The Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables

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1 The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka 1998) showed that cetai positively depedet MT 2 ) adom vaiables satisfy the Simes Iequality. The multivaiate-t distibutio does ot satisfy this popety, so othe meas ae ecessay to show that it also satisfies the Simes iequality. A coollay was give i Saka 1998) to hadle this distibutio, but thee is a eo. I this pape a diect poof is give to show the multivaiate-t does satisfy the Simes iequality. Keywods: Multivaiate-t distibutio, Simes Iequality, positive depedece, MT 2 1 Suppoted by NSA Gat H Suppoted by NSF Gat DMS

2 1. Itoductio. This pape is motivated by the eed to have a claificatio of a esult o Simes 1986) iequality i Saka 1998). While establishig this iequality fo positively depedet multivaiate adom vaiables, as oigially cojectued i Simes 1986), Saka 1998) assumed a specific type of positive depedece, e.g., the depedece that is chaacteized by the MT 2 popety see Theoem 3.1 of Saka, 1998). Havig ealized that the stadad multivaiate-t with the associated multivaiate omal havig o-egative coelatio does ot have this popety, Saka 1998) made a attempt to pove the Simes iequality fo this distibutio ude cetai sig estictios though a coollay Coollay 3.1). Ufotuately the poof of this coollay does ot appea to be coect, as we will poit out i this pape. Thus, the claim that the multivaiate-t distibutio satisfies the Simes iequality does ot follow fom the agumet of Remak 3.2 i Saka 1998). A caeful study of Saka s 1998) poof of the Simes iequality would eveal, ad also it follows fom the poofs of the FDR false discovey ate) cotol of the method of Bejamii ad Hochbeg 1995) i the positive depedece case Bejamii ad Yekutieli, 2001; Saka, 2002), that the Simes iequality will cotiue to hold fo distibutios with a weake vesio of depedece coditio tha the MT 2 coditio, such as the positive depedet though stochastic odeig DS) coditio of Block, Savits ad Shaked 1985), which is essetially the same as the positive egessio depedece o subset RDS) coditio cosideed i Bejamii ad Yekutieli 2001) ad Saka 2002). Howeve, as we will show i this pape, the stadad multivaiate-t with o-egative coelatios does ot satisfy the DS o RDS coditio. Thus, the Simes iequality fo the stadad multivaiate-t with o-egative coelatios does ot follow fom the DS o RDS coditio eithe. 2

3 Nevetheless, the Simes iequality is ideed tue fo the stadad multivaiate-t with oegative coelatios ude cetai sig estictios Bejamii ad Yekutieli, 2001; Saka, 2002, 2008). I this pape we povide a diect poof, oe based o the pape of Saka 2002). The pape is ogaized as follows. I the ext sectio, we make some commets o the esults of Saka 1998) towads povig the Simes iequality fo the multivaiate-t distibutio i Sectio Commets o Saka 1998) I this sectio, we make some impovemets to Saka 1998) befoe givig poofs of the Simes iequality i the ext sectio fo the multivaiate-t distibutio. Fist, we peset a moe steamlied poof of his Lemma 2.1 usig agumets simila to those used i moe ecet papes elated to FDR methodologies; see also Saka 2008). Secod, the Simes iequality is peseted ude a weake fom of positive depedece moe suitable fo multivaiate t distibutios. I additio, we give a example showig that the poof of Coollay 3.1 of Saka 1998) is ot coect, ad so his agumet showig that the multivaiate-t satisfies Simes iequality does ot apply. Lemma 2.1. Let X 1) X ) be the odeed compoets of a adom vecto X X 1,..., X ) ad a 1 a be eal umbes. The, X 1) > a 1,, X ) > a F i a ) + i1 i1 j1 [ IXi a j+1 ) E j + 1 IX i a j ) j X i) j) > a j+1,, X i) ] 1) > a X i, 1) whee fo each i 1,...,, X i) 1) X i) 1) deotes the odeed compoets of the 1)- dimesioal adom vecto X i) obtaied by elimiatig X i fom X. 3

4 oof. Let R max X ) a, so that R X ) a, X +1) > a +1,..., X ) > a, fo 1,...,, ad R 0 X 1) > a 1,..., X ) > a. The I R > 0 1 I R ) i1 1 i1 1 whee R i) 1 max X i) 1) a ad R i) I X i a ) I 1 I X i a ) I R ) 1 i1 ) X ) a, X +1) > a +1,..., X ) > a 1 I X i a ) I R i) ) 1 1., defied aalogously to R, i.e., X i) 1) a, X i) ) > a +1,..., X i) 1) > a fo 2,...,, R i) 1 0 X i) 1) > a 2,..., X i) 1) > a. Sice IR i) 1 1 IR i) 1 1 IR i) 1 2, fo 1,...,, with IR i) 1 < 0) 0 ad IR i) 1 1 1, it follows that I R > 0 1 I i1 1 I i I X i a ) + I i1 i I X i a ) + I i1 i1 1 [ IXi a ) R i) ) I 1 1 Xi a ) + I R i) 1 2 ) IXi a ) IX i a +1 ) + 1 R i) 1 1 ) R i) 1 1 ) [ IX i a ) i1 X i) ) > a +1,..., X i) ) 1) > a 4 ]. I X i a ) IX ] i a +1 ) + 1

5 Usig IX 1) > a 1,..., X ) > a ) 1 IR > 0), ad takig expectatios gives the equied esult. Lemma 2.2 Simes Iequality). Fo the adom vecto X i Lemma 2.1, i) we have, X 1) > a 1,..., X ) > a 1 X 1 a, fo ay fixed a 1 a such that j 1 X 1 a j is odeceasig i j 1,...,, if X i) j) > a 1,, X i) 1) > a 1 X i i, fo evey j 1,..., 1; ad ii) we have X 1) b 1,..., X ) < b X 1 b 1, fo ay fixed b 1 b such that j 1 X 1 > b j+1 is odeceasig i j 1,...,, if X i) 1) < b 1,, X i) j) < b 1 X i b i i b i, fo evey j 1,..., 1. oof. The fist pat of the lemma follows fom the followig iequality: [ E I X i) j) > a j+1,..., X i) 1) > a I X i a j+1 ) j + 1 X i) j) > a j+1,..., X i) 1) > a X i a j+1 ) X i a j j + 1 I X ] i a j ) j X i a j ) j 0, Fo ay fixed pai of i ad j. The secod pat follows fom the fist pat. 5

6 Example 2.1. The followig example shows that the poof of Coollay 3.1 of Saka 1998) is ot coect. Let gx, y) 1/y exp x/y) fo all x ad y > 0. Also let qz) λ exp λz) fo all z > 0 whee λ > 0. It is easy to show that g x, y) λy λy + x) 2 which is clealy T 2 i x ad y. Fo 2 ad ay a 1 a 2 it ca be show that T 1 a 1 ) 1/2 T 1 a 2 ). Now fo a 1 1, a 2 3 ad λ 1, afte some calculatios it ca be show that T 1) 1, T 2) 3 < X 1 1, X 2 3 fo some hy), ad thus 3.8) of Saka 1998) does ot hold. Howeve, we have ot bee able to show whethe the statemet of the coollay is coect o ot. 3. Multivaiate-t Distibutio I this sectio we show that the stadad multivaiate-t with oegative coelatios satisfies Simes iequality ude cetai sig estictios. The stadad multivaiate-t distibutio is give i Tog 1990, Chapte 9) as follows. Let R ρ ij ) be a symmetic matix which is eithe positive defiite o positive semidefiite with ρ ii 1 fo i 1, 2,...,. I the followig we will always assume positive defiiteess. Let Z Z 1,..., Z ) have a multivaiate omal distibutio with mea vecto 0 ad coelatio matix R ad let the uivaiate adom vaiable S be such that i) S is idepedet of Z ad ii) νs 2 has a chi-squae distibutio with ν degees of feedom. The the multivaiate T is defied by T T 1,..., T ) 6 Z1 S,..., Z ). S

7 This is called The Multivaiate-t with Commo Deomiato by Johso ad Kotz 1972) see vaious othe multivaiate-t distibutios defied thee) ad is also the basic multivaiate-t give by 1.1) of Kotz ad Nagaajah 2004). I Saka 1998) this is called the multivaiate-t of Duett ad Sobel 1954) type, but Johso ad Kotz 1972) seem to attibute this distibutio to Coish ad Lauet also. It is ot had to show that fo ρ ij 0, i j, the multivaiate-t has seveal positive depedece popeties. See Sectio of Tog 1990) whee vaious lowe ad uppe othat iequalities ae deived ad it is also show that the distibutio is associated. Howeve it does ot possess highe ode depedece popeties such as MT 2, as show i Sampso 1983), ad also is ot positive depedet though stochastic odeig Block, Savits ad Shaked, 1985), which we will show. This latte popety is essetially the same as RDS of Bejamii ad Yekutieli 2001). We ow pove the Simes iequality fo the multivaiate-t distibutio ude cetai sig estictios i.e., the ae egative ad the b j ae positive) by establishig the followig theoem. Theoem 3.1. Let T be the multivaiate-t distibutio as give above. The, i) fo fixed a 1 a < 0, we have T 1) > a 1,..., T ) > a 1 T 1 a, 2) if j 1 T 1 a j is odeceasig i j 1,..., ; ad ii) fo fixed 0 < b 1 b, we have T 1) b 1,..., T ) b T 1 b 1, 3) if j 1 T 1 > b j+1 is odeceasig i j 1,...,. 7

8 oof. The theoem will be poved by showig that T satisfies the equied coditios i Lemma 2.2 fo the Simes iequality to hold. Recall that T i Z i /S. Let S i Z 2 i + S2 ) 1/2 ad U i Z i /S i. Sice T i is equivalet to U i 1+a 2 i )1/2, usig the otatio of Lemma 2.1, we have the followig: E T i) j) [ [ E T i) j) ψ Z i, S) I > a j+1,..., T i) 1) > a T i > a j+1,..., T i) 1) > a Z i, S U i U i U i ) 1+a 2 i )1/2 )] 1+a 2 i )1/2 ) 1+a 2 i )1/2 I U i )] 1+a 2 i )1/2 whee ψz i, S) is the coditioal pobability withi the expectatio i the secod lie above. It is ot had to show ψz i, S) is stochastically odeceasig i Z i ad S, due to the sig costais o the a j s. Sice ψz i, S) ψu i S i, 1 U 2 i )1/2 S i ), ad U i ad S i ae idepedet, this coditioal pobability becomes [ ) E E ψ U i Si, 1 U i 2)1/2 Si U i I U i whee E φ U i ) I U i U i ) 1+a 2 i )1/2 ), 1+a 2 i )1/2 ) φu i ) E ψ U i Si, 1 Ui 2 ) 1/2 Si U i )] 1+a 2 i )1/2 ), 1+a 2 i )1/2 U i, ad is stochastically odeceasig i U i. Thus, the above atio is odeceasig i, which follows fom the followig stadad lemma. Lemma 3.1. Fo ay adom vaiable X, the coditioal expectatio EgX) X c is odeceasig i c if gx) is odeceasig i x. 8

9 This poves the fist pat of the theoem. The secod pat follows fom the fist pat. The above poof of Theoem 3.1 is a slight vaiatio of that based o a idea biefly outlied i Saka 2002) ad elaboated i Saka 2008). Example. This example shows that the bivaiate-t distibutio fo ρ 0 does ot satisfy the positive depedece popety of stochastic iceasig. Let T 1, T 2 ) have a bivaiate-t distibutio with ρ 0. The it is ot had to show that T 2 > t 2 T 1 t 1 is deceasig i t 1 fo t 2 1. Refeeces 1. Bejamii, Y. ad Hochbeg, Y. 1995). Cotollig the false discovey ate: A pactical ad poweful appoach to multiple testig. Joual of Royal Statistical Society Se. B 57, Bejamii, Y. ad Yekutieli, D. 2001). The cotol of the false discovey ate i multiple testig ude depedecy. Aals of Statistics 29, Block, H.W., Savits, T.H. ad Shaked, M. 1985). A cocept of egative depedece usig stochastic odeig. Statistics ad obability Lettes 3, Duett, C.W. ad Sobel, M. 1954). A bivaiate geealizatio of Studet s t distibutio with tables fo cetai special cases. Biometika, 41, Johso, N.L. ad Kotz, S.1972). Distibutios i Statistics: Cotiuous Multivaiate Distibutios,Wiley, New Yok. 6. Kotz, S. ad Nagaajah, S. 2004). Multivaiate t Distibutios ad Thei Applicatios, Cambidge Uivesity ess, Cambidge. 9

10 7. Sampso, A.R. 1983). ositive depedece popeties of elliptically symmetic distibutios. J. Multivaiate Aalysis 13, Saka, S.K. 1998). Some pobability iequalities fo odeed MT 2 adom vaiables: a poof of the Simes cojectue. Aals of Statistics 26, Saka, S. K. 2002). Some esults o false discovey ate i stepwise multiple testig pocedues. Aals of Statistics 30, Saka, S. K. 2008). O the Simes iequality ad its geealizatio. IMS Collectios Beyod aametics i Itediscipliay Reseach: Festschift i Hoo of ofesso aab K. Se 1, Simes, R.J. 1986). A impoved Bofeoi pocedue fo multiple tests of sigificace. Biometika 73, Tog, Y.L. 1990). The Multivaiate Nomal Distibutio, Spige-Velag, New Yok. 10