-sertions hold true.(a) If z ∗ from (13) does not lie in the support of F Z , i. e., if F Z ( z ∗ ) = 0 or F Z ( z ∗ ) = 1 , then γ min = γ min = 1 and, consequently, FDR ϑ,ψ ( ϕ LSU ) = m q/m .(b) Assume that π = lim m →∞ m /m exists. If z ∗ = z ∗ ( m ) is such that F Z ( z ∗ ( m )) → or F Z ( z ∗ ( m )) → as m → ∞ , then lim m →∞ FDR ϑ,ψ ( ϕ LSU ) = π q. Part (b) of Corollary 3.2 motivates a deeper consideration of asymptotic orhigh-dimensional multiple tests, i. e., the case of m → ∞ , under our general odnar and Dickhaus/Copula-based FDR Control setup. This approach has already been discussed widely in previous literature.For instance, it was called ”asymptotic multiple test” by Genovese and Wasser-man (2002). The case m → ∞ was also considered by Finner and Roters (1998),Storey (2002), Genovese and Wasserman (2004), Finner, Dickhaus and Roters(2007, 2009), Jin and Cai (2007), Sun and Cai (2007), and Cai and Jin (2010),among others.Very interesting connections can be drawn between Archimedean p -valuecopulae and infinite sequences of H -exchangeable p -values. More precisely, letus assume an infinite sequence ( P i ) i ∈ N of p -values which are absolutely con-tinuous and uniformly distributed on [0 ,
1] under the respective null hypoth-esis H i . Furthermore, we let F i denote the cdf. of P i under ϑ and assumethat F ( P ) , . . . , F m ( P m ) , . . . are exchangeable random variables, entailing that P , . . . , P m , . . . themselves are exchangeable under the global hypothesis H . Se-quences of H -exchangeable p -values have already been investigated by Finnerand Roters (1998) and Finner, Dickhaus and Roters (2007) in special settings.Moreover, the assumption of exchangeability is also pivotal in other areas ofstatistics, let us mention Bayesian analysis and validity of permutation tests.The problem of exchangeability in population genetics has been discussed byKingman (1978).For ease of notation, let ˜ P i = F i ( P i ) for i ∈ N . Because ˜ P , . . . , ˜ P m , . . . is anexchangeable sequence of random variables, it exists a random variable Z withdistribution function F Z such that the joint distribution of ˜ P , . . . , ˜ P m is for anyfixed m ∈ N given by F ˜ P ,..., ˜ P m ( p , . . . , p m ) = (cid:90) F ˜ P | Z = z ( p ) × . . . × F ˜ P m | Z = z ( p m ) dF Z ( z ) , (20)see Olshen (1974) and equation (3.1) of Kingman (1978). Moreover, assumingthat Z ∈ (0 , ∞ ) with probability 1, we obtain for any i ∈ N from Marshall andOlkin (1988), p. 834, that p i = F ˜ P i ( p i ) = (cid:90) exp (cid:0) − zψ − ( p i ) (cid:1) dF Z ( z ) , where ψ denotes the Laplace transform of Z , i. e., ψ ( t ) = E [exp( − tZ )].Theorem 3.3 establishes a connection between the finite-dimensional marginaldistributions of H -exchangeable p -value sequences and Archimedean copulae. Theorem 3.3.
Assume that the elements in the infinite sequence ( P i ) i ∈ N areabsolutely continuous and H -exchangeable. Furthermore, let the following twoassumptions be valid.(i) The random variable Z from (20) takes values in (0 , ∞ ) with probability .(ii) It holds F ˜ P i | Z = z ( p i ) = exp (cid:0) − zψ − ( p i ) (cid:1) , z ∈ (0 , ∞ ) . (21) odnar and Dickhaus/Copula-based FDR Control Then, for any m , p = ( p , . . . , p m ) (cid:62) (cid:55)→ ψ (cid:32) m (cid:88) i =1 ψ − ( p i ) (cid:33) is a copula of P , ..., P m , where ψ ( t ) = E [exp( − tZ )] . The final result of this section is an immediate consequence of Theorem 3.3and Corollary 3.1.
Corollary 3.3.
Under the assumptions of Theorem 3.3, it holds:a) Any m -dimensional marginal distribution of the sequence ( P i ) i ∈ N possessesthe MTP property, m ≥ .b) The linear step-up test ϕ LSU , applied to p , . . . , p m , controls the FDR atlevel q .
4. Examples: Parametric copula families
In this section, we apply the theoretical results of Section 3 to several parametricfamilies of Archimedean copulas.
The generator of the independence copula is given by ψ ( t ) = exp( − t ). Substi-tuting ψ − ( x ) = − ln( x ) in (11), we get γ min = min k ∈{ ,...,m } (cid:26) exp (ln ( kq/m )) kq/m (cid:27) = 1and, hence, ∀ ϑ ∈ Θ :
F DR ϑ,ψ ( ϕ LSU ) = m ( ϑ ) m q under the assumption of independence. This result is in line with the previousfinding reported in (3). The generator of the Clayton copula is given by ψ ( t ) = (1 + ηt ) − /η , η ∈ (0 , ∞ ) , (22)leading to ψ − ( x ) = ( x − η − /η and to the probability density function (pdf) f Z ( z ) = 1 η f Γ(1 /η, ( z/η ) = 1Γ (1 /η ) η − /η z /η − exp ( − z/η ) (23) odnar and Dickhaus/Copula-based FDR Control of Z , where Γ denotes Euler’s gamma function and f Γ( α,β ) the pdf of the gammadistribution with shape parameter α ∈ (0 , ∞ ) and scale parameter β ∈ (0 , ∞ ).For the Clayton copula, z ∗ is given by z ∗ = log mη − (cid:16) ( q/m ) − η − q − η (cid:17) = η log m ( m/q ) η − (1 /q ) η . (24)In Figure 1, we plot F Z ( z ∗ ) as a function of η for m = 20 and q = 0 .
05. Itis worth mentioning that the Clayton copula converges to the independencecopula for η →
0. In this case we get z ∗ → f Z ( z ∗ ) tends to the Diracdelta function concentrated in 1. As a result, we observe that F Z ( z ∗ ) → η → ϕ LSU approaches m q/m . As η increases, F Z ( z ∗ ) steeplydecreases and takes values very close to zero for large values of η . Consequently,it is expected that the FDR of ϕ LSU is close to m q/m for large values of η ,too. For η of moderate size, however, the FDR of ϕ LSU can be much smallerthan m q/m . This is shown in Figure 2 below and discussed in detail there. h F Z ( z * ) . . . . . . . . . Fig 1 . The value F Z ( z ∗ ) as a function of η for m = 20 and q = 0 . under the assumptionof a Clayton copula.odnar and Dickhaus/Copula-based FDR Control The quantity γ min for the Clayton copula is calculated by γ min = 1 − η − (cid:90) z ∗ exp (cid:0) − zψ − ( q/m ) (cid:1) q/m f Γ(1 /η, ( z/η ) dz + η − (cid:90) z ∗ exp (cid:0) − zψ − ( q ) (cid:1) q f Γ(1 /η, ( z/η ) dz = 1 − I C + I C , (25)where I C = η − /η Γ (1 /η ) mq (cid:90) z ∗ z /η − exp (cid:18) − zη (cid:18)(cid:18) mq (cid:19) η − (cid:19) − zη (cid:19) dz = η − /η Γ (1 /η ) mq (cid:90) z ∗ z /η − exp (cid:18) − zη (cid:18) mq (cid:19) η (cid:19) dz = F Γ(1 /η,η − ( m/q ) η ) ( z ∗ ) = F Γ(1 /η, ( η − ( m/q ) η z ∗ )= F Γ(1 /η, (cid:18) m η ln mm η − (cid:19) and, similarly, I C = F Γ(1 /η,η − (1 /q ) η ) ( z ∗ ) = F Γ(1 /η, ( η − (1 /q ) η z ∗ ) = F Γ(1 /η, (cid:18) ln mm η − (cid:19) . Hence, from Theorem 3.2 we get for all ϑ ∈ Θ that
F DR ϑ,η ( ϕ LSU ) ≥ m qm (cid:18) F Γ(1 /η, (cid:18) ln mm η − (cid:19) − F Γ(1 /η, (cid:18) m η ln mm η − (cid:19)(cid:19) . Next, we consider the sharper upper bound for the FDR in the case of Claytoncopulae in detail. As outlined in the discussion around Theorem 3.1, a so-calledDirac-uniform configuration (cf., e. g., Blanchard et al. (2013) and referencestherein) is assumed for P in case of m < m . Namely, the p -values ( P i : i ∈ I ( ϑ ))are assumed to be P ϑ -almost surely equal to 0. Under assumptions (I1)-(I2)from Theorem 2.1, Dirac-uniform configurations are least favorable (provideupper bounds) for the FDR of ϕ LSU , see Benjamini and Yekutieli (2001). Inthe case of dependent p -values, such general results are yet lacking, but it isassumed that Dirac-uniform configurations yield upper FDR bounds for ϕ LSU also under dependency, at least for large m (cf., e. g., Finner, Dickhaus andRoters (2007)).Under a Dirac-uniform configuration, the sharper upper bound for the FDRof ϕ LSU is expressed by (see Theorem 3.1) b ( m, ϑ, η ) = qm m (cid:88) i =1 m − (cid:88) k = m +1 E (cid:34)(cid:32) exp (cid:0) − Zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − Zψ − ( q k ) (cid:1) q k (cid:33) × (cid:0) G ik ( Z ) − G ik ( z ∗ k ) (cid:1) [ z ∗ k , ∞ ) ( Z ) (cid:105) , (26) odnar and Dickhaus/Copula-based FDR Control where z ∗ k is given in (8) and the random set D ( i,z ) Y ; k the probability of whichis given by G ik ( z ) can under Dirac-uniform configurations equivalently be ex-pressed as D ( i,z ) Y ; k = (cid:110) exp (cid:0) − zψ − ( q k +1 ) (cid:1) ≤ Y ( i )( k ) , ..., exp (cid:0) − zψ − ( q m ) (cid:1) ≤ Y ( i )( m − (cid:111) . The last equality follows from the fact that Y ( i )( k ) , ..., Y ( i )( m − almost surely cor-respond to p -values associated with true null hypotheses, i. e., F P ( i )( k ) ( x ) = ... = F P ( i )( m − ( x ) = x . Moreover, since each of the Y ( i )( k ) , ..., Y ( i )( m − is obtained by the same isotonictransformation from the corresponding element in the sequence P ( i )( k ) , ..., P ( i )( m − ,we get that Y ( i )( k ) , ..., Y ( i )( m − is an increasing sequence of independent and iden-tically UNI[0 , G ik ( z ) = P ϑ,η ( D ( i,z ) Y ; k ) for k ∈ { m + 1 , . . . , m − } can be calculated recursively, for in-stance by making use of Bolshev’s recursion (see, e. g., Shorack and Wellner(1986), p. 366).In general, Bolshev’s recursion is defined in the following way. Let 0 ≤ a ≤ a ≤ . . . ≤ a n ≤ U (1) ≤ U (2) ≤ . . . ≤ U ( n ) be the orderstatistics of independent and identically UNI[0 , P n ( a , . . . , a n ) = P ( a ≤ U (1) , . . . , a n ≤ U ( n ) ). Then, the probability¯ P n ( a , . . . , a n ) is calculated recursively by¯ P n ( a , . . . , a n ) = 1 − n (cid:88) j =1 (cid:18) nj (cid:19) a jj ¯ P n − j ( a j +1 , . . . , a n ) . (27)Application of (27) with n = m − a j = (cid:26) j ∈ { , . . . , k − m − } exp (cid:0) − zψ − ( q j + m +1 ) (cid:1) for j ∈ { k − m , . . . , m − } for k ∈ { m + 1 , . . . , m − } as well as numerical integration with respect to thedistribution of Z over [ z ∗ k , ∞ ] lead to a numerical approximation of the sharperupper bound for the FDR of ϕ LSU under Dirac-uniform configurations.In Figure 2 we present the lower bound (dashed red line), the upper bound(dashed blue line), the sharper upper bound (solid green line), and the simulatedvalues of the FDR of ϕ LSU (solid black line) as a function of the parameter η ofa Clayton copula. We put m = 20, q = 0 .
05, and m = 16. The p -values whichcorrespond to the false null hypotheses have been set to zero. The simulatedvalues are obtained by using 10 independent repetitions. We observe that theFDR of ϕ LSU starts at m q/m = 0 .
04 for η = 0 and decreases to a minimumof approximately 0 .
023 at η ≈ .
7. This value is much smaller than the nominallevel q , offering some room for improvement of ϕ LSU for a broad range of values odnar and Dickhaus/Copula-based FDR Control of η . After reaching its minimum, the FDR of ϕ LSU increases and tends to 0 . η increases. This behavior of the FDR of ϕ LSU is as expected from the valuesof F Z ( z ∗ ), as discussed around Figure 1.In contrast to the ”classical” upper bound, the sharper upper bound repro-duces the behavior of the simulated FDR values very well. It provides a goodapproximation of the true values of the FDR of ϕ LSU for all considered valuesof η . In particular, it is much smaller than the ”classical” upper bound for mod-erate values of η . Consequently, application of the sharper upper bound can beused to improve the power of the multiple testing procedure by adjusting thenominal value of q depending on η . If η is unknown, we propose techniques forpre-estimating it in Section 5. It is also remarkable that the difference betweenthe sharper upper bound and the corresponding simulated FDR-values is notlarge. In contrast, the empirical standard deviations of the sharper upper bound(over repeated simulations) are about five times smaller than the correspondingones for the simulated values of the FDP of ϕ LSU (see Figure 3). While thesestandard deviations are always smaller than 0 .
028 for the sharper upper bound,they are around 0 .
14 for almost all of the considered values of η in case of thesimulated FDP-values. Finally, we note that the lower bound seems not to beinformative in this particular model class. It is close to zero even for moderatevalues of η . h F DR FDR, Clayton Copula . . . . Upper BoundSharper Upper BoundLower BoundSimulated Values
Fig 2 . Lower bound (dashed red line), upper bound (dashed blue line), the sharper upper bound(solid green line), and simulated values of the FDR of ϕ LSU (solid black line) as functions of η for a Clayton copula. We put m = 20 , q = 0 . , and m = 16 . Simulated values are basedon independent pseudo realizations of Z .odnar and Dickhaus/Copula-based FDR Control h S t anda r d D e v i a t i on Standard Deviations, Clayton Copula . . . . . . . Sharper Upper BoundSimulated Values
Fig 3 . Empirical standard deviations of the sharper upper bound (solid green line), and ofFDP ϑ,η ( ϕ LSU ) (solid black line) as functions of the parameter η of a Clayton copula. Weput m = 20 , q = 0 . , and m = 16 . Simulated values are based on independent pseudorealizations of Z . The generator of the Gumbel copula is given by ψ ( x ) = exp (cid:16) − x /η (cid:17) , η ≥ , (28)which leads to ψ − ( x ) = ( − ln x ) η and a stochastic representation Z d = (cid:18) cos (cid:18) π η (cid:19)(cid:19) η Z , η > , (29)for Z , where the random variable Z has a stable distribution with index ofstability 1 /η and unit skewness. The cdf of Z is given by (cf. Chambers, Mallowsand Stuck (1976), p. 341) F Z ( z ) = 1 π (cid:90) π exp (cid:16) − z − / ( η − a ( v ) (cid:17) dv with a ( v ) = sin ((1 − η ) v/η ) (sin( v/η )) / ( η − (sin v ) η/ ( η − , v ∈ (0 , π ) . odnar and Dickhaus/Copula-based FDR Control Although (29) in connection with F Z characterizes the distribution of Z com-pletely, the integral representation of F Z may induce numerical issues withrespect to implementation. Somewhat more convenient from this perspective isthe following result. Namely, Kanter (1975) obtained a stochastic representationof Z , given by Z = ( a ( U ) /W ) η − , (30)where U and W are stochastically independent, W is standard exponentially dis-tributed and U ∼ UNI(0 , π ). We used (30) for simulating Z and, consequently, Z . h F Z ( z * ) . . . . Fig 4 . The value F Z ( z ∗ ) as a function of η for m = 20 and q = 0 . under the assumptionof a Gumbel copula. The graph was obtained via simulations by generating independentpseudo realizations of Z according to (29) and (30) . For the Gumbel copula we get z ∗ = ln m (cid:0) − ln qm (cid:1) η − ( − ln q ) η = ln m (cid:16) ln mq (cid:17) η − (cid:16) ln q (cid:17) η . (31)In Figure 4, we plot F Z ( z ∗ ) as a function of η for m = 20 and q = 0 .
05. Asimilar behavior as in the case of the Clayton copula is present. If η = 1 thenthe Gumbel copula coincides with the independence copula. Hence, F Z ( z ∗ ) = 1and, consequently, the FDR of ϕ LSU is equal to m q/m in this case. As η increases, F Z ( z ∗ ) decreases and it approaches 0 for larger values of η . Hence, odnar and Dickhaus/Copula-based FDR Control FDR ϑ,η ( ϕ LSU ) tends to m q/m as η becomes considerably large. For moderatevalues of η , FDR ϑ,η ( ϕ LSU ) can again be much smaller than m q/m , in analogyto the situation in models with Clayton copulae.Recall from (17) that γ min = 1 − E (cid:2) g ( Z ) [0 ,z ∗ ] ( Z ) (cid:3) , (32)where g ( Z ) = g (cid:0) ψ − ( q/m ) | Z (cid:1) − g (cid:0) ψ − ( q ) | Z (cid:1) . For the Gumbel copula, weobtain g ( Z ) = exp (cid:0) − Zψ − ( q/m ) (cid:1) q/m − exp (cid:0) − Zψ − ( q ) (cid:1) q = exp (cid:16) − Z (cid:16) ln mq (cid:17) η (cid:17) q/m − exp (cid:16) − Z (cid:16) ln q (cid:17) η (cid:17) q . The expectation in (32) cannot be calculated analytically. However, it can easilybe approximated with Monte Carlo simulations by applying the stochastic rep-resentations (29) and (30) for any fixed η >
1. This leads to a numerical valueon the left-hand side of the chain of inequalities m qm (cid:0) − E (cid:2) g ( Z ) [0 ,z ∗ ] ( Z ) (cid:3)(cid:1) ≤ FDR ϑ,η ( ϕ LSU ) ≤ m qm . (33)The sharper upper bound from Theorem 3.1 can be calculated by using Bol-shev’s recursion similarly to the discussion around (27), but here with ψ as in(28). Figure 5 displays the lower bound (dashed red line), the upper bound(dashed blue line), the sharper upper bound (solid green line), and simulatedvalues of FDR ϑ,η ( ϕ LSU ) (solid black line) as functions of η . Again, we chose m = 20, q = 0 .
05, and m = 16. The p -values corresponding to the false nullhypotheses were all set to zero, as in the case of Clayton copulae. The simulatedvalues were obtained by generating 10 independent pseudo realizations of Z .Similarly to the case of the Clayton copula, the curve of simulated FDRvalues has a U -shape. It starts at m q/m = 0 .
04 and drops to its minimumof approximately 0 .
024 for values of η around 6 .
6. For such values of η , theblack curve is considerably below the classical upper bound of 0 .
04. In contrast,the sharper upper bound gives a much tighter approximation of the simulatedFDR values in such cases and reproduces the U -shape over the entire range ofvalues for the parameter η of the Gumbel copula. As a result, its applicationcan be used to improve power by adjusting the nominal value of q and therebyincreasing the probability to detect false null hypotheses. Moreover, as in thecase of Clayton copulae, the empirical standard deviations of the sharper upperbound are much smaller than those of the simulated values of the FDP (seeFigure 6). The lower bound from (33) (corresponding to the dashed red curvein Figure 5) has been obtained by approximating the expectation in (32) viasimulations. As in the case of the Clayton copula, the lower bound is not tooinformative for the model class that we have considered here (Dirac-uniformconfigurations). odnar and Dickhaus/Copula-based FDR Control h F DR FDR, Gumbel Copula . . . . Upper BoundSharper Upper BoundLower BoundSimulated Values
Fig 5 . Lower bound (dashed red line), upper bound (dashed blue line), the sharper upper bound(solid green line), and simulated values of the FDR of ϕ LSU (solid black line) as functionsof the parameter η of a Gumbel copula. We put m = 20 , q = 0 . , and m = 16 . Simulatedvalues are based on independent pseudo realizations of Z .odnar and Dickhaus/Copula-based FDR Control h S t anda r d D e v i a t i on Standard Deviations, Gumbel Copula . . . . . . . Sharper Upper BoundSimulated Values
Fig 6 . Empirical standard deviations of the sharper upper bound (solid green line), and ofFDP ϑ,η ( ϕ LSU ) (solid black line) as functions of the parameter η of a Gumbel copula. Weput m = 20 , q = 0 . , and m = 16 . Simulated values are based on independent pseudorealizations of Z .odnar and Dickhaus/Copula-based FDR Control
5. Empirical copula calibration
In the previous section we studied the influence of the copula parameter η onthe FDR of ϕ LSU under several parametric families of Archimedean copulae.It turned out that adapting ϕ LSU to the degree of dependency in the databy adjusting the nominal value of q based on the sharper upper bound fromTheorem 3.1 is a promising idea, because the unadjusted procedure may lead toa considerable non-exhaustion of q , cf. Figures 2 and 5. Due to the decision ruleof a step-up test, this also entails suboptimal power properties of ϕ LSU whenapplied ”as is” to models with Archimedean p -value copulae.In practice, however, often the copula parameter itself is an unknown quan-tity. Hence, the outlined adaptation of q typically requires some kind of pre-estimation of η before multiple testing is performed. Although this is not in themain focus of the present work, we therefore outline possibilities for estimating η and for quantifying the uncertainty of the estimation in this section.One class of procedures relies on resampling, namely via the parametric boot-strap or via permutation techniques if H , . . . , H m correspond to marginal two-sample problems. Pollard and van der Laan (2004) provided an extensive com-parison of both approaches and argued that the permutation method reproducesthe correct null distribution only under some conditions. However, if these con-ditions are met, the permutation approach is often superior to bootstrapping(see also Westfall and Young (1993) and Meinshausen, Maathuis and B¨uhlmann(2011)). Furthermore, it is essential to keep in mind that both bootstrap andpermutation-based methods estimate the distribution of the vector P under theglobal null hypothesis H . Hence, the assumption that η does not depend on ϑ is an essential prerequisite for the applicability of such resampling methodsfor estimating η . Notice that the latter assumption is an informal descriptionof the ”subset pivotality” condition introduced by Westfall and Young (1993).The resampling methods developed by Dudoit and van der Laan (2008) candispense with subset pivotality in special model classes, but for the particulartask of estimating the copula parameter this assumption seems indispensable.Estimation of η and uncertainty quantification of the estimation based onresampling is generally performed by applying a suitable estimator ˆ η to the re-(pseudo) samples. In the context of Archimedean copulae the two most widelyapplied estimation procedures are the maximum likelihood method (see, e. g.Joe (2005), Hofert, M¨achler and McNeil (2012)) and the method of moments(referred to as ”realized copula” approach by Fengler and Okhrin (2012)).Hofert, M¨achler and McNeil (2012) considered the estimation of the parame-ter of an Archimedean copula with known margins by the maximum likelihoodapproach. To this end, they derived analytic expressions for the derivatives ofthe copula generator for several families of Archimedean copulae, as well asformulas for the corresponding score functions. Using these results and assum-ing a regular model, an elliptical asymptotic confidence region for the copulaparameter η can be obtained by applying general limit theorems for maximumlikelihood estimators (see Hofert, M¨achler and McNeil (2012) for details and thecalculations for different types of Archimedean copulae). odnar and Dickhaus/Copula-based FDR Control In the context of the method of moments, Kendall’s tau is often considered.For a bivariate Archimedean copula with generator ψ of marginally UNI[0 , P and P , it is given by τ P ,P = 4 (cid:90) (cid:90) F ( P ,P ) ( u, v ) dF ( P ,P ) ( u, v ) −
1= 1 − (cid:90) ψ − (0)0 t [ ψ (cid:48) ( t )] dt, (34)cf. McNeil and Neˇslehov´a (2009).The right-hand side of (34) can analytically be calculated for some familiesof Archimedean copulae. For instance, for a Clayton copula with parameter η it is given by τ ( η ) = η/ (2 + η ), while it is equal to τ ( η ) = ( η − /η for aGumbel copula with parameter η (see Nelsen (2006), p. 163-164). Based on suchmoment equations, Fengler and Okhrin (2012) suggested the ”realized copula”method for empirical calibration of a one-dimensional parameter η of an m -variate Archimedean copula. The method considers all m ( m − / m underlying random variables, replaces the population versions of τ ( η ) by the corresponding sample analogues, and finally aggregates the resulting m ( m − / g ij ( η ) = ˆ τ ij − τ ( η ) for 1 ≤ i < j ≤ m and define q ( η ) = ( g ij ( η ) :1 ≤ i < j ≤ m ) (cid:62) , where ˆ τ ij is the sample estimator of Kendall’s tau (see, e. g.,Nelsen (2006), Section 5.1.1). The resulting estimator for η is then obtained byˆ η = arg min η (cid:8) q ( η ) (cid:62) Wq ( η ) (cid:9) (35)for an appropriate weight matrix W ∈ R ( m ) × ( m ). An application of the realizedcopula method to resampled p -values generated by permutations in the contextof multiple testing for differential gene expression has been demonstrated byDickhaus and Gierl (2013). Multivariate extensions of Kendall’s tau and centrallimit theorems for the sample versions have been derived by Genest, Neˇslehov´aand Ben Ghorbal (2011). These results can be used for uncertainty quantificationof the moment estimation of η by constructing asymptotic confidence regions.
6. Discussion
We have derive a sharper upper bound for the FDR of ϕ LSU in models withArchimedean copulae. This bound can be used to prove that ϕ LSU controls theFDR for this type of multivariate p -value distributions, a result which is in linewith the findings of Benjamini and Yekutieli (2001) and Sarkar (2002). Since cer-tain models with H -exchangeable p -values fall into this class at least asymptot-ically (see Theorem 3.3), our findings complement those of Finner, Dickhaus andRoters (2007) who investigated infinite sequences of H -exchangeable p -valuesin Gaussian models. While our general results in Section 3 qualitatively extendthe theory, our results in Section 4 regarding Clayton and Gumbel copulae are odnar and Dickhaus/Copula-based FDR Control quantitatively very much in line with the findings for Gaussian and t -copulaereported by Finner, Dickhaus and Roters (2007). Namely, over a broad class ofmodels with dependent p -values, the FDR of ϕ LSU as a function of the depen-dency parameter has a U -shape and becomes smallest for medium strength ofdependency among the p -values. This behavior can be exploited by adjusting q in order to adapt to η . We have presented an explicit adaptation scheme basedon the upper bound from Theorem 3.1. To the best of our knowledge, this kindof adaptation is novel to FDR theory.It is beyond the scope of the present work to investigate which parametricclass of copulae is appropriate for which kind of real-life application. Relatedly,the problem of model misspecification (i. e., quantification of the approximationerror if the true model does not belong to the class with Archimedean p -valuecopulae and is approximated by the (in some suitable norm) closest memberof this class) could not be addressed here, but is a challenging topic for futureresearch. One particularly interesting issue in this direction is FDR control forfinite sequences of H -exchangeable p -values.Finally, we would like to mention that the empirical variance of the falsediscovery proportion was large in all our simulations, implying that the randomvariable FDP ϑ,η ( ϕ LSU ) was not well concentrated around its expected valueFDR ϑ,η ( ϕ LSU ). This is a known effect for models with dependent p -values (see,e. g., Finner, Dickhaus and Roters (2007), Delattre and Roquain (2011), Blan-chard et al. (2013)) and provokes the question if FDR control is a suitablecriterion under dependency at all. Maybe more stringent in dependent modelsis control of the false discovery exceedance rate, meaning to design a multipletest ϕ ensuring that FDX ϑ,η ( ϕ ) = P ϑ,η (FDP ϑ,η ( ϕ ) > c ) ≤ γ , for user-definedparameters c and γ . In any case, practitioners should be (made) aware of the factthat controlling the FDR with ϕ LSU does not necessarily imply that the FDP fortheir particular experiment is small, at least if dependencies among P , . . . , P m have to be assumed as it is typically the case in applications. In contrast, theempirical standard deviations of our proposed sharper upper bound are aboutfive times smaller than the empirical standard deviations of the simulated valuesof the FDP of ϕ LSU . This provides an additional (robustness) argument for theapplication of the results presented in Theorem 3.1 in practice.
7. Proofs
Proof of Theorem 3.1
Following Benjamini and Yekutieli (2001), an analytic expression for the FDRof ϕ LSU is given byFDR ϑ,η ( ϕ LSU ) = m (cid:88) i =1 m (cid:88) k =1 k P ϑ,η (cid:110) A ( i ) k (cid:111) , (36)where A ( i ) k = { P i ≤ q k ∩ C ( i ) k } denotes the event that k hypotheses are rejectedone of which is H i (a true null hypothesis) and C ( i ) k is the event that k − odnar and Dickhaus/Copula-based FDR Control hypotheses additionally to H i are rejected. It holds that ( C ( i ) k : 1 ≤ k ≤ m ) aredisjoint and that (cid:83) mk =1 C ( i ) k = [0 , m − .Let D ( i ) k = (cid:83) kj =1 C ( i ) j for k = 1 , . . . , m denote the event that the number ofrejected null hypotheses is at most k . In terms of P ( i ) introduced in Theorem3.1, the random set D ( i ) k is given by D ( i ) k = { q k +1 ≤ P ( i )( k ) , . . . , q m ≤ P ( i )( m − } . (37)Next, we prove that P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k (cid:17) q k ≤ P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k (cid:17) q k +1 (38) − (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z ) . To this end, we consider the function T introduced in Theorem 3.1, whichtransforms a possible realization of the original p -values P into a realizationof Y for Z = z , where Y = ( Y , . . . , Y m ) (cid:62) and Z are as in (5). Because eachcomponent of this multivariate transformation is a monotonically increasingfunction which fully covers the interval [0 , , m into itself. Let C ( i,z ) Y ; k and D ( i,z ) Y ; k denote theimages of the sets C ( i ) k and D ( i ) k under T for given Z = z . Then(a) C ( i,z ) Y ; k are disjoint, i. e., C ( i,z ) Y ; k ∩ C ( i,z ) Y ; k = ∅ for 1 ≤ k (cid:54) = k ≤ m ,(b) D ( i,z ) Y ; k = (cid:83) kj =1 C ( i,z ) Y ; j ,(c) D ( i,z ) Y ; m = (cid:83) mj =1 C ( i,z ) Y ; j = [0 , m − .Statements (a) - (c) follow directly from the facts that each T j is a monotonicallyincreasing function and T is a one-to-one transformation with image equal to[0 , m . Moreover, we obtain D ( i,z ) Y ; k = (cid:110) ∀ k ≤ j ≤ m − Y ( i )( j ) ≥ exp (cid:18) − zψ − (cid:18) F P ( i )( j ) ( q j +1 ) (cid:19)(cid:19) (cid:111) , (39)where Y ( i ) is the ( m − Y = ( Y , . . . , Y m ) T by deleting Y i . The last equality shows that D ( i,z ) Y ; k ⊆ D ( i,z ) Y ; k for z ≤ z and,hence, that G ik , given by G ik ( z ) = P ϑ,η (cid:16) D ( i,z ) Y ; k (cid:17) , is an increasing function in z . odnar and Dickhaus/Copula-based FDR Control Returning to (38), we obtain P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k (cid:17) q k +1 − P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k (cid:17) q k = (cid:90) P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k | Z = z (cid:17) q k +1 − P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k | Z = z (cid:17) q k dF Z ( z )= (cid:90) P ϑ,η ( P i ≤ q k +1 | Z = z ) P ϑ,η (cid:16) D ( i ) k | Z = z (cid:17) q k +1 − P ϑ,η ( P i ≤ q k | Z = z ) P ϑ,η (cid:16) D ( i ) k | Z = z (cid:17) q k dF Z ( z )= (cid:90) (cid:32) P ϑ,η (cid:0) Y i ≤ exp (cid:0) − zψ − ( q k +1 ) (cid:1)(cid:1) q k +1 − P ϑ,η (cid:0) Y i ≤ exp (cid:0) − zψ − ( q k ) (cid:1)(cid:1) q k (cid:33) P ϑ,η (cid:16) D ( i,z ) Y ; k (cid:17) dF Z ( z )= (cid:90) (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) G ik ( z ) dF Z ( z ) . (40)Next, we analyze the difference under the last integral. It holds thatexp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k = exp (cid:0) − log q k +1 − zψ − ( q k +1 ) (cid:1) − exp (cid:0) − log q k − zψ − ( q k ) (cid:1) = exp (cid:0) − log q k − zψ − ( q k ) (cid:1) × (cid:0) exp (cid:0) − log q k +1 + log q k − zψ − ( q k +1 ) + zψ − ( q k ) (cid:1) − (cid:1) . The last expression is nonnegative if and only if − log q k +1 + log q k − zψ − ( q k +1 ) + zψ − ( q k ) ≥ . Hence, for z ≥ z ∗ k with z ∗ k given in (8), the function under the integral in (40) odnar and Dickhaus/Copula-based FDR Control is positive and for z ≤ z ∗ k it is negative. Application of this result leads to P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k (cid:17) q k +1 − P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k (cid:17) q k = (cid:90) z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) G ik ( z ) dF Z ( z )+ (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) G ik ( z ) dF Z ( z ) ≥ G ik ( z ∗ k ) (cid:90) z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) dF Z ( z )+ (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) G ik ( z ) dF Z ( z ) . Because of (cid:90) (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) dF Z ( z )= (cid:90) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 dF Z ( z ) − (cid:90) exp (cid:0) − zψ − ( q k ) (cid:1) q k dF Z ( z )= ψ (cid:0) ψ − ( q k +1 ) (cid:1) q k +1 − ψ (cid:0) ψ − ( q k ) (cid:1) q k = q k +1 q k +1 − q k q k = 0we get (cid:90) z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) dF Z ( z )= − (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) dF Z ( z )and, consequently, P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k (cid:17) q k +1 − P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k (cid:17) q k ≥ − G ik ( z ∗ k ) (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) dF Z ( z )+ (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) G ik ( z ) dF Z ( z )= (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × (41)( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z ) , odnar and Dickhaus/Copula-based FDR Control which is obviously positive since both the differences under the integral in (41)are positive. This completes the proof of (38).Using (38), we get for all 1 ≤ k ≤ m − P ϑ,η (cid:16) P i ≤ q k ∩ D ( i ) k (cid:17) q k + P ϑ,η (cid:16) P i ≤ q k +1 ∩ C ( i ) k +1 (cid:17) q k +1 ≤ P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k (cid:17) q k +1 + P ϑ,η (cid:16) P i ≤ q k +1 ∩ C ( i ) k +1 (cid:17) q k +1 − (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z )= P ϑ,η (cid:16) P i ≤ q k +1 ∩ D ( i ) k +1 (cid:17) q k +1 − (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z )and, consequently, starting with D ( i )1 = C ( i )1 and proceeding step-by-step for all k ≤ m −
1, we obtain m (cid:88) k =1 P ϑ,η (cid:110) P i ≤ q k +1 ∩ C ( i ) k (cid:111) q k ≤ P ϑ,η (cid:110) P i ≤ q m ∩ D ( i ) m (cid:111) q m − m − (cid:88) k =1 (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z )= 1 − m − (cid:88) k =1 (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z ) . odnar and Dickhaus/Copula-based FDR Control Hence, FDR ϑ,η ( ϕ LSU ) = m (cid:88) i =1 m (cid:88) k =1 k P ϑ,η (cid:110) A ( i ) k (cid:111) = m (cid:88) i =1 qm m (cid:88) k =1 P ϑ,η (cid:110) P i ≤ q k +1 ∩ C ( i ) k (cid:111) q k ≤ m (cid:88) i =1 qm − m (cid:88) i =1 qm m − (cid:88) k =1 (cid:90) ∞ z ∗ k (cid:32) exp (cid:0) − zψ − ( q k +1 ) (cid:1) q k +1 − exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:33) × ( G ik ( z ) − G ik ( z ∗ k )) dF Z ( z )= m m q − b ( m, ϑ, ψ ) , where b ( m, ϑ, ψ ) is defined in Theorem 3.1. This completes the proof of thetheorem. Proof of Theorem 3.2
Straightforward calculation yieldsFDR ϑ,η ( ϕ LSU ) = m (cid:88) i =1 m (cid:88) k =1 k (cid:90) P ϑ,η (cid:110) A ( i ) k | Z = z (cid:111) dF Z ( z )= m (cid:88) i =1 m (cid:88) k =1 k (cid:90) P ϑ,η { P i ≤ q k | Z = z } × P ϑ,η (cid:110) C ( i ) k | Z = z (cid:111) dF Z ( z )= m (cid:88) i =1 qm m (cid:88) k =1 (cid:90) P ϑ,η { P i ≤ q k | Z = z } q k × P ϑ,η (cid:110) C ( i ) k | Z = z (cid:111) dF Z ( z ) , where the random events A ( i ) k and C ( i ) k are defined in the proof of Theorem 3.1.Moreover, making use of the notation C ( i,z ) Y ; k introduced in the proof of Theorem odnar and Dickhaus/Copula-based FDR Control ϑ,η ( ϕ LSU ) byFDR ϑ,η ( ϕ LSU ) = m (cid:88) i =1 qm m (cid:88) k =1 (cid:90) P ϑ,η (cid:110) P i ≤ exp (cid:16) − zψ − (cid:16) F P i (cid:16) kqm (cid:17)(cid:17)(cid:17)(cid:111) q k × P ϑ,η (cid:110) C ( i,z ) Y ; k (cid:111) dF Z ( z )= m (cid:88) i =1 qm m (cid:88) k =1 (cid:90) exp (cid:0) − zψ − ( q k ) (cid:1) q k P ϑ,η (cid:110) C ( i,z ) Y ; k (cid:111) dF Z ( z ) ≥ m (cid:88) i =1 qm m (cid:88) k =1 (cid:90) min k ∈{ ,...,m } (cid:40) exp (cid:0) − zψ − ( q k ) (cid:1) q k (cid:41) × P ϑ,η (cid:110) C ( i,z ) Y ; k (cid:111) dF Z ( z ) , where the latter inequality follows from Y (cid:96) ∼ UNI[0 ,
1] for all 1 ≤ (cid:96) ≤ m andthe fact that each H i is a true null hypothesis.Now, it holds thatFDR ϑ,η ( ϕ LSU ) ≥ m (cid:88) i =1 qm (cid:90) min k ∈{ ,...,m } exp (cid:16) − zψ − (cid:16) kqm (cid:17)(cid:17) kq/m × m (cid:88) k =1 P ϑ,η (cid:110) C ( i,z ) Y ; k (cid:111) dF Z ( z )= m (cid:88) i =1 qm (cid:90) min k ∈{ ,...,m } exp (cid:16) − zψ − (cid:16) kqm (cid:17)(cid:17) kq/m dF Z ( z )= (cid:90) min k ∈{ ,...,m } exp (cid:16) − zψ − (cid:16) kqm (cid:17)(cid:17) kq/m dF Z ( z ) m (cid:88) i =1 qm = (cid:90) min k ∈{ ,...,m } exp (cid:16) − zψ − (cid:16) kqm (cid:17)(cid:17) kq/m dF Z ( z ) m qm . This completes the proof of the theorem. odnar and Dickhaus/Copula-based FDR Control Proof of Theorem 3.3
We plug (21) into (20) and obtain F ˜ P ,..., ˜ P m ( p , . . . , p m ) = (cid:90) m (cid:89) i =1 exp (cid:0) − zψ − ( p i ) (cid:1) dF Z ( z )= (cid:90) exp (cid:32) − z m (cid:88) i =1 ψ − (cid:0) F ˜ P i ( p i ) (cid:1)(cid:33) dF Z ( z )= ψ (cid:32) m (cid:88) i =1 ψ − (cid:0) F ˜ P i ( p i ) (cid:1)(cid:33) , since the last integral is the Laplace transform of Z at (cid:80) mi =1 ψ − (cid:0) F ˜ P i ( p i ) (cid:1) .Noticing that P , . . . , P m are obtained by componentwise increasing transfor-mations of ˜ P , . . . , ˜ P m we conclude the assertion. Acknowledgments
This research is partly supported by the Deutsche Forschungsgemeinschaft viathe Research Unit FOR 1735 ”Structural Inference in Statistics: Adaptation andEfficiency”.
References
Benjamini, Y. and
Hochberg, Y. (1995). Controlling the false discovery rate:A practical and powerful approach to multiple testing.
J. R. Stat. Soc. Ser.B Stat. Methodol. Benjamini, Y. and
Yekutieli, D. (2001). The control of the false discoveryrate in multiple testing under dependency.
Ann. Stat. Blanchard, G. , Dickhaus, T. , Roquain, E. and
Villers, F. (2013). Onleast favorable configurations for step-up-down tests.
Statistica Sinica, forth-coming.
Cai, T. T. and
Jin, J. (2010). Optimal rates of convergence for estimating thenull density and proportion of nonnull effects in large-scale multiple testing.
Ann. Stat. Chambers, J. M. , Mallows, C. L. and
Stuck, B. W. (1976). A method forsimulating stable random variables.
J. Am. Stat. Assoc. Delattre, S. and
Roquain, E. (2011). On the false discovery proportionconvergence under Gaussian equi-correlation.
Stat. Probab. Lett. Dickhaus, T. and
Gierl, J. (2013). Simultaneous test procedures in terms ofp-value copulae.
Proceedings on the 2nd Annual International Conference onComputational Mathematics, Computational Geometry & Statistics (CMCGS2013) odnar and Dickhaus/Copula-based FDR Control Dudoit, S. and van der Laan, M. J. (2008).
Multiple testing procedureswith applications to genomics.
Springer Series in Statistics. New York, NY:Springer.
Fengler, M. R. and
Okhrin, O. (2012). Realized Copula SFB 649Discussion Paper No. 2012-034, Sonderforschungsbereich 649, Humboldt-Universit¨at zu Berlin, Germany. available at http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2012-034.pdf.
Finner, H. , Dickhaus, T. and
Roters, M. (2007). Dependency and falsediscovery rate: Asymptotics.
Ann. Stat., Finner, H. , Dickhaus, T. and
Roters, M. (2009). On the false discoveryrate and an asymptotically optimal rejection curve.
Ann. Stat. Finner, H. and
Roters, M. (1998). Asymptotic comparison of step-down andstep-up multiple test procedures based on exchangeable test statistics.
Ann.Stat. Genest, C. , Neˇslehov´a, J. and
Ben Ghorbal, N. (2011). Estimators basedon Kendall’s tau in multivariate copula models.
Aust. N. Z. J. Stat. Genovese, C. and
Wasserman, L. (2002). Operating characteristics and ex-tensions of the false discovery rate procedure.
J. R. Stat. Soc., Ser. B, Stat.Methodol. Genovese, C. and
Wasserman, L. (2004). A stochastic process approach tofalse discovery control.
Ann. Stat. Guo, W. and
Rao, M. B. (2008). On control of the false discovery rate underno assumption of dependency.
J. Stat. Plann. Inference
Hofert, M. , M¨achler, M. and
McNeil, A. J. (2012). Likelihood inferencefor Archimedean copulas in high dimensions under known margins.
J. Multi-variate Anal.
Jin, J. and
Cai, T. T. (2007). Estimating the null and the proportion of nonnulleffects in large-scale multiple comparisons.
J. Am. Stat. Assoc.
Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method forcopula-based models.
J. Multivariate Anal. Kanter, M. (1975). Stable densities under change of scale and total variationinequalities.
Ann. Probab. Kingman, J. F. C. (1978). Uses of exchangeability.
Ann. Probab. Marshall, A. W. and
Olkin, I. (1988). Families of multivariate distributions.
J. Am. Stat. Assoc. McNeil, A. J. and
Neˇslehov´a, J. (2009). Multivariate Archimedean copulas, d -monotone functions and (cid:96) -norm symmetric distributions. Ann. Stat. Meinshausen, N. , Maathuis, M. H. and
B¨uhlmann, P. (2011). Asymptoticoptimality of the Westfall-Young permutation procedure for multiple testingunder dependence.
Ann. Stat. M¨uller, A. and
Scarsini, M. (2005). Archimedean copulae and positive de-pendence.
J. Multivariate Anal. Nelsen, R. B. (2006).
An introduction to copulas. 2nd ed.
Springer Series inStatistics. New York, NY: Springer. odnar and Dickhaus/Copula-based FDR Control Olshen, R. (1974). A note on exchangeable sequences.
Z. Wahrscheinlichkeit-stheor. Verw. Geb. Pollard, K. S. and van der Laan, M. J. (2004). Choice of a null distributionin resampling-based multiple testing.
J. Stat. Plann. Inference
Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multipletesting procedures.
Ann. Stat. Shorack, G. R. and
Wellner, J. A. (1986).
Empirical processes with appli-cations to statistics . Wiley Series in Probability and Mathematical Statistics:Probability and Mathematical Statistics . John Wiley & Sons Inc., New York.MR838963 (88e:60002)
Storey, J. D. (2002). A direct approach to false discovery rates.
J. R. Stat.Soc., Ser. B, Stat. Methodol. Sun, W. and
Cai, T. T. (2007). Oracle and adaptive compound decision rulesfor false discovery rate control.
J. Am. Stat. Assoc.
Troendle, J. F. (2000). Stepwise normal theory multiple test procedures con-trolling the false discovery rate.
Journal of Statistical Planning and Inference Westfall, P. H. and
Young, S. S. (1993).