Generation of multivariate random variables with known marginal. distribution and a specified correlation matrix

Transcription

1 Genertion o multivrite rnom vriles with known mrginl istriution n speciie correltion mtri Isiro R. Cru_Mein, Mucio Osório_ánche n ernno Grcí_Páe Astrct An lgorithm or generting correlte rnom vriles with known mrginl istriutions n speciie correltion is provie. This lgorithm, in which the speciiction o the joint multivrite istriution is not necessr, is wiel pplicle n llows the genertion o positivel or negtivel correlte rnom vriles rom n pir o mrginl continuous istriutions. It is known ct tht the mimum correltion coeicient etween two rnom vriles epens on the chrcteristics o their istriutions, in this pper, progrm coe or otining upper n lower limits o the correltion coeicient or two given mrginl istriutions is given. An emple or generting smple rom trivrite istriution with positive n negtive correltions n gmm mrginls is provie. Kewors: Rnom vector genertion, Multivrite istriutions, Bivrite gmm istriution, Bivrite et istriution, Mrginl istriution.. Introuction. A consierle mount o work hs een one or genertion o univrite rnom vriles with some speciic istriution. or the ivrite cse, the genertion o ivrite rnom vriles is tightl connecte with the prolem o constructing ivrite istriutions with speciie mrginls, this prolem hs een iscusse Plcket 967, Instituto Tecnológico e onor, Cinco e erero 88, C. Oregón on. CP Méico niversi Autónom e inlo,

2 Mri 967, Morn 969, Long n Krstoowics 995 mong others. Generting pirs o correlte rnom vriles is not strightorwr ecuse or most o the istriutions there is not nturl multivrite etension, or emple, or the et istriution, Louks 984 consiers the -imensionl Dirichlet istriution s the ivrite et istriution, ut Mgnussen 004 consiers tht the conition tht the sum o the two et rnom vriles re less or equl to one is ver restrictive, this uthor lso consiers tht the miture pproch given Michel n chucn 00, which gives ivrite et or gmm istriution where oth components hve the sme prmeters, s result, this ct limits the ppliction omin o their lgorithm. The lgorithm propose in this pper, uses the populr technique o introucing shre rnom vriles or the genertion o correlte rnom vriles n oes not require the ormultion o the joint multivrite istriution. This lgorithm is reil pplicle or generting rnom vector o heterogeneous continuous component vriles Prk, Description o the metho. The ollowing well known results re use or the genertion o correlte rnom vriles, in the irst step n multivrite istriution cn e use ut the multinorml istriution is more convenient.. Depenence mong vriles in multivrite stnr norml istriution is completel speciie the correltion mtri this propert mkes the multivrite norml istriution iel or generting epenence mong multivrite vriles with istriutions other thn norml.. The cumultive istriution unction CD or n continuous univrite istriution hs uniorm istriution 0,, in the intervl 0, ]. This propert llows

3 to otin rnom smple or n continuous istriution rom rnom norml smple using the uniorm istriution s rige. 3. A correlte multivrite norml smple will generte, in most cses, correlte multivrite smple or n given continuous istriution mens o the ppliction o the quntile unction to the CD s o norml smple. Nevertheless, it will e necessr to clirte the correltion o the norml smple to otin trget correltion on the speciie mrginl istriutions. 3. Nottion n einitions. The populr technique o introucing shre rnom vriles m e use or generting correltion coeicient or rnom norml vriles n V. Let T, T n Z e three uilir inepenent norml vriles with ero men n vrince given : T N 0, T N 0, Z N 0, I the trget correltion is positive, eine vriles n V s: V T Z 0, N T Z 0, N The correltion coeicient etween vriles n V is equl to E stns or the epecte vlue opertor: E V E T T T Z Z T Z E Z I the trget correltion is negtive, eine vriles n V s: T Z 0, N 3

4 V T Z 0, N The correltion coeicient etween vriles n V is: E V E T T T Z Z T Z E Z Let T =, V e the trnspose o column rnom vector which hs ivrite stnr norml istriution see, or emple, Csell n Berger 990] given 3: u, v Ep u u v v / 3 Where Ep is the eponentil unction n is the correltion prmeter or rnom norml vriles n V. Let T =, e the trnspose o column vector, whose components n, ollow the speciie mrginl PD given 4: ; 4 ; 4 where i, i =, is column vector o t prmeters. or emple, or the gmm T i mil, t=, n, ; i n i i i re the scle n shpe prmeters respectivel o the gmm istriution. The corresponing CDs or vriles n re given equtions 5: ; t 5 t; 0 ; t 5 t ; 0 Vriles n cn e otine rom vriles n V with the element-wise oneto-one trnsormtion 6: u v 6 6 4

5 5 where r stns or the stnr norml CD given 7. r t t Ep r 0 / / 7 or the genertion o rnom smples or vriles n it is not necessr to speci their joint istriution, ollowing Morn s 7] metho or otining ivrite gmm istriution, Cru_Mein n lr_góme 007 present the ivrite ensit unction or n, given : ; ;, v u v u Ep 8 where u n v re oservtions o vriles: n V which re eine s the inverse trnsormtion or 6 n 6 respectivel. 4. Otining trget correltion coeicient A trget correltion mong rnom vriles n cn e otine selecting n pproprite vlue etween vriles n V. Correltion is: V E / A secon-orer Tlor series pproimtion or trnsormtion 6 t =0 will give n pproimte reltionship etween n. 3 ' ] ] 0] 0] ] 0] ' ] 0] 0 9 Then, V E E

6 E V E V E V where n re the mein n men, respectivel, or vrile Z=, ;, n re eine in eqution 9. As E V =0, E V =0 n E V, it ollows tht: 0 Epression 0 ss tht epens on the smmetr o the PDs 4 n 4, i one PD is smmetric the irst prouct in 0 isppers. Tle shows tht the seconorer Tlor series pproimtion or the correltion coeicient 0 is not ccurte or pproimting the correltion coeicient or ivrite gmm istriution. The pproimtion or the correltion coeicient or ivrite et istriution is even worse. Tle. Tlor pproimtion o secon-orer or the correltion coeicient r r T r r T Correltion coeicient or the ivrite norml Men o the correltion coeicient or two gmm s.5, 3 or 000 simultions o smples o 00 oservtions n its Tlor pproimtion r T. Men o the correltion coeicient or gmm.5, 3 n gmm 5, 5 or 000 simultions o smples o 00 oservtions n its Tlor pproimtion r T. 6

7 Tle. Chrcteristics o the correltion coeicient or some ivrite istriutions. Prmeter Norml mrginls niorm mrginls Two gmm s.5, 3 Men r s r s Men r s r s Men r s r s Men o the correltion coeicient or 000 smples o sie n=00 tnr evition o the correltion coeicient or 000 smples o sie n=00 Tle n tle 3 give the correltion men r s or 000 simultions o smples o 00 oservtions or ierent vlues o correltion or the norml ivrite smples or the ollowing our ivrite istriutions: Two Gmm s.5, 3, Gmm.5, 3 n Gmm 0, 3, 3 Two Bet s 0., 0. n 4 Bet 0., 0. n Bet 5, 5. The PD or the gmm istriution is given in n the PD or the et istriution is eine in. 7

8 Z, Ep / Z, B, The trget correltion coeicient etween rnom vriles n cn e otine with the help o tle similr to tle. 5. Emple uppose tht smple rom three vriles, n 3 with Gmm istriutions with prmeters.5, 3,.5, 3 n 0, 3 n correltion mtri C is neee: 3 C= or this correltion mtri, we require to eine three stnr norml vriles:, V n W. = T + Z + Z V = T - Z + Z 3 W = T 3 + Z + Z 3 Vrile Z will generte correltion coeicient equl to -0.4 or n ; vrile Z correltion coeicient equl to 0. etween n 3. inll, Z 3 will generte correltion coeicient equl to 0.55 or vriles n 3. sing tles n 3, it is possile to select our or ive points in the vicinit o the trget correltion, terwrs regression cn e it in orer to estimte the correltion coeicient tht gives the trget correltion, in this w correltion o is selecte or =-0.4; correltion 8

9 o or mens tht: =0. n correltion coeicient o or Z N 0, 0.639, Z N 0, n Z 3 N 0, Thereore: T N 0, , T N 0, n T 0, N =0.55, this Tle 3. Correltion coeicients chrcteristics or some ivrite istriutions. Prmeter Gmm.5, 3 n Gmm 0, 3 Two Bet s 0., 0. Bet 0., 0. n Bet5, 5 Men r r Men r r Men r r Men o the correltion coeicient or 000 smples o sie n=00 tnr evition o the correltion coeicient or 000 smples o sie n=00 9

10 Vriles, V n W re otine with the lst vriles. inll trnsormtion 6, 6 llows to otin vriles, n 3. In orer to ssess the lgorithm perormnce, 000 simultions o smples o 00 oservtions or these vriles were conucte, proviing the stnr evition in prenthesis n the verge o the correltion coeicient presente in mtri, C : 3 C = Concluing remrks A simple metho or generting positivel or negtivel correlte rnom vriles with known mrginl istriutions n speciie correltion ws presente. This metho, which uses the populr technique o introucing shre rnom vriles, genertes speciie correltion coeicient etween stnr norml vriles n clirtion it is possile to otin the trget correltion etween the esire mrginl istriutions. An emple is presente to illustrte how this metho cn e pplie to otin multivrite smples with speciie correltions. Reerences Csell G. n Berger R. L ttisticl inerence Belmont A: Duur Press. Cru_Mein I. R., n lr Góme M n mili e istriuciones ivris sus plicciones en Hirologí. Agrocienci 4, Johnson M. E., n Tenenein A. 98. A ivrite istriution mil with speciie mrginls, J. Amer. ttist. Assoc. 76,

11 Long D., n Krstoowics R A mil o ivrite ensities constructe rom mrginls, J. Amer. ttist. Assoc. 90, Louks imple metho or computer genertion o ivrite et rnom vriles, J. ttist. Comput. imul. 0, Mgnussen An lgorithm or generting positivel correlte et-istriute rnom vriles with known mrginl istriutions n speciie correltion, Computtionl ttistics & Dt Anlsis, 46, Mri K. V ome contriutions to contingenc-tpe ivrite istriutions, Biometrik, 54, Michel J. R. n chucn W. R. 00. The miture pproch or simulting ivrite istriutions with speciie correltions, The Americn ttisticin, 56, Morn, P. A. P ttisticl Inerence with ivrite gmm istriutions, Biometrik, 56, Plckett R. L A clss o ivrite istriutions, J. Amer. ttist. Assoc. 60, Prk C. P Construction o rnom vectors o heterogeneous component vriles uner speciie correltion structures, Computtionl ttistics & Dt Anlsis, 46,