Computation of the Multivariate Normal Integral over a Complex Subspace

Transcription

1 Aled Mathematcs htt://dxdoog/0436/am Publshed Ole May 0 (htt://wwwcrpog/oual/am) Comutato of the Multvaate Nomal Itegal ove a Comlex ubsace Katlos Joseh Kachashvl Mutazm Abbas Hashm 3 Abdul alam chool of Mathematcal ceces GC Uvesty Lahoe Paksta Vekua Isttute of Aled Mathematcs bls tate Uvesty bls Geoga 3 A Uvesty Multa Camus Multa Paksta Emal: katlos55@yahoocom mutazmabbas@gmalcom Receved Jauay 5 0; evsed Mach 30 0; acceted Al 6 0 ABRAC he comutato of the multvaate omal tegal ove a Comlex ubsace s a challege esecally whe the tegato ego s of a comlex atue uch tegals ae met wth fo examle the geealzed Neyma-Peaso cteo codtoal Bayesa oblems of testg may hyotheses ad so o he Mote-Calo methods could be used fo the comutato but at ceasg dmesoalty of the tegal the comutato tme ceases uustfedly heefoe a method of comutato of such tegals by sees afte educto of dmesoalty to oe wthout fomato loss s offeed below he calculato esults ae gve Keywods: Multvaate Nomal Itegal; Radom Vaable; Pobablty; Momets; ees Itoducto At testg may hyotheses wth efeece to the aametes of multvaate omal dstbuto the oblem of comutato of multvaate omal tegals ove a Comlex ubsace of the followg fom ases [] x H d x () whee s the umbe of tested hyotheses H : θ θ suosg that samle x x x was bought about by dstbuto x θ m x x x ; H whee θ m s the vecto of dstbuto aametes ad s the accetace ego of hyothess H fom samle sace R x R whch has the followg fom : x k x H k x H whee 0 k uch egos of hyotheses accetace ase fo examle the geealzed Neyma-Peaso cteo ad also codtoal Bayesa oblems of testg may hyotheses [3] he dmesoalty of these tegals ofte eaches seveal tes whe actcal oblems ae solved Fo examle ecologcal oblems the umbe of cotolled aametes accodg to whch the decso () s made s qute ofte equal to seveal tes [4]; the a defece oblems atcula the oblems of tackg of flyg obects usg ada measuemet fomato the dmesoalty of the oblem s equal to the multlcato of the umbe of flyg obects by the umbe of suveys made by the ada set [5] ad so o O the othe had the tme fo soluto of these oblems s ofte lmted ad at tmes t lays a decsve ole esecally at solvg the defece oblems It s kow that the comlexty of ealzato ad the obtaed accuacy of umecal methods of comutato of multdmesoal tegals deed heavly o the dmesoalty of these tegals ad the comlexty of the tegato ego cofguato I the cosdeed case the tegato egos ae ocovex ad qute comlex heefoe t s dffcult to ealze the umecal methods ad to ovde the desed accuacy of calculato eve whe the dmesoalty of tegal s geate tha o equal to thee [6] he methods of comutato of the multvaate omal tegal o the hyeectagle offeed [7-] ae usutable fo ths case because of the comlexty of the tegato ego Deste the coveece ad the smlcty of comutatos the Mote Calo method s comute tme cosumg esecally at lage dmesoalty of tegals [334] heefoe the method of aoxmate comutato of tegal () fo a vey shot eod of tme s tocal may alcatos of mathematcal statstcs [56] Coyght 0 cres

2 490 K J KACHIAHVILI M A HAHMI he am of the eset ae s the develomet of the method of comutato of obablty tegal () wth the desed accuacy a mmum of tme Poblem tatemet Let us cosde the case whe the obablty dstbuto desty of the vecto x looks lke x H / / W x a W xa π ex whee a a a W Fo obablty dstbuto desty (3) let us ewte decso-makg ego () as x : C ex y 0 (4) whee / C k π W C / k π W y xa W xa (5) Radom vaables y ae squaed foms of the omally dstbuted adom vecto ad f hyothess H s tue the mathematcal exectatos ae equal to Ey H a a W a a (6) tace WW heefoe f hyothess H s tue the adom vaable y has ocetal dstbuto wth the degee of feedom ad wth the aamete of ocetalty equal to (6) [78] It s obvous that at ad hyothess H s tue the adom vaable y has the cetal dstbuto wth the degee of feedom Let us wte dow () as follows x H dx P C exy H 0 (7) he task cossts the comutato of obablty (7) he method of ts aalytcal comutato s ot kow so fa Fo ts comutato t s ossble fo examle to use (3) a modfed Mote-Calo method (wth the uose of educg the comutato tme) [3] hough at lage t stll takes a good deal of tme he method of comutato of obablty (7) f hyotheses ae fomulated wth efeece oly to the mathematcal exectato of omally dstbuted adom vecto s offeed [3] hs method s usutable hee as the adom vaable ex C y (8) whch fomulates tegato ego (4) [3] s the weghted sum of log-omally dstbuted adom qua ttes; C ad y ae detemed by fomulae (5) I ou case s the weghted sum of the exoets of egatve quadatc foms of the omally dstbuted adom vecto wth coelated comoets Let us cosde the case I ths case egos () take the fom x: x H k x H x: x H k x H Wth takg to accout obablty destes (3) fo these egos we deve : x x W x x x a W a W x x x W x x W x a W a W x W : whee l W k a W a a W a W l W k a W a a W a W Let us desgate xw xxw x a W a W x xw xxw x a W a W x he fally fo the equed egos we shall obta x : : x Each of adom vaables ad s the sum of thee adom vaables oe of whch s dstbuted by the omal law ad the two othes ae dstbuted by the law heefoe the obablty dstbuto laws of adom vaables ad have ot closed foms hus at e at testg two hyotheses wth esect to all aametes of multvaate omal dst- Coyght 0 cres

3 K J KACHIAHVILI M A HAHMI 49 buto ( cotadstcto to the case whe hyotheses ae fomulated wth esect to oly the vecto of mathematcal exectato [3]) the cal comlexty of the cosdeed oblem does ot decease 3 Comutato of Pobablty Itegal (7) by ees Let us use the exaded fom of eesetato of the quadatc fom (8) [89] he x a x a t t t t C ex t t tt t t whee t t ae the coeffcets detemed uamb- guously by the elemets of matx W (see fomula (3)) Let z H be the codtoal desty of obablty dstbuto of the adom vaable he fo (7) we obta 0 (9) z H dz (0) Hee the fte teval s take as the doma of defto of adom vaable because of the sgs of coeffcets C fom (5) As was metoed above the obablty dstbuto law of the adom vaable has ot a closed fom Let us cosde the ootuty of aoxmatg ths desty by sees Fo ths easo we eed the momets of the adom vaable [9-] Let us cosde the oblem of obtag of these momets Wth ths uose let us calculate the tal momet of the th ode of adom vaable ovded that hyothess H s tue ex C C E ex y y H E H E C y H () Exesso y y s the sum of coelated Quadatc Foms dstbuted by ocetal obablty dstbuto laws Because of coelato the oety of eoducblty of the dstbuto does ot take lace [8] ad cosequetly the mathematcal exectato () has ot a closed fom Let us use owe sees exaso of the exoet ex y y y y y 0! y y y! 0 ; () Let us use the exaded eesetato of quadatc fom (9) ad be satsfed wth the fst M tems of exaso () he exesso fo calculato of momets () ca be eeseted as follows M C C! { }; tt t t t t m x a E H whee 0 m ad (3) m Exesso (3) cotas oduct momets [790] of the ( M ) odes of omalzed comoets of the coelated omally dstbuted adom obsevato vecto heefoe they ae ot equal to zeo [8] A lot of woks ae dedcated to the oblem of comutato of oduct momets [see fo examle -8] I [] the followg oblem was solved Let x x x be adom vaables wth mutually dee det dstbutos ad let X x hee s foud the obablty that X les betwee A ad B e PA X B by usg the cetal lmt theo em accodace wth whch the adom vaable l X l x s aoxmately dstbuted by the o- mal law he bette s ths aoxmato the bgge s he vaace of the oduct of two adom vaables was studed by Baett (955) ad Goodma (960) the case whe they do ot eed to be deedet hellad (95) studed the case whe the dstbuto of k x was (aoxmately) logathmc-omal he au- tho cosdeed the case whe x x x ae adom vaables wth mutually deedet dstbutos Fo fdg the obablty that X x les betwee A ad e P A X B the cetal lmt theoem s used to aoach the obablty dstbuto of the adom vaable l X by omal dstbuto ad ths aoach s bette at ceasg I wok [5] o assum- B to s made about the dstbuto of cussed the case whe the k x hee s ds- K adom vaables Coyght 0 cres

4 49 K J KACHIAHVILI M A HAHMI x x xk K ae mutually deedet ad the case whe they do ot have to be deedet ad thee ae obtaed the vaace fomulae hese esults ae geealzatos of the esults eseted [4] I [6] ae gve exact fomulae fo the mathematcal ex x X x X x X ad ectato of k k x Xx X xk Xkxh Xh k h whee x s the samle mea of the th chaacte a samle of elemets fom a oulato of N elemets ad X s the coesodg oulato mea Fomulae fo estmatg these oduct momets fom the samle wee also gve hese estmatos ae slghtly based I [7] the ubased estmate of the 4-vaate oduct momet was obtaed Asymtotc esults fo the 3-vaate ad 4-vaate oduct momets ad the estmates wee also obtaed I [8] s deved a fomula fo the oduct momet m m EX X m m tems of the ot suvval fucto whe X X s a o-egatve adom vecto Fom the gve evew (of couse comlete because ths s ot the am of ths ae) of the woks dedcated to the study of oduct momets t s see that the oblem cosdeed hee dffes fom them heoem 3 he tal momet of the th ode of adom vaable detemed by (9) ovded that hyothess H s tue ca be calculated wth ay secfed accuacy by the fomula! M C C ; t t J t t t t 0 ; 0 whee d 3 (4) J mod K β ; β ad K ae the matces of egevectos ad egevalues of the vese covaace matx of omalzed adom vaables x a ; d 0 ae the coeffcets detemed by the tems of matces β ad K ad vecto b ; ad ae th e tal ad cetal momets of t he fst ad odes esectvely defed by fomulae () (3) ad (4) x a Poof If hyothess s tue the values H ae coelated omally dstbut ed adom vaables wth the aametes x a a a E H b x a V H v ; x a x a cov H (5) v hus fo calculato of momets (3) t s equed to calculate the oduct momets of -dmesoal M omally dstbuted adom vectos fo whch the comoets of the vectos of mathematcal exectatos ad the covaace matces ae calculated by fomulae (5) Let us desgate b b b v v v v v v V (6) v v v ad the coesodg adom vecto by y y y e x a x a x a y y y Fo calculato of codtoal oduct momets of the -ode we have m m m m m m E y y y H y y y whee f y y y H dydy d y (7) f y y y s the -dmesoal omal obablty dstbuto desty wth the vecto of mathematcal exectatos ad the covaace matx calculated by fomulae (6) It s kow that the value of tegal (7) s vaat to lea tasfomato of the comoets of vecto x [8] wth the accuacy of Jacoba of asfomato Let us desgate the matxes of egevectos ad ege- values of matx V by β ad K k K s a dago ets of esectvely It should be oted out that al matx he the como - Coyght 0 cres

5 K J KACHIAHVILI M A HAHMI 493 dmesoal adom vecto Z β K y b ( 8) wll be ucoelated ad wll have stadad omal dstbuto of obabltes [30] Fom (8) we wte y K β Z b Let us toduce the followg desgato γ K β he fo the elemets of the vecto y we obta the followg exesso Usg tasfomato (9) fo mathematcal exectato (7) we obta y z b (9) t t t E y y y H m m m J m E t t z t b H whee mod J K β s the Jacoba (0) asfomato (8) Let us ase to the owes the lea foms the ghthad sde of exesso (0) ad gou the detcal tems he (0) ca be wtte as m m m E y y y H J 0 ; of d E z H () whee the coeffcets of the detcal tems (0) ae desgated by d 0 ; ; the tems of the vecto Z ae detemed as z It s kow that [9] K y b E z H 0 () whee ad ae the tal ad cetal momets of ad odes esectvely of a dom vaable z Afte smle oute tasfomatos fo the cosdeed case we obta z! f! 0 f s odd s eve (3) Hee V z C H z C D b b C b D K D K b (4) akg advatage of atos () () fo comutato of the momets (3) we obta exesso (4) Pobablty tegal (0) ca be comuted wth the hel of Edgewot s sees [9-] usg fomula (4) fo comutato of the tal momets of adom vaable (9) I atcula the cosdeed case usg wellkow techques of obtag Edgewot s sees [30] we have 0 * d πˆ 0 z H z H z 3 *3 * 4 *4 * 3/ z 3z z 6z 3 3! 4! 0 3 *6 *4 * 3 z 5z 45z 5 6! 5 *5 *3 * 3 4 5z 0z 5z ! 7! z z 05z 05z *7 *5 6 4 *6 *4 * z 5z 45z *3 * 3 3 9! *9 *7 *5 *3 * 3 *8 *6 *4 * 80 z 36z 378z 60z 945z 35 6! 8! z z z z 00 0! z 45z 630z *0 *8 * z *4 * 3 z 3! z 66z 485z 3860z * *0 *8 *6 5975z 6370z 0395 z *4 * * 3 4 (5) Coyght 0 cres

6 494 K J KACHIAHVILI M A HAHMI * whee z k k s the k th sem-vaat of the adom vaable ovded hyothess H s tue (the comutato of sem-va- ats s ot dffcult kowg all tal momets clude- g k (see fo examle [])); s the secod cetal momet; x s stadad omal desty e xexx π atsfyg the fst seve tems exaso (5) the absolute value of calculato eo of the obablty tegal s calculated by the fomula ; 6! 6 *6 *4 * z 5z 45z *8 *6 *4 * z 8z 0z 40z 05 8! z 45z 630z 350z 0! z 945 *0 *8 *6 *4 * 3 3/! z 66z 485z 3860z 5975z * *0 *8 *6 *4 z 6370z 0395 * * 4 he vaable C exy s cotuous ad uambguously defed fo evey value of x heefoe the adom vaable s cotuous he chaac- testc fucto of the adom vaable ad ts devatves of ay ode exst as the momets of ay ode of ths adom vaable exst At the same tme the chaactestc fucto s ufomly cotuous Cosequetly the dstbuto fucto of ths adom vaable exsts ad s cotuous [] heoem 3 he dstbuto fucto of adom vaable exsts ad s uquely detemed by momets (4) Poof Fo ovg ths theoem t s ecessay to show that all momets exst ad the followg codto takes lace [9] lm su he fact that all momets exst s obvous fom fomula (4) as by usg t t s ossble to calculate the momets of ay ode wth ay secfed accuacy he values of these momets exst ad ae fte Whe solvg the actcal oblems coeffcets k k take o the values bouded above; coelato matces W ae ostvely detemed matces the detemats of whch dffe fom zeo heefoe coeffcets C ae bouded-above quattes hee takes lace y y d x Eex y y H e N xa W whee y y s the sum of quadatc foms of o mally dstbuted -dmesoal vecto x at dffeet vectos of mathematcal exectatos ad covaace matces heefoe at chagg comoets of the vecto x fom u to the quadatc fom y y takes the values f om 0 to ad the valu e of fucto e ectvely heefoe [8] y y Eex y y H N xa W x vaes fom to 0 es- d hus takg to accout (5) ad () we ca wte dow C C π W A H whee A s the maxmum by absolute value amog coeffcets k Assume the we have Let us desgate π H A W W m {} m W he A π Wm H A π Wm If π m A W the cult to be covced that Let A C ad t s ot dff- 0 at W m whee π C ad C Hece / C C C C 0 C Coyght 0 cres

7 K J KACHIAHVILI M A HAHMI 495 at whch oves the theoem 4 Comutato Results he accuacy of ths algothm deeds heavly o M - the umbe of used tems exaso () I ode to cease the accuacy of aoxmato of the exoet fo gve M ad geeal the elablty of comutato the tasks of hyotheses testg t s exedet to efom fst the omalzato of tal data by fomulae: x xc d c d c d c whee c d ae the mmum ad maxmum values of the th aamete fo the gve set of the cosdeed hyotheses e c m a d max a { } { } [3] I ths case the values of the aametes of the algothm M 5 ad seve tems exaso (5) ovded the absolute eo of comutato of tegal () that does ot exceed 0005 fo comuted examles (see below) hs fact was establshed by modelg fo the obsevato vecto wth ocoelated comoets Ufotuately by ow the cosdeed algothm has bee ealzed oly fo such a case [3] he esults of smulato showed that the tme of executo of the task (decso-makg ad comutato of the sutable value of the sk fucto) by usg the Mote-Calo method made u sec 45 se c ad 3 sec fo the umbe of hyotheses 3 4 ad 5 esectv ely the dmesoalty of the obseved vecto beg equal to 8 all cases he tested hyotheses ad coelato matx fo the case 5 ae gve table s of Fgue ad Fgue esectvely Fgues ae es eted as the sutable foms of the task of hyotheses test of the statstcal softwae whch the aoate methods ae ealzed [3] Fo othe values of thee ae chose the sutable sub-sets of the tables of these Fgues I the fst colum of the table of Fgue s gve the vecto of measuemets ad the othe colums ae gve hyothetcal values of mathematcal exectato of ths vecto Meaw hle whe usg the method offeed hee the comutato tme dd ot actcally chage ad the esults wee obtaed fo the tme otceably less tha sec I both cases obablty tegals (7) wee comuted wth the accuacy of 0005 I Fgues 3 ad 4 ae gve the deedeces of the tegal comutato tme o the accuacy ad umbe of tested hyotheses esecttvely At solvg may actcal oblems esecally mltay oblems [53] the dmesoalty of the tegals Fgue he fom of eteg the tested hyotheses ad a measuemet vecto Fgue he fom of eteg the covaace matx lke () ofte s equal to seveal tes ad dffeece betwee the comutato tme ecessay fo the cosdeed methods s sgfcatly loge tha the above metoed case [4] wheeas the comutato tme fo solvg the defece oblems ae of geat motace he theoetcal vestgato of the deedece of the accuacy of comutato of tegal () o M-the umbe of tems exaso (3) s a challegg task heefoe at ogam ealzato of the offeed algothm ad geeal algothms of such a kd t s wothwhle to Coyght 0 cres

8 496 K J KACHIAHVILI M A HAHMI = 8 = 5 me of comutato (sec) Accuacy of comutato 0050 Fgue 3 Deedece of the tegal comutato tme o the accuacy = 8 me of comutato (sec) Numbe of hyotheses Fgue 4 Deedece of the tegal comutato tme o the umbe of hyotheses Coyght 0 cres

9 K J KACHIAHVILI M A HAHMI 497 make aamete M ad the umbe of tems exaso (5) exteal aametes of the ogam hs allows establshg the otmal values fo each cocete case by exemetato deedg o the desed accuacy of comutato 5 Cocluso he method of comutato of the obablty tegal fom the multvaate omal desty ove the Comlex ubsace by usg sees ad the educto of dmesoalty of the multdmesoal tegal to oe wthout losg the fomato was develoed he fomulae fo comutato of oduct momets of omalzed omally dstbuted adom vaables wee also deved he exstece of the obablty dstbuto law of the weghted sum of exoets of egatve quadatc foms of the omally dstbuted adom vecto was ustfed he ootuty of ts uambguous detemato by the comuted momets was oved REFERENCE [] homso O the Dstbuto of ye II Eos Hyothess estg Aled Mathematcs Vol No do:0436/am00 [] C R Rao Lea tatstcal Ifeece ad Its Alcato d Edto Joh Wley & os Ltd New Yok 006 [3] K J Kachashvl Geealzato of Bayesa Rule of May mle Hyotheses estg Iteatoal Joual of Ifomato echology & Decso Makg Vol No do:04/ [4] A V Pmak V V Kafaov ad K J Kachashvl ystem Aalyss of A ad Wate Qualty Cotol Naukova Dumka Kev 99 [5] A I Potaov A G Vogadov I A Gotsky ad E E Petsov About Decso-Makg of Pesece of Obects at Gou Measuemets Questos of Rado-Electocs Vol [6] P J Davd ad P Rabovtz Methods of Numecal Itegato Comute cece ad Aled Mathematcs d Edto Academc Pess Ic Olado 984 [7] A Gez Numecal Comutato of Multvaate Nomal Pobabltes Joual of Comutatoal ad Gahcal tatstcs Vol [8] A Gez Comaso of Methods fo the Comutato of Multvaate Nomal Pobabltes Comutg cece ad tatstcs Vol [9] A Gez ad F Betz Numecal Comutato of Multvaate t-pobabltes wth Alcato to Powe Calculato of Multle Cotasts Joual of tatstcal Comutato ad mulato Vol 63 No do:0080/ [0] Joe Aoxmatos to Multvaate Nomal Rectagle Pobabltes Based o Codtoal Exectatos Joual of the Ameca tatstcal Assocato Vol [] I H loa ad Joe Lattce Methods fo Multle Itegato Claedo Pess Oxfod 994 [] V Havasslou D McFadde ad P Ruud mulato of Multvaate Nomal Rectagle Pobabltes ad he Devatves: heoetcal ad Comutatoal Results Joual of Ecoometcs Vol 7 No do:006/ (94)076-6 [3] J O Bege tatstcal Decso heoy ad Bayesa Aalyss ge New Yok 985 [4] K J Kachashvl Bayesa Algothms of May Hyothess estg Gaatleba bls 989 [5] D V Ldley he Use of Po Pobablty Dst- butos tatstcal Ifeece ad Decsos Poceedgs of the 4th Bekeley ymosum o Mathematcal tatstcs ad Pobablty Vol [6] L eey ad J B Kadae Accuate Aoxmatos fo Posteo Momets ad Magal Destes Joual of the Ameca tatstcal Assocato Vol [7] A tuat J K Od ad Aols Kedall s Advaced heoy of tatstcs Classcal Ifeece ad the Lea Model 6th Edto Vol A Oxfod Uvesty Pess Ic New Yok 999 [8] W Adeso A toducto to Multvaate tatstcal Aalyss 3d Edto Wley & os Ic New Jesey 003 [9] A tuat J K Od ad Aols Kedall s Advaced heoy of tatstcs Dstbuto heoy 6th Edto Vol Oxfod Uvesty Pess Ic New Yok 994 [0] H Came Mathematcal Methods of tatstcs Pceto Uvesty Pess Pceto 999 [] M Kedall ad A tuat Dstbuto heoy Vol Chales Gfft & Comay Lmted Lodo 966 [] G D hellad Estmatg the Poduct of eveal Radom Vaables Joual of the Ameca tatstcal Assocato Vol [3] H A R Baett he Vaace of the Poduct of wo deedet Vaables ad Its Alcato to a Ivestgato Based o amle Data Joual of the Isttute of Actuaes Vol [4] L A Goodma O the Exact Vaace of Poducts Joual of the Ameca tatstcal Assocato Vol [5] L A Goodma he Vaace of the Poduct of K Radom Vaables Joual of the Ameca tatstcal Assocato Vol 57 No [6] N Nath O Poduct Momets fom a Fte Uvese Joual of the Ameca tatstcal Assocato Vol 63 No [7] N Nath Moe Results o Poduct Momets fom a Fte Uvese Joual of the Ameca tatstcal Assocato Vol 64 No [8] Nadaaah ad K Mtov Poduct Momets of Multvaate Radom Vectos Commucatos tatstcs Coyght 0 cres

10 498 K J KACHIAHVILI M A HAHMI heoy ad Methods Vol 3 No do:008/a [9] Kotz N Balaksha ad N L Johso Cotuous Multvaate Dstbutos Models ad Alcatos Vol d Edto Joh Wley & os Ltd New Yok 000 do:000/ [30] G zego Othogoal Polyomals Ameca Mathe- matcal ocety New Yok 959 [3] K J Kachashvl ad D I Melkdzhaa Do he oftwae Package fo tatstcal Pocessg of Exemetal Ifomato Iteatoal Joual Ifomato echology & Decso Makg Vol 9 No do:04/ [3] K J Kachashvl ad A Mueed Codtoal Bayesa ask of estg May Hyotheses tatstcs 0-0 do:0080/ Coyght 0 cres