Some probability inequalities for multivariate gamma and normal distributions. Abstract

Transcription

1 -- Soe probably equales for ulvarae gaa ad oral dsrbuos Thoas oye Uversy of appled sceces Bge, Berlsrasse 9, D-554 Bge, Geray, e-al: Absrac The Gaussa correlao equaly for ulvarae zero-ea oral probables of syercal -recagles ca be cosdered as a equaly for ulvarae gaa dsrbuos ( he sese of Krshaoorhy ad Parhasarahy [5]) wh oe degree of freedo. Is geeralzao o all eger degrees of freedo ad suffcely large o-eger degrees of freedo was recely proved []. Here, hs equaly s parly exeded o saller o-eger degrees of freedo ad parcular - a weaer for - o all fely dvsble ulvarae gaa dsrbuos. A furher ooocy propery - soees called ore PLOD (posvely lower orha depede) - for creasg correlaos s proved for ulvarae gaa dsrbuos wh eger or suffcey large degrees of freedo. Keywords ad phrases: probably equales, ulvarae oral dsrbuo, ulvarae gaa dsrbuo, Gaussa correlao equaly, posve lower orha depedece, ooocy properes of ulvarae gaa dsrbuos aheacs Subec Classfcao: 6E5. Iroduco ad Noao Soe specal oaos: For ay ( ) -arx A he subarx of A wh row ad colu dces, {,..., }, s deoed by A. I a slar way A s he subarx wh row dces ad colu dces. For a arx wh soe dces, A or, dces eas always ( A dces ), e.g. ( ) ( ). For wo ( ) -arces A ( a ) B ( b ) eas a b for all,, ad "A B" s equvale o " A B ad A B". A ull-arx s deoed by O ad a ull-vecor sply by. The ras- pose of a arx A s deoed by A. The leer T sads always for a pos. def. (or soees pos. se-def.) arx T dag(,..., ). The probably desy fuco (pdf) of a uvarae gaa dsrbuo wh he ( ) ( ( )) exp( ),,, shape paraeer s g x x x x ad G ( x) s he correspodg cuulave dsrbuo fuco (cdf). The o-ceral gaa cdf wh he o-ceraly paraeer y s gve by G ( x, y) exp( y) G ( x) y!. / ecely, [] a sple proof was gve for he Gaussa correlao equaly (GCI) s arrow for for ulvarae ceered oral probables of syercal -recagles. Ths equaly assers for all he eves A { X x },,...,, ha P A P A P A, (.)

2 -- where ( X,..., X } has a oral N(, ) - dsrbuo. W. l. o. g., he covarace arx ca be chose as a correlao arx. Accordg o [] he equaly (.) ples also he ore geeral for of he GCI: P( C C) P( C) P( C) for all covex ad cerally syerc ses C, C. The equaly (.) - wh - ca be read as a equaly for he eves B where { Y y x }, ( Y,..., Y ) has a -varae gaa dsrbuo ( he sese of Krshaoorhy ad Parhasarahy [5]) wh he degree of freedo ad he assocaed correlao arx, here deoed as he (, ) - dsrbuo. I ca be defed ore geerally by s Laplace rasfor (L) I T, (.) wh he dey arx, I T dag(,..., ),,...,, ad a se A ( ) of adssble posve values, where adssbly eas, ha he Laplace verso of (.) yelds acually a (, ) - pdf, whch s always rue for. Uforuaely, A ( ) s o exacly ow for he os correlao arces. Equvale codos for a fely dvsble (f. dv.) L posve are adssble. I T are foud [] ad [3]. I hs case, all Exedg he uvarae o-ceral gaa deses o fucos g ( x, y) o x, y, soe xure represeaos ca be gve for he fucos wh he L (.): The o-sgular correlao arx s called -facoral f, (.3) / / D AA D D I BB wh a pos. def. arx D dag( d,..., d ) ad a ( ) -arx A of he al possble ra ad wh colus a, whch ay be real or also pure agary. Wh such a -facoral he fuco g ( x,..., x; ) : E d g ( d x, bs b ) (.4) wh he rows b of B (.3) ad a expecao referrg o a W(, I ) -Wshar (or pseudo-wshar) arx, or, has he L (.), (see [7] or [8]). Wh a real arx A (.3) he S fucos (.4) are obvously probably desy fucos. Ther Laplace rasfors are obaed wh he L of he o-ceral gaa deses followed by egrao over S. Wh ad a real colu A a all are adssble. Ay o-sgular has always a a os ( ) -facoral represeao wh D I, where s he lowes egevalue of, ad a real arx A of ra. Hece, all oeger values are also adssble. I he res of hs paper a - facoral correlao arx eas always a wh a represeao as (.3), bu oly wh a real arx A. Le Q ( q ) be he arx c c ( r ) wh c ax{ r,..., } ad c ( ), (.5) where s he lowes egevalue of. The elees o s dagoal. Accordg o Grffhs [3], he L Q Q I has a specral or Q ad oly egave I T (wh a o-sgular ) s f. dv. f ad

3 -3- oly f ( ) q q q (.6),, 3, for all subses {,..., } {,...,}, 3, ( I T s always f. dv.). The, wh Z dag z z z c we oba he seres expaso (see also sec. [6]) (,..., ), ( ), I T QZ I QZ QZ exp r( QZ ) q( ;,..., ) z ( ) (.7),.., wh coeffces q( ;,..., ), whch are all o-egave uder he codos (.5), (.6). By Laplace verso we ge for all he (, ) -cdf G ( x,..., x ; ) q( ;,..., ) G (c x ),,.., (.8) whch s a covex cobao of producs of uvarae gaa-cdfs. Oly wh he codo c ( ) he seres (.8) s also abs. coverge for ay o-sgular ad for all, x,..., x, bu he, represes he cdf of a (, ) - dsrbuo oly for adssble - values. Also wh he correlao arx Q, havg he elees,.., G ( x,..., x ; ) q( ;,..., ) G ( r x ) q r r r / ( ), a seres represeao (.9) ca be derved wh oly o-egave coeffces uder he codo (.6). Uder hs codo, he specral or of Q Q I s aga less ha, sce he assupo of he corary leads o a coradco. The seres (.9) s also abs. coverge for ay o-sgular, ad x,...,. x I [] acually a ore geeral heore was proved for (, ) -dsrbuos wh or, where he specal case ples he Gaussa correlao equaly. Ths heore ad s shor proof are reproduced seco, sce he proof shows he beefs, bu also he laos of he appled ehod, whch serves also as a odel for he slar proofs of he hree followg heores seco 3 a 4. The frs a of he uderlyg paper s a exeso of heore o saller o-eger values of, parcular o f. dv. (, ) -dsrbuos. Ths wll be accoplshed parly by heore. Theore 3 deals wh -facoral ( ) - correlao arces wh. A furher ooocy propery - soees called ore PLOD - for creasg correlaos s proved seco 4 for ulvarae gaa dsrbuos wh eger degrees of freedo ad a leas for all real values. I parcular wh, heore 4 coprses a geeralzed for of a coecure of Šdá seco 3 of []. For he ulvarae ceered oral dsrbuo a global verso of he local heore of Bølve ad Joag-Dev [] s obaed. If all would be adssble, he heore ad heore 4 would also hold for all ad, case of fely dvsble ulvarae gaa dsrbuos, also for all. Theore ad heore 3 would be dspesable as specal cases of heore.

4 -4-. Proof of he Gaussa correlao equaly exeded o soe ulvarae gaa dsrbuos Theore. Le O. The, for be a o-sgular correlao arx wh ( ) -subarces ad,, he cdf G ( x,..., x; ) of he (, ) - dsrbuo s creasg for all posve ubers x,..., x, ad a leas for all real values. Ths ples he equaly G ( x,..., x ; ) G ( x,..., x ; ) G ( x,..., x ; ) G ( x ), (.) ad parcular, for, he Gaussa correlao equaly. Proof. All he arces ( ),, are o-sgular correlao arces. Wh he L I T T I T T, {,...,}, of G ( x,..., x; ),,,,, {,..., }, {,...,},,, r ra( ) ad he o-egave egevalues (whch are squared caocal cor-,,,, relaos) of,,..., r, we oba he o-egave coeffces / /,,, r,,,,, (,, ),,, c ( ) :,. Therefore, H (,..., ; ) : I T T c ( ) I T T T s he L of * ( ) H( x,..., x ; ) c ( ) G ( x,..., x ; ), x whch s always posve for ay posve ubers x,...,, x sce he coeffces c () cao decally vash because of he posve ra of. Fally, he dey H( x,..., x; ) G( x,..., x; ) follows fro he egrao of over [, ], usg he uqueess of he Laplace verso ad Fub s crero, sce r * H (,..., ; ) x * ( G (,..., ; ) (,..., ; ) ) (,..., ; ) x x G x x e dx H d x (,..., ; ), H x x d e dx whch cocludes he proof.

5 ears. For decal x -5- x ad posve correlao eas r,, ( ) r r ( ) r he approxao G ( x,..., x; ) ( ) ( ( )! ) G ( x,..., x; ) G ( x,..., x; ) ( rr ) r ) c ( x;,, r ) c ( x;,, r ), wh where he ( he orgal arcle oly r r r r r r ) ad ( )! ( ) ( ;,, ) ( (( ),( ) )) ( ) ( ), ( ) L c x r G r x r r y y g y dy ( ) L are he geeralzed Laguerre polyoals, s proposed [9]. Ths approxao s recoeded parcular for sall exceedace probables G ( x,..., x; ). The error of hs approxao eds o zero wh a decreasg varably of he correlaos wh, ad. Soe uercal exaples wh very accurae coservave approxaos of hs ype are foud [9] ad [4]. 3. Soe supplees o heore For f. dv. (, ) -dsrbuos a heore slar o heore s proved here, bu oly for a paro of wh. The equaly (3.) heore eas, ha such a (, ) -rado vecor has posvely lower orha depede copoes for all. A sple specal case s he followg oe: For a oefacoral ( ) -correlao arx I wh correlaos r aa,, ad real ubers a (,) we oba fro (.4) (,..., ; ) (( ),( ) ) g ( ) G x x G a x a a y y dy for all. Sce he uvarae o-ceral gaa cdf G ( x, y) s decreasg y, he equaly G ( x,..., x ; ) G ( x,..., x ; ) G ( x,..., x ; ) follows fro Kball s equaly (see e.g. sec.. [3]). The equaly s src for all posve ubers ad x,...,, x f O. Wh he proof of heore we use he crero of Bapa [] for he fe dvsbly of he L I T wh a o-sgular. A sgaure arx S dag( s,.., s ) has oly he values s o s dagoal. A real o-sgular ( ) - arx A s a -arx f A has oly o-posve off-dagoal elees ad has oly o-egave elees. The, accordg o Bapa, I T s f. dv. f ad oly f here exss a sgaure arx S for whch ( SS) s a -arx. A

6 Theore. Le I T. The, for all -6- r r, r, r be a o-sgular ( ) - correlao arx wh a f. dv. L r r,, he L I T s f. dv. ad he (, ) cdf G ( x,..., x; ) s a creasg fuco of for all posve ubers, x,..., x, whch ples - G ( x,..., x ; ) G ( x,..., x ; )G ( x ) G ( x ). (3.) If (,..., ) [,] ad dag r (,..., ), he G ( x,..., x; ) s a o-decreasg fuco for each. Proof. Accordg o Bapa [] here exss a sgaure arx S wh S S beg a -arx. Wh Q q Q q SS q ad q Q Q ( q q) we oba Q Q Q q Q qq q Q q Q q q Q q wh oly o-posve off-dagoal elees ad oly o-egave elees Q, ad he sae way wh (3.) Q Q q S S ad q q Q Q ( q q) q, he verse arx Q Q Q q Q qq q Q q. Q q q Q q All he off-dagoal elees he lef upper bloc (3.) are o-posve ad s s a -arx oo ad I T s f. dv. for all [,]. Now, Q q Therefore,. Q I T T I T T T, ( ), where he suao exeds over all dex ses { }, {,..., }. Wh where c ( ),, r r r has he copoes r,, hs leads by Laplace verso o c ( ) G ( x,..., x; ) G( x,..., x; ),, (3.3) x where he dey s usfed as he proof of heore. Because of r, he coeffces c () cao decally vash. The dervaves (3.3) are posve for all posve ubers x,..., x because of he represeao (.8) of a f. dv. (, ) - cdf. The reag pars of he heore are obvous cosequeces of (3.3).

7 -7- ears. Ths proof cao be exeded drecly he sae way o a paroed correlao arx as heore wh, sce such cases I T s frequely o f. dv. for all (,). A exaple s gve by he correlao arces The arx , s a -arx, bu he elee 3 r.5 I s posve ad he reag elees r,,.5 are egave. Hece,.5 s o -arx ad 4.5T s o f. dv., whch s also recogzed fro he crero of Grffhs [3]. However, for 4, we ca also chose sce ( ), bu for larger values of s o sure f such cases he fucos G ( x,..., x; ) represe he cdf of a probably dsrbuo for all (,) ad sall. Cosequely, he posvy of he paral dervaves of G ( x,..., x; ) (3.3) cao always be guaraeed. The followg heore exeds heore o -facoral ( ) - correlao arces wh. For he specal case wh see he rear o Kball s equaly a he begg of hs seco. Theore 3. Le D AA be a o-sgular ( ) -correlao arx, 4, wh ( ) -subarces, ra ( ), a pos. def. dagoal arx D ad a real ( ) -arx A wh he al possble ra,. Furherore, le DB BB be a represeao of wh a pos. def. dagoal arx D B ad a real ( ) -arx B wh he al possble ra, (, ), ( eas I ). The, for,, he fucos G ( x,..., x; ) are creasg o (,) for all fxed posve ubers x,..., x ad a leas for all oeger values ax(, ( 3, 4)) or. Ths ples for all posve ubers x,..., x he equaly.5 G ( x,..., x ; ) G ( x,..., x ; ) G ( x,..., x ; ) (3.4) a leas for ax(, ( 3, 4)) or. ears. The fuco G( x,..., x; ) wh he L I T T s o ecessarly he cdf of a probably dsrbuo for all (,). However, wh he proof G ( x,..., x; ) s show o be a cdf. Forula (.4) shows G ( x,..., x; ) o be he cdf of a (, ) - dsrbuo f or. E.g., for 6 a cera fraco of he (66) -correlao arces s 3-facoral, sce he (63) - arces A provde 5 free paraeers for he 5 scalar producs represeg he 5 correlaos. Hece, he equaly (3.4) holds for such a leas for ad all values. Slar cosderaos are possble for hgher desos. Proof. Wh D D, ( ) D - dagoal, A A wh D D ( ) D B D A A A D ( ) ad he ( ( )) -arx A ( ) ad, we oba D A A O ( ) /. B

8 -8- Hece, G ( x,..., x; ) all real values ( ) wh (or ) are adssble o oba he cdf of a (, ) -dsrbuo. If we ca use a dffere represeao I A A,, wh he lowes egevalue of, beg posve sce s o sgular, ad wh a ( ) - arx A of ra r. The, ( ) 4 s adssble. Therefore, all he dervaves G ( x,..., x; ), x,..., x, x are posve ad he res of he proof ca be ae fro he proof of heore. 4. A furher ooocy propery of soe ulvarae gaa-dsrbuos Theore 4. Le ( r ) ad ( r, ) be o-sgular ( ) -correlao arces,, wh, all r, ad r for all off-dagoal elees or, he cdf G ( x,..., x; ( )) The, for all posve ubers x,...,, x. s creasg o [,], whch ples he equaly G ( x,..., x ; ) G ( x,..., x ; ) (4.) ears. I parcular for he equaly (4.) provdes a equaly for ulvarae ceral oral probables of syercal -recagles. I Šdá [] a local verso of hs laer equaly was proved for he specal case wh decal correlaos r, r. I a slar heore of Bølve ad Joag-Dev [] for ulvarae oral dsrbuos he correlaos ca be couously creased o correlaos of a as log as reas a -arx, whereas heore 4 does o eed such a resrco for. Theore 4 o- geher wh he heore of Bølve ad Joag-Dev - geeralzed by covoluos o eger values - shows ha (4.) reas rue uder he assupos,, all r, all r, ad r for all, sce heore 4 ca be appled o he par, ( r, ), where r, f r, ad r, r, oherwse, wh a suffcely sall. Sce he (, ) -dsrbuo s vara uder he rasfora- os SS wh ay sgaure arces S, heore 4 holds also for a par, of correlao arces f here exss a sgaure arx S for whch SS ad SS sasfy he above assupos. Proof. All he arces ( ) Q, Q ( q ) O,, are o-sgular correlao arces ad G ( x,..., x; ) wh he L a leas for or. Fro ad s he cdf of a (, ) I T T -dsrbuo, (4.3) I T T, {,...,},,,,,,, Q I Q ( ) wh he cardaly of ad he egevalues, of, ( ) :,,,, Q,, we oba he coeffces c ( c ( ) f ). (4.4) The sus, are o-creasg (decreasg, f here s a leas oe, ),, equal o r( Q ) f., ad hey are

9 The L -9- I T s f. dv. accordg o he crera [] ad [3]. Therefore, all he Laplace rasfors, of he correspodg argal dsrbuos are f. dv. oo. The, accordg o Bapa [], here I T exss always a sgaure arx S, whch geeraes a -arx S S, codo r, for all, ples S I. Therefore, all he off-dagoal elees ealg S, S O. The, are o-pos- ve ad r( Q, ) sce Q O wh oly zeros o s dagoal. I parcular, s r( Q, ) f q ad {, }. Now we oba fro (4.3) ad (4.4) he L H (,..., ; ) : I T T c ( ) I T T T of * ( ) H( x,..., x; ) c ( ) G ( x,..., x; ) x wh o decally vashg coeffces c ( ). The dey H( x,..., x; ) G( x,..., x; ) follows as he proof of heore. I would be of eres for soe ulple sascal ess o apply he followg (hypohecal) equaly G ( x,..., x; ) G ( x,..., x; ) (4.5) ( parcular for sall exceedace probables p G ( x,..., x; )) wh, decal values x x ad a ( ) - correlao arx, 3, wh decal correlaos r, where r s he ea of he correlaos r fro. For a o-egave correlao ea r a leas a local verso of (4.5) (.e. for all suffcely close o ) s rue uder he codo (,, r, x) c ( 4) c ( 3) c wh 3 ( f ( f ) c ( ) ryf ) F g ( y) dy, c r y f F g ( y) dy, where 3 f c r yf ( ryf ) F g ( y) dy, (( ),( ) ), (( ),( ) ), F G r x r ry f G r x r ry,, x (see heore 3 [9]). Ths codo ca frequely be verfed oly by a plo of he egrad. I [9] also a Taylor approxao s gve for G ( x,..., x; H) by a Taylor polyoal T (,,r, x; H) of d degree wh he devaos h r r, useful for sall values h ad larger values of x. I s 3 4 T (,,r, x; H) F g ( y) dy ( c c ) H ( c c ) H, H h, H h h, {, } {, }. 4, I parcular, for he applcao o he probably P{ax Z z} wh a oral N(, ) -rado vecor Z we fd wh

10 -- z ry z ry r yz r yz (( r) (( r) r r F erf erf, S sh, C cosh he coeffces z( r) y ( r) 3 z( r) y ( r) 3/ 3 c r y rs zc F dy ( ) exp, 5/ 3 ( ) exp F, c r y rs zc S dy 3 3/ 5/ 4 z ( 3 r) y 4 4 ( r F ) c ( r) exp S dy. ore geeral Taylor approxaos of d degree for G ( x,..., x; ) are also foud [9] ad sec. A.5 [4]. efereces []. B. Bapa, Ife dvsbly of ulvarae gaa dsrbuos ad -arces, Sayā 5 (989), [] E. Bølve ad K. Joag-Dev, ooocy of he probably of a recagular rego uder a ulvarvarae oral dsrbuo, Scad. J. Sas. 9 (98), [3]. C. Grffhs, Characerzao of fely dvsble ulvarae gaa dsrbuos, J. ulvarae Aal. 5 (984), 3-. [4] T. Dchaus ad T. oye, A survey o ulvarae ch-square dsrbuos ad her applcaos esg ulple hypoheses, Sascs 49 (5), [5] A. S. Krshaoorhy ad. Parhasarahy, A ulvarae gaa ype dsrbuo, A. ah. Sa. (95), [6] T. oye, Expasos for he ulvarae ch-square dsrbuo, J. ulvarae Aal. 38 (99), 3-3. [7] T. oye, O soe ceral ad o-ceral ulvarae ch-square dsrbuos, Sas. Sca 5 (995), [8] T. oye, Iegral represeaos ad approxaos for ulvarae gaa dsrbuos, A. Is. Sas. ah. 59 (7), [9] T. oye, Soe upper al approxaos for he dsrbuo of he axu of correlaed ch-square or gaa rado varables, Far Eas J. Theor. Sa. 43 (3), [] T. oye, A sple proof of he Gaussa correlao coecure exeded o soe ulvarae gaa dsrbuos, Far Eas J. Theor. Sa. 48 (4), [] G. Schecha, T. Schluprech ad J. Z, O he Gaussa easure of he erseco, A. Probab. 6 (998), [] Z. Šdá, O ulvarae oral probables of recagles: her depedece o correlaos, A. ah. Sa. 39 (968), [3] Y. L. Tog, Probably Iequales ulvarae Dsrbuos, Acadec Press, New Yor, 98.