On the Simes inequality and its generalization

Transcription

1 IMS Collections Beyond Paametics in Intedisciplinay Reseach: Festschift in Hono of Pofesso Panab K. Sen Vol ) c Institute of Mathematical Statistics, 2008 DOI: / On the Simes inequality and its genealization Sanat K. Saa Temple Univesity Abstact: The Simes inequality has eceived consideable attention ecently because of its close connection to some impotant multiple hypothesis testing pocedues. We evisit in this aticle an old esult on this inequality to claify and stengthen it and a ecently poposed genealization of it to offe an altenative simple poof. 1. Intoduction Let X 1:n X n:n denote the n odeed values of a set of andom vaiables X 1,...,X n. Assuming that the X i s ae continuous with a common cdf F, Simes [36] consideed the following inequality involving the odeed values U 1:n U n:n of U0, 1) andom vaiables U i F X i ), i 1,...,n, 1.1) P U i:n iα/n, i 1,...,n 1 α, while poposing a modification of the Bonfeoni test fo the intesection of null hypotheses. He conjectued that the inequality holds unde positive dependence of the X i s equivalently, the U i s), having poved the equality unde the independence and numeically veified the inequality fo some specific positively dependent multivaiate distibutions. Saa and Chang [29] poved this conjectue fo a class of positively dependent multivaiate distibutions that ae conditionally iid with a distibution that is stochastically inceasing in the value of the conditioning vaiable, genealizing the wo in Hochbeg and Rom [10] and Samuel-Cahn [20] who fist attempted to pove the conjectue in the bivaiate case. Saa [22] late established the conjectue fo a class of multivaiate totally positive of ode two MTP 2 ) distibutions that is lage than the one consideed in Saa and Chang [29]. A caeful study of Saa s [22] poof of the conjectue of couse eveals that it holds fo a slightly lage class of positively dependent multivaiate distibutions. This lage class is chaacteized by the following condition: Condition 1.1. EφX 1,...,X n ) X i is nondeceasing o noninceasing) in X i fo each i 1,...,n, and any nondeceasing o noninceasing) function φx 1,..., X n ). Suppoted by NSF Gant DMS Depatment of Statistics, Fox School of Business and Management, Temple Univesity, Philadelphia, PA 19122, USA, sanat@temple.edu AMS 2000 subject classifications: 62G30, 62H15. Keywods and phases: multivaiate totally positive of ode two, positive dependence though stochastic odeing, pobability inequalities, Simes test, symmetic multivaiate nomal, symmetic multivaiate t. 231

2 232 S. K. Saa By a nondeceasing o noninceasing function of moe than one vaiable, in Condition 1.1 and elsewhee in the pape, we mean this to be so coodinatewise. The condition has been efeed to as the positive dependence though stochastic odeing PDS) condition by Bloc et al. [3]. Thus, Theoem 3.1 in Saa [22] establishing the Simes conjectue can be ephased in its slightly impoved fom as follows, which we will efe to as the Simes inequality. Simes Inequality. Let X 1,...,X n be a set of PDS continuous andom vaiables with F i as the maginal cdf of X i, i 1,...,n. Then, fo any fixed <a 1 a n <, 1.2) P X 1:n a 1,...,X n:n a n 1 1 n n F i a n ), i1 if j 1 F i a j ) is nondeceasing in j 1,...,n fo all i 1,...,n. The equality in 1.2) holds when j 1 F i a j ) is constant in j fo each i and the X i s ae independent. Since the PDS popety is invaiant unde co-monotone tansfomations of the X i s, the Simes inequality can be equivalently descibed with 1.2) eplaced by the following: 1.3) P X 1:n b 1,...,X n:n b n 1 n n F i b 1 ), i1 fo any fixed <b 1 b n < such that j 1 [1 F i b n j+1 )] is nondeceasing in j 1,...,n fo all i 1,...,n. The equality in 1.3) holds when j 1 [1 F i b n j+1 ] is constant in j fo each i and the X i s ae independent. The Simes inequality holds special impotance in hypothesis testing. Besides theoetically validating the Type I eo ate contol of the Simes 1986) test, which has now been fequently used in place of the Bonfeoni test in many scientific investigations Dmitieno et al. [7], Hommel et al. [11], Meng et al. [16], Neuhause et al. [17], Rosenbeg et al. [19], Someville et al. [37] and Westfall and Kishen [38]), it povides theoetical basis fo the familywise eo ate FWER) contol unde positive dependence of the commonly used Hochbeg 1988) pocedue fo multiple testing, see, fo example, Saa [22] and Saa and Chang [29]. Most impotantly, it is closely lined to the inequality establishing the false discovey ate FDR) contol of the Benjamini and Hochbeg [1] pocedue Benjamini and Yeutieli [2] and Saa [23, 24]), which as a FDR pocedue has eceived the most attention so fa in the multiple testing liteatue; see, fo example, Saa [25, 26, 28] and Saa and Guo [30, 31] fo efeences. The fact that the Simes inequality holds unde Condition 1.1 is a by-poduct of the FDR contol of the Benjamini-Hochbeg pocedue. Of couse, the authos of the papes dealing with the Simes inequality wee not awae of this condition being defined ealie as the PDS condition and efeed to it as a special case of the positive egession dependence on subset PRDS) condition unde which the Benjamini-Hochbeg pocedue contols the FDR. In this aticle, we will stat with a genealized fom of the Simes inequality that Saa [27] has ecently obtained. Fist, we will povide an altenative simple poof of this genealization. Then, we will go bac to the oiginal Simes inequality to claify and stengthen an ealie esult in Saa [22].

3 Simes inequality 233 Saa [27] has genealized the Simes inequality by poviding a lowe bound fo the pobability P X :n a,...,x n:n a n,foafixed1 n, intemsof the th ode joint distibutions of the X i s in an attempt to genealize cetain multiple testing pocedues. He poved this genealization fo MTP 2 distibutions. We impove this wo in this pape Section 2) by offeing an altenative simple poof using a condition that is weae than the MTP 2 condition. To establish the Simes conjectue, that is, the oiginal Simes inequality 1), fo both multivaiate and absolute-valued multivaiate cental t distibutions when the associated multivaiate nomal has the same coelations, Saa [22] made use of a coollay Coollay 3.1) attempting to extend his main theoem Theoem 3.1) to cetain scale mixtues of MTP 2 distibutions. Unfotunately, while the main theoem is coect, thee is a flaw in his poof of the coollay, as noted by Heny Bloc in a pivate communication. We will evisit this coollay in this aticle Section 3) to claify and at the same time stengthen it. Moe specifically, we will povide diect poofs of the Simes inequality fo i) multivaiate t distibution when the associated multivaiate nomal has nonnegative coelations and ii) fo absolute-valued multivaiate t distibution when the associated coelation matix has a moe geneal stuctue than just having equal coelations. These poofs will be based on ideas boowed fom Saa [23], although one can see Benjamini and Yeutieli [2] fo a diffeent poof, of couse moe complicated and given fo a moe geneal esult. 2. Genealized Simes inequality In this section, we give an altenative simple poof of the genealized Simes inequality in Saa [27]. This simplification is achieved by offeing a simple poof of a suppoting lemma on pobability distibution of odeed andom vaiables and by using the following condition fo the distibution of X that is weae, yet moe diectly applicable, than the MTP 2 condition. The following notation is used in the condition: X i1,...,i ) i, i 1,...,n, ae the components of the set X 1,...,X n ) \X i1,...,x i. Condition 2.1. Fo evey i 1,...i 1,...,n of size and any nondeceasing o noninceasing) function φ, EφX i1,...,i ) 1,...,X i1,...,i ) n ) maxx i1,..., X i z is nondeceasing o noninceasing) in z. Theoem 2.1 Genealized Simes inequality). Unde Condition 2.1, we have 2.1) P X :n a,...,x n:n a n 1 ) 1 n F i1...i a n ), 1 i 1< <i n with F i1...i x) P max 1 j X ij x, fo any fixed 1 n, wheea a n ae such that j ) 1Fi1...i a j ) is nondeceasing in j,...,n fo evey i 1,...,i 1,...,n of cadinality. The equality in 2.1) holds unde the independence and when j ) 1Fi1...i a j ) is constant in j,...,n fo evey i 1,...,i 1,...,n of cadinality. The theoem will be poved using the following lemma. As mentioned above, this lemma, although poved befoe in Saa [27], will be given an altenative simple poof hee.

4 234 S. K. Saa Lemma 2.1. Given an inceasing set of constants a 1 a n,let 2.2) R n max 1 i n i : X i:n a i. Then, 2.3) P R n ) 1 i 1< <i n P max X i j a 1 j n +1 1 i 1< <i n E P [ ) 1 R i1,...,i ) 1 n X i1,...,x i I ) 1 ]) I max X i j a, 1 j max X i j a 1 1 j whee 2.4) R i1,...,i ) n max i : X i1,...,i ) i:n a +i, 1 i n with X i1,...,i ) i:n, i 1,...,n, being the n odeed values of the set X 1,..., X n ) \X i1,...,x i. Poof. Given that R n, whee n, R n can be expessed as R n n i1 IX i a ). Hence, we have 2.5)! R n R n 1) R n +1) I max X i j a ) 1 j 1 i 1 i n 1 i 1< <i n I max 1 j X i j a ). Theefoe, fo n, wehave 2.6) IR n ) R nr n 1) R n +1) IR n ) 1) +1) ) 1 I max X i j a,r n ), 1 j 1 i 1< <i n which yields 2.7) I R n ) n I R n ) n ) 1 n ) 1 1 i 1 i n 1 i 1 i n ) I max X i j a,r n 1 j ) I max X i j a,r i1,...,i ) n. 1 j

5 Simes inequality 235 Suppessing the supescipt in R n and using IR n ) IR n ) IR n +1)fo,...,n in 2.7), we get 2.8) I R n ) 1 i 1 i n ) I max X i j a 1 j [ ) 1 ) 1 I max X i j a 1 1 j n 1 i 1 i n +1 ) 1 I I R n ) max X i j a 1 j ) ]. Taing expectations of both sides in 2.8), we get the lemma. Rema 2.1. Befoe we poceed to pove Theoem 2.1, it is impotant to note that fomula 2.3) can altenatively be witten as 2.9) P R n ) ) 1 n 1 i 1< <i n P max X i j a n 1 j n +1 1 i 1< <i n E P 0 R i1,...,i ) n < X i1,...,x i Poof of Theoem 2.1. Since ) 1 ]) 1 I max X i j a 1. 1 j [ ) 1 I max X i j a 1 j 2.10) P X :n a,...,x n:n a n 1 P R n, the theoem follows fom the fact that, fo evey fixed +1,...,n and i 1,...,i 1,...,n, the expectation unde the multiple summation in the ight-hand side of 2.9) is geate than o equal to 0, which can be poved as follows. Define ψ, Z) P 0 R i1,...,i ) n < max X i j Z. 1 j Then, the above expectation is [ ) 1 ) 1 E ψ, Z) I Z a ) I Z a 1)] 1 [ ) 1 ) 1 E ψ, Z)I Z a ) I Z a 1)] 1 [ E ψ ) 1 ) 1,Z)I Z a ) P Z a ) P Z a 1)] P Z a ) 1 0, as ψ, Z) is a nondeceasing function of Z, because of Condition 2.1 and 0 R i1,...,i ) n < being a nondeceasing set, and ) 1 1IZ 1) a 1 )is also a nondeceasing function of Z.

6 236 S. K. Saa Rema 2.2. One can get an equivalent statement of Theoem 2.1, genealizing 1.3), the equivalent vesion of the Simes inequality 1.2), by eplacing each X i by, say -X i. But, since unlie the PDS condition, Condition 2.1 is not invaiant unde co-monotone tansfomations, one needs to have a condition diffeent fom Condition 2.1 in this altenative statement of Theoem 2.1. It is not difficult to see the following altenative vesions of Condition 2.1 and Theoem 2.1: Condition 2.1*. Fo evey i 1,...i 1,...,n of size and any nondeceasing o noninceasing) function φ, EφX i1,...,i ) 1,...,X i1,...,i ) n ) minx i1,..., X i z is nondeceasing o noninceasing) in z. Theoem 2.1*. Unde Condition 2.1*, we have 2.11) P X 1:n b 1,...,X n +1:n b n +1 ) 1 n G i1...i b 1 ), 1 i 1< <i n with G i1...i x) P min 1 j X ij x, fo any fixed 1 n, wheeb 1 b n +1 ae such that j ) 1[1 Gi1...i b n j+1 )] is nondeceasing in j,...,n fo evey i 1,...,i 1,...,n of cadinality. The equality in 2.11) holds unde the independence and when j 1[1 ) Gi1...i b n j+1 )] is constant in j,...,n fo evey i 1,...,i 1,...,n of cadinality. Rema 2.3. Condition2.1 o 2.1*,fo 1 < n, is moe estictive than the PDS condition. So, Theoem 2.1 o 2.1* holds fo a smalle class of distibutions than the one fo which the Simes oiginal inequality holds. In fact, we have the following lemma poviding a class of distibutions fo which the genealized Simes inequality holds. This will be poved using popeties of MTP 2 distibutions discussed in Kalin and Rinott [12]. Lemma 2.2. Both Conditions 2.1 and 2.1* ae satisfied when X 1,...,X n ) has a symmetic MTP 2 distibution. Poof. Let X 1,...,X n ) fx 1,...,x n ), which is symmetic and MTP 2,andY maxx 1,...,X. The joint density of Y,X +1,...,X n is given by 2.12) gy, x +1,...,x n ) fy, x 2,...,x,x +1,...,x n ) Ix i y) dx i. i2 i2 This is MTP 2,sincef and i2 Ix i y) aebothmtp 2, thei poduct is MTP 2, and so is the above integal. Theefoe, Y,X 1,...,X n ) satisfies the positive egession dependence condition, which implies that E [ φx +1,...,X n ) maxx 1,...,X z ] is nondeceasing noninceasing) in z fo any nondeceasing noninceasing) function φ, which is Condition 2.1 in this case. With Y minx 1,...,X,Y,X +1,...,X n ) is also jointly MTP 2, which can be poved as befoe by eplacing i2 Ix i y) by i2 Ix i y) in 2.12) and using the fact that i2 Ix i y) is also MTP 2. So, Condition 2.1* is also satisfied.

7 Simes inequality 237 Lemma 2.2 now yields the following coollay to both Theoems 2.1 and 2.1*. Coollay 2.1. Suppose that X 1,...,X n ) have a symmetic MTP 2 distibution. a) Let F x) be the common cdf of the maximum of any of the nx i s and a a n be such that j 1F ) a j ) is nondeceasing in j,...,n.then, 2.13) P X :n a,...,x n:n a n 1 F a n ). The equality holds unde the independence and when j 1F ) a j ) is constant in j,...,n. b) Let G x) be the common cdf of the minimum of any of the nx i s and b 1 b n +1 be such that j 1[1 ) G b n j+1 )] is nondeceasing in j,...,n. Then, 2.14) P X 1:n b 1,...,X n +1:n b n +1 G b 1 ), The equality holds unde the independence and when j 1[1 ) G b n j+1 )] is constant in j,...,n. Poof. The expession 2.10) in this case educes to 2.15) P X :n a,...,x n:n a n ) n n 1 F a n )+ +1 [ ) 1 ) 1 1 E ψ, Y ) IY a ) IY a 1)], whee ψ, Y )Eφ, X +1,...,X n ) Y, with φ, X +1,...,X n ) as the indicato function of the event 0 R 1,...,) n < 2.16) X 1,...,) :n a,...,x 1,...,) n :n a n. Since Condition 2.1 is now satisfied because of Lemma 2.2) and the indicato function φ, is a nondeceasing function of X +1,...,X n ), ψ, Y ) is a nondeceasing function of Y, which poves 2.13) as in the poof of Theoem 2.1. The pat b) of the coollay can be similaly poved by noting that P X 1:n b 1,...,X n +1:n b n +1 ) n n G b 1 )+ +1 [ ) 1 ) ) E ψ, Y ) IY b n +1) IY b n +2)], whee now Y minx 1,...,X and ψ, Y )E φ X +1,...,X n ) Y, fo some noninceasing function φ,. The inequality 2.14) follows because ψ, Y ) is noninceasing in Y.

8 238 S. K. Saa The covaiance matix Σ of a symmetic multivaiate nomal distibution with a common non-negative coelation satisfies each of the popeties: i) the offdiagonals of Σ 1 ae non-negative and ii) the off-diagonals of DΣ 1 D ae non-negative fo some diagonal matix D with diagonal enties ±1, which ae the conditions, espectively, fo multivaiate nomal, N n μ, Σ), and absolute-valued zeo-mean multivaiate nomal, N n 0, Σ), tobemtp 2 ; see, fo example, Kalin and Rinott [12, 13]. Thus, we have the following: Poposition 2.1. The genealized Simes inequality holds fo both symmetic and absolute-valued zeo-mean symmetic multivaiate nomal distibutions with a common non-negative coelation. If the above distibutions ae studentized based on an independent chi-squae andom vaiable, the esulting multivaiate and absolute-valued multivaiate t distibutions may not etain the MTP 2 popety. So, it becomes unclea if the genealized Simes inequality still holds fo these distibutions, although the Simes oiginal inequality does. In fact, as we will show in the next section, the associated covaiance matices fo these t distibutions do not have to be symmetic fo the oiginal Simes inequality to hold. 3. Simes inequalities fo t distibutions In this section, we will evisit the Simes inequalities fo multivaiate and absolutevalued multivaiate t distibutions to claify and stengthen pevious elated wo. To be moe specific, we have the following theoem. Theoem 3.1. Let T i Z 1 X i, i 1,...,n,wheeX 1,...,X n ) N n 0, Σ) with the diagonal enties of Σ being 1, and is independent of Z χ ν / ν. Then, the Simes inequality 1.2) [o 1.3)] holds i) fo the T i s if a n 0 o b 1 0) and the off-diagonals of Σ ae non-negative and ii) fo the T i s if the off-diagonals of DΣ 1 D ae non-negative fo some diagonal matix D with diagonal enties ±1. Befoe we poceed to pove this theoem, it is impotant to e-emphasize that it is the PDS condition that dives the Simes inequality, and hence thee ae distibutions, not necessaily MTP 2, fo which the inequality holds. A case in point is multivaiate nomal with nonnegative coelations. Its PDS popety follows easily fom the fact that the conditional means given any X i ae inceasing in that X i, even though it is not MTP 2 unless the off-diagonals of Σ 1 ae also nonnegative [12]. Fo absolute-valued multivaiate nomal distibution, of couse, the PDS popety does not follow that easily unless the MTP 2 popety is invoed, and that holds when the off-diagonals of DΣ 1 D ae non-negative fo some diagonal matix D with diagonal enties ±1 [12, 13]. Having poved the Simes inequality fo MTP 2 distibutions, Saa [22] attempted to pove it in Coollay 3.1) fo scale-mixtues of cetain symmetic MTP 2 distibutions befoe discussing that the inequality holds fo symmetic multivaiate t and absolute-valued symmetic multivaiate t distibutions. Unfotunately, as noted in the intoduction, thee is a flaw in his poof of the coollay. Nevetheless, while the tuth of the coollay becomes an open issue at this point, it is impotant to emphasize that the Simes inequalities fo these t distibutions that the coollay intends to pove still hold, as noted when dealing with simila inequalities aising in the context of the FDR contol of the Benjamini-Hochbeg pocedue Benjamini and Yeutieli [2] and Saa [23]). In Benjamini and Yeutieli [2], a geneal esult

9 Simes inequality 239 on the PDS popety of cetain scale-mixtues of PDS distibutions is given, fom which one can see that the Simes inequality can be extended fom a multivaiate o absolute-valued multivaiate nomal to the coesponding multivaiate o absolutevalued multivaiate t distibution. While this esult is impotant in its own ight, its poof, howeve, seems complicated. In fact, one can avoid it while extending the Simes inequality fom multivaiate nomal to the coesponding multivaiate t distibution and, instead, apply an independence esult of nomal distibution. This has been biefly pointed out in Saa [23], of couse, in the context of the FDR. We elaboate this point hee in the context of Simes inequality, theeby claifying and stengthening the inequalities fo symmetic multivaiate t and absolute-valued symmetic multivaiate t distibutions discussed in Saa [22]. We ae now giving altenative and diect poofs of these inequalities with covaiance matices that ae not necessaily symmetic. Poof of Theoem 3.1. Fom 2.10), we have 3.1) P T 1:n a 1,...,T n:n a n n n [ ITi a ) 1 F a n )+ E ψ T i ) i1 2 IT ] i a 1 ), 1 whee F is the common cdf of each T i and ψ T i )P T i) 1:n 1 a,...,t i) n 1:n 1 a n T i. We will pove in the following that each expectation unde the double summation in 3.1) is geate than o equal to zeo if F a i )/i is nondeceasing in i as long as a n 0, which will pove the Simes inequality 1.2). Let us conside the expectation fo i n, and assume without any loss of geneality that Z χ ν. Then, this expectation is given by 3.2) 1:n 1 a Z,...,X n) n 1:n 1 a nz X n,z ]). E P X n) [ IXn a Z) IX n a 1 Z) 1 Let 3.3) gx, z) P X n) 1:n 1 a z,...,x n) n 1:n 1 a nz X n x, whee z>0. Then, the expectation in 3.2) can be ewitten in tems of independent andom vaiables Zn Z 2 + Xn 2 and Tn T n / 1+Tn 2 as follows )[ IT E g ZnT n,z n 1 Tn 2 n a ) IT n a ] 1) 3.4) [ IT E h Tn) n a ) IT n a 1) 1 ], whee a a / 1+a 2 and ) 3.5) htn)e g ZnT n,z n 1 T n 2 T n. 1

10 240 S. K. Saa Since the X i s ae PDS and a i s ae assumed negative, the pobability in 3.3) is nondeceasing in x, z), implying that 3.5) is a nondeceasing function of Tn as long as Tn < 0. Hence, the expectation in 3.4) is geate than o equal to 3.6) E ht n)it n a ) P T n a E ht n)it n a ) P T a [ P Tn a [ P Tn a P ] Tn a 1 1 ], P T n a 1 1 which is geate than o equal to zeo. Thus, the Simes inequality 1.2) holds. The vesion 1.3) of the Simes inequality can be similaly poved with positive b i s. To pove Theoem 3.1ii), we continue with the same aguments as above eplacing X i o T i )by X i o T i ). The X i s ae PDS because of being MTP 2 unde the assumed condition on the covaiance matix. So, the function gx, z) now is nondeceasing in x and is noninceasing in z, implying that the function h T n ) continues to be an inceasing function of T n. The est of the aguments emains same, completing the poof. 4. Concluding emas The esults discussed in this aticle basically ae pobability inequalities fo the odeed components of a cetain type of positively dependent andom vaiables. Moe specifically, they povide bounds fo joint pobabilities of the odeed components of a set of andom vaiables in tems of lowe dimensional maginal distibutions unde a fom of positive dependence among the vaiables. Ou pimay focus in this pape has been on these inequalities, athe than on discussing about the elated Simes tests validated by these inequalities and thei use in multiple testing pocedues. Reades can see Cai and Saa [4, 5], Hochbeg and Libeman [9], Kummenaue and Hommel [15], Rødland [18], Samuel-Cahn [21], Sen [32], Sen and Silvapulle [33], Seneta and Chen [34], Silvapulle and Sen [35] fo the Simes test and Saa [27] fo its genealization, in addition to those cited befoe. Given two independent andom samples X 1,...,X n )andy 1,...,Y n )fomtwo continuous populations F and G espectively, Lemma 2.1has a potential application in developing a nonpaametic test fo testing the null hypothesis that F and G ae equal vesus G is stochastically lage than F. Specifically, one can conside the statistic T n max X i:n Y i:n, 1 i n whee X 1:n X n:n and Y 1:n Y n:n ae the ode statistics coesponding to these samples, the pobability distibution of which unde the null hypothesis can be explicitly obtained using this lemma. A class of tests based on U-statistics with enels based on sub-sample maximas is poposed in Deshpande and Kocha [6] and Kocha [14]. Pehaps one can popose and study tests based on a combination of some o all membes of this class to incease the efficiency. The esults obtained in this pape may be useful in finding the p-values of such tests. Refeences [1] Benjamini, Y. and Hochbeg, Y. 1995). Contolling the false discovey ate: A pactical and poweful appoach to multiple testing. J. Roy. Statist. Soc. Se. B

11 Simes inequality 241 [2] Benjamini, Y. and Yeutieli, D. 2001). The contol of the false discovey ate in multiple testing unde dependency. Ann. Statist [3] Bloc, H. W., Savits, T. H. and Shaed, M. 1985). A concept of negative dependence using stochastic odeing. Statist. Pobab. Lett [4] Cai, G. and Saa, S. K. 2005). Modified Simes citical values unde independence. Technical epot, Temple Univ. Available at sanat/epots/modified-simes Independence.pdf. [5] Cai, G. and Saa, S. K. 2006). Modified Simes citical values unde positive dependence. J. Statist. Plann. Infeence [6] Deshpande, J. V. and Kocha, S. C. 1980). Some competitos of tests based on powes of ans fo the two-sample poblem. Sanhyā Se.B [7] Dmitieno, A., Offen, W. and Westfall, P. 2003). Gateeeping stategies fo clinical tials that do not equie all pimay effects to be significant. Statist. Med [8] Hochbeg, Y. 1988). A shape Bonfeoni pocedue fo multiple tests of significance. Biometia [9] Hochbeg, Y. and Libeman, U. 1994). An extended Simes test. Statist. Pobab. Lett [10] Hochbeg, Y. and Rom, D. 1995). Extensions of multiple testing pocedues based on Simes test. J. Statist. Plann. Infeence [11] Hommel, G., Lindig, V. and Faldum, A. 2005). Two-stage adaptive designs with coelated test statistics. J. Biopham. Statist [12] Kalin, S. and Rinott, Y. 1980). Classes of odeings of measues and elated coelation inequalities. I. Multivaiate totally positive distibutions. J. Multivaiate Anal [13] Kalin, S. and Rinott, Y. 1981). Total positivity popeties of absolute value multinomal vaiables with applications to confidence inteval estimates and elated pobabilistic inequalities. Ann. Statist [14] Kocha, S. C. 1978). A class of distibution-fee tests fo the two-sample slippage poblem. Comm. Statist. A Theoy Methods [15] Kummenaue, F. and Hommel, G. 1999). The size of Simes global test fo discete test statistics. J. Statist. Plann. Infeence [16] Meng, Z., Zayin, D., Kanoub, M., Seeuma, G., St Jean, P. and Ehm, M. 2001). Identifying susceptibility genes using linage and linage disequilibium analysis in lage pedigees. Gen. Epidem. 21 S453 S458. [17] Neuhause, M., Steinijans, V. and Betz, F. 1999). The evaluation of multiple clinical endpoints, with application to asthma. Dug Inf. J [18] Rødland, E. A. 2006). Simes pocedue is valid on aveage. Biometia [19] Rosenbeg, P. S., Che, A. and Chen, B. E. 2006). Multiple hypothesis testing stategies fo genetic case-contol association studies. Stat. Med [20] Samuel-Cahn, E. 1996). Is the Simes impoved Bonfeoni pocedue consevative? Biometia [21] Samuel-Cahn, E. 1999). A note about a cuious genealization of Simes theoem. J. Statist. Plann. Infeence [22] Saa, S. K. 1998). Some pobability inequalities fo odeed MTP 2 andom vaiables: a poof of the Simes conjectue. Ann. Statist [23] Saa, S. K. 2002). Some esults on false discovey ate in stepwise multiple

12 242 S. K. Saa testing pocedues. Ann. Statist [24] Saa, S. K. 2004). FDR-contolling stepwise pocedues and thei false negatives ates. J. Statist. Plann. Infeence [25] Saa, S. K. 2006). False discovey and false nondiscovey ates in singlestep multiple testing pocedues. Ann. Statist [26] Saa, S. K. 2007a). Stepup pocedues contolling genealized FWER and genealized FDR. Ann. Statist. To appea. [27] Saa, S. K. 2007b). Genealizing Simes test and Hochbeg s stepup pocedue. Ann. Statist. To appea. [28] Saa, S. K. 2007c). Two-stage stepup pocedues contolling FDR. J. Statist. Plann. Infeence. Toappea. [29] Saa, S. K. and Chang, C.-K. 1997). The Simes method fo multiple hypothesis testing with positively dependent test statistics. J. Ame. Statist. Assoc [30] Saa, S. K. and Guo, W. 2006). Pocedues contolling genealized false discovey ate. Technical epot, Temple Univ. Available at temple.edu/ sanat/epots/genealizedfdr.pdf. [31] Saa, S. K. and Guo, W. 2007). On genealized false discovey ate. Unpublished manuscipt. [32] Sen, P. K. 1999). Some emas on Simes-type multiple tests of significance. J. Statist. Plann. Infeence [33] Sen, P. K. and Silvapulle, M. J. 2002). An appaisal of some aspects of statistical infeence unde inequality constaints. J. Statist. Plann. Infeence [34] Seneta, E. and Chen, J. 2005). Simple stepwise tests of hypotheses and multiple compaisons. Intenat. Statist. Rev [35] Silvapulle, M. J. and Sen, P. K. 2004). Constained Statistical Infeence. Wiley, New Yo. [36] Simes, R. J. 1986). An impoved Bonfeoni pocedue fo multiple tests of significance. Biometia [37] Someville, M., Wilson, T., Koch, G. and Westfall, P. 2005). Evaluation of a weighted multiple compaison pocedue. Pham. Statist [38] Westfall, P. H. and Kishen, A. 2001). Optimally weighted, fixed sequence and gateeepe multiple testing pocedues. J. Statist. Plann. Infeence