Petru P. Blaga-Reducing of variance by a combined scheme based on Bernstein polynomials

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1 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals REUCG OF VARACE BY A COMBE SCHEME BASE O BERSTE POYOMAS by Ptru P Blaa Abstrat A obd sh o th otrol varats ad whtd uor sal thods or rdu o vara s vstatd by us o th ultvarat Brst orator o th ut hyrub Th w obtad stators or th rado ural trato ar urally oard wth th rud Mot Carlo otrol varats ad whtd uor sal stators TROUCTO t trals a b statd by robablst osdratos ad ths ar rathr wh ultl trals ar ord Th tral s trrtd as th a valu o a rta rado varabl whh s a uow aratr To stat ths aratr th dt tral o rards th sal a o th sal ro a sutabl rado varabl Ths sal a s a ubasd stator or th dt tral ad s rrd as th rud Mot Carlo stator Grally ths thod s ot ast-ovr rato to th volu o sal ad y dds o th vara o th stator whh s rssd by th vara o th trad Cosqutly or rov th y o Mot Carlo thod t ust rdu as uh as ossbl th vara o th tratd uto Thr s a lot o rodurs or rdu o th vara th Mot Carlo thod th ollow w aroah th rdu o vara by a obd sh o th otrol varats ad whtd uor sal thods us th ultvarat Brst orators o th ut hyrub ural rts ar osdrd oaratvly wth th rud Mot Carlo otrol varats ad whtd uor sal stats 2 CRUE MOTE CARO METHO t X b a -dsoal rado varabl hav th robablty dsty uto ρ :R R th rado ural trato th ultdsoal tral ρ ; ρ d () [ ] R 47

2 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals s trrtd as th a valu o th rado varabl ( X) whr th uto :R R usually blos to 2 ρ ( R ) othr words ρ 2 d sts ad R thror th a valu [ ρ ; ] sts Us a bas statstal thqu th a valu v by () a b statd by ta ddt sals (rado ubrs) wth th robablty dsty uto ρ Ths rado ubrs ar rardd as valus o th ddt dtally dstrbutd rado varabls X sal varabls wth th oo robablty uto ρ ρ; or th sal a o rado varabls W us th sa otato [ ] ( X ) Th stator [ ; ] ad rstvly or ts valu [ ] [ ρ; ] ( X ) [ ] [ ρ; ] ( ) ρ satss th ollow rorts: ( [ ρ ; ]) [ ρ ] (ubasd stator o [ ; ] E ; Var ( [ ; ]) [ ; ] [ ρ; ] ρ ρ ) ρ (wth robablty ) Ta to aout ths rsults th rud Mot Carlo trato orula s dd by [ ρ ; ] ρ d ( ) (2) R t ust rar that () th doa o trato s oly aartly th whol -dsoal Eulda sa Thus t s ossbl that th dsty ρ whr s a ro o th -dsoal Eulda sa R thror th tral () bos 48

3 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals [ ; ] ρ ρ d ad th rud Mot Carlo thod ust b trrtd a arorat ar For al th trato ro s th ut hyrub [ ] ad ρ th th rud Mot Carlo stator s [ ] X (3) whr th sal varabls X ar ddt uorly dstrbutd o 3 COTRO VARATES METHO Cotrol varats s a thqu or rdu o varato ad t ossts th slt o th tral () to two arts [ ] ρ h d ρ [ h ] R R ρ ; d (4) whh ar tratd saratly th rst by athatal thory ad th sod by Mot Carlo thod Th uto h ust b sly ouh to b trat thortally ad s to absorb ost o ts varato Ths thod was ald [7] osdr th uto h v by th ultvarat Brst olyoal o th hyrub hav th dr th varabl : B ( ) ( ) ( ) (5) 49

4 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals W hav usd th ollow otatos ( ) ( ) R ( ) [ ] ad rstvly or th Brst bass ths ar th stat o th tral [ ] d s v [ ] [ ( ) B ( ; )] ( ) (6) whr ar ddt uorly dstrbutd ots th hyrub 4 WEGHTE UFORM SAMPG METHO Ths thod was v [4] rosdrd [6] ad rtly [2] t was oard wth othr thods or rdu o th vara t us osdr th tral whr th boudd [ ] d V d V R ro has th volu V Th rud Mot Carlo stator or [ ] V s [ ] X wth th sal varabls X ddt uorly th ro ths hav th oo dsty robablty uto 5

5 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals 5 V ρ Th thod o whtd uor sal ossts th osdr o uto R : suh that d ad th orrsod sal uto [ ] [ ] w w ; X X whr X ar th sa abov sal varabls [] was osdrd th uto ro th whtd uor sal thod v by th ultvarat Brst olyoal orrsod to th trad dd by (5) ; B K whr th ostat K s suh that d Fro ths odto w hav that

6 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals Fally th rado ural trato orula s v by w [ ] ( ) ( ) ( ) ( ) ( ) (7) Th rado ots hyrub ar ddt uorly dstrbutd th 5 COMBE SCHEME Th rdu thqu o vara by roosd obd sh ossts th slt o tral ro th otrol varats thod (4) wth th trato doa ad ( ) h B [ ] ( ) whr B ( ; ) d (8) th to aly th whtd uor sal thod or th tral ro th rht sd o (8) a suh way th uto ro whtd uor sal thod s v by ultvarat Brst olyoal aly whr th ostat K s suh that Fro ths odto w hav that K B ( ; ) d 52

7 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals 53 Fally th rado ural trato orula s v by [ ] [ ] [ ] vw (9) Th rado ots ar ddt uorly dstrbutd th hyrub 6 UMERCA EXPERMETS ural als ar osdrd th udsoal ad bdsoal 2 ass or th stator (9) wth th trad v by ad y y rstvly Th ural rsults otad th ollow two tabls oar th stats obtad by th obd sh (9) whtd uor sal thqu (7) otrol varats (6) ad th rud Mot Carlo thod (3)

8 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals Eah tabl otas: th sal volu th dr ( or 2 ) Brst olyoals th stat v by th obd sh vw [ ] o th th rato v w vw valus o th rror stats ad W also rar that th statos ro ah row o tabls rrst th a valus o hudrd o sals [ ] lo vw [ ; ] v w vw 27 6 [ ] lo vw [ ] v w vw

9 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals Rrs Blaa PP Brst orators to rdu o vara rado ural trato Prods o th tratoal Syosu o ural Aalyss ad Aroato Thory Cluj-aoa May 9-22 Cluj Uvrsty Prss Gorbahva B Sobol M Truzov A Vara-rdu ultlrs th outato o trals by th Mot Carlo thod (Russa) Zh Vyhsl Mat Mat Fz Vol4 (2) Harsly J M Hadsob C Mot Carlo Mthods Mthu-Joh Wly ad Sos odo-w Yor 964 4Hadsob C Rars o a Mot Carlo trato thod ur Math Vol6 (964) ortz GG Brst Polyoals Uvrsty o Toroto Prss Toroto 953 6Powll MJ Swa J Whtd uor sal-a Mot Carlo thqu or rdu vara Jst Math Al Vol2 (966) Rosbr Brst olyoals ad Mot Carlo trato SAM J ural Aalyss Vol4 (967) Stau Coa Gh Blaa P ural Aalyss ad Aroato Thory Vol Cluj Uvrsty Prss 22 55

10 Ptru P Blaa-Rdu o vara by a obd sh basd o Brst olyoals Author: Ptru P Blaa Babş-Bolya Uvrsty o Cluj-aoa Roaa blaa@athubblujro 56