An Extended Mixture Inverse Gaussian Distribution

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1 Avlble ole t htt://wwwssstjscssructh Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty A Eteded Mture Iverse Guss Dstrbuto Chookt Pudrommrt * Fculty o Scece d Techology, Su Sudh Rjbht Uversty U-thog Nok Rod, Dust, Bgkok 10300, Thld Corresodg uthor E-ml: *chooktu@ssructh Abstrct: Ths er rooses eted mture verse Guss (EMIG) dstrbuto whch s med betwee the verse Guss dstrbuto d the legth bsed verse Guss (LBIG) dstrbuto The Brbum- Suders (BS) dstrbuto d LBIG dstrbuto re reseted s secl cses o the EMIG dstrbuto The roertes o ths dstrbuto re dscussed whch clude the shes o the robblty desty uctos, dstrbuto uctos survvl uctos d hzrd rte uctos, me d vrce The EMIG hs two rmeters d t s show tht mmum lkelhood estmto (MLE) c be obted by solvg equto A lcto o the model to rel dt set s lyzed usg the ew dstrbuto, whch shows tht the EMIG dstrbuto c be used qute eectvely lyzg rel dt by usg Akke s ormto crtero (AIC) sttstcs d goodess o t tests Keywords: Mture verse Guss dstrbuto, Legth bsed verse Guss dstrbuto, Brbum- Suders dstrbuto 1 Itroducto The verse Guss (IG) dstrbuto s cotuous o-egtve rdom vrble whch s rght skewed dstrbuto d t lys mortt role relblty lyss Jorgese et l (1991), Gut d Akm (1995), d Heze et l (00) studed the IG dstrbuto Let 1 IG( b, ), e 1 hs IG dstrbuto wth the rmeters > 0, > 0 d ts robblty desty ucto (d) d dstrbuto ucto re gve by 3/ 1 1 b 1 1;, b e, b 1 b F ( 1 1;, b) Φ b Φ b e, b 1 b 1 where Φ () s the dstrbuto ucto o the stdrd orml dstrbuto The legth bsed verse Guss (LBIG) dstrbuto s weghted dstrbuto by me o IG dstrbuto whch hs receved cosderble tteto due to ts vrous lctos deret bomedcl res, such s mly hstory o dseses, erly detecto o dseses, ltecy erods o AIDS etc (Akm d Gut, 199; Gut d Akm,1995) The LBIG dstrbuto ws studed by Jorgese et l (1991), Akm d Gut (199) d Gut d Akm (1995) Let LBIG( b, ), the hs LBIG dstrbuto wth the rmeters > 0, b > 0 d ts d d dstrbuto ucto re gve s ollows b 1/ 1 b 1 b ;, b e, b b b b F ( ;, b) Φ e Φ b b The mture verse Guss (MIG) dstrbuto, lso kow s the weghted verse Guss dstrbuto or the three-rmeters geerlzed verse Guss dstrbuto or, whch s med betwee the IG dstrbuto d the LBIG dstrbuto whch ws studed by Jorgese et l (1991), d Gut d Akm, (1995) Let MIG(, b, ), the hs MIG dstrbuto wth the rmeters > 0, b > 0 d 0 1, whch d s gve by ;, b, ;, b 1 ;, b, 1 where 1 s rdom vrble o IG dstrbuto d s rdom vrble o LBIG dstrbuto The, the d o MIG c be wrtte the orm: 1 b b 1 b 3/ 1/ 1 b b 1 b 3/ 1/ 1 b e, 0, b Vol03 No1 DOI: /ssstj0161 1

2 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty > 0 d 0 1 For MIG dstrbuto hs severl terestg secl cses I rtculr, the MIG dstrbuto becomes the LBIG dstrbuto whe, the IG dstrbuto whe d Brbum-Suders (BS) dstrbuto (Gut, 011) whe Although, the MIG dstrbuto hs my desrble roertes lctos, rmeter estmto my stll hve roblems whch hve metoed tht dg the ecet tl guesses d solvg the o-ler equtos smulteously re o-trvl ssues (Jorgese et l,1991; Gut d Akm,1995) Thereore, order to solve such roblems, ew weght rmeter s cosdered We roose eteded mture verse Guss dstrbuto whch s obted by ddg ew weght rmeter to the mture verse Guss dstrbuto I ths er, we reset eteded mture verse Guss (EMIG) dstrbuto Severl roertes o the ew dstrbuto cludg the robblty desty uctos, dstrbuto uctos, survvl uctos, hzrd rte uctos, cumults d momets re rovded I ddto, we use mmum lkelhood estmto or rmeter estmto d reset the comrso lyss betwee the eteded mture verse Guss dstrbutos bsed o rel dt set usg Akke s Iormto Crtero (AIC) sttstcs d goodess o t tests where > 0, b 0 05 Mterl d Methods, b, 1 > 0 Proo: From Deto 1, the d o the EMIG dstrbuto c be obted by 3/ b 1 b e 1 b b where > 0 d b 1/ 1 1 b 1 b e 1 b b 3/ 1/ 1 b b 1 b e 1b b Corollry 1 I 0 the EMIG dstrbuto reduces to LBIG dstrbuto wth rmeter = 0 d b > 0 wth d gve by 1/ 1 b 1 e b b Corollry I reduces to BS dstrbuto wth rmeter =1 d b > 0 wth d gve by 1the EMIG dstrbuto 3/ 1/ 1 b b 1 b e b b Some rmeters o the EMIG dstrbuto d ther robblty desty uctos re rovded Fgure 1 I ths rt, we troduce the deto o the eteded mture verse Guss dstrbuto deoted by EMIG( b, ) We beg wth geerl deto o the EMIG dstrbuto whch wll cosequetly revel ts robblty desty ucto Deto 1 Let 1 d be deedet rdom vrbles such tht 1 IG( b, ) d LBIG( b, ) The the ew rdom vrble s sd to hve EMIG dstrbuto wth rmeter > 0 d b > 0 the d o s deed by 1 ;, b ;, b ;, b Theorem 1 Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The d o s gve by 1 b b 1b 3/ 1/ 1 b e, 0 b Fgure 1 The robblty desty uctos o the EMIG dstrbuto or some vlues o rmeters () d (b) b 05

3 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty Theorem Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The dstrbuto ucto o s gve by b 1 F Φ b 1 b e 1 Φ, 0 b where Φ () s the dstrbuto ucto o the stdrd orml dstrbuto Proo: Let s cotuous o-egtve rdom vrble, the the dstrbuto ucto o s gve by 0 F t dt I the dstrbuto ucto o dstrbuto, whch s eressed by F s EMIG 3/ b 1 t b e dt 1 0 b t b t 1/ 1 1 b 1 t b e dt 1 0 b t b t Usg the dstrbuto ucto o IG d LBIG dstrbutos, the dstrbuto ucto o EMIG becomes b b F( ) Φ Φ e 1 b b 1 b b Φ e Φ 1 b b b 1 b Φ e 1 Φ b 1 b The dstrbuto uctos o the EMIG wth some rmeter vlues re show Fgure Fgure Dstrbuto uctos o EMIG or some vlues o rmeters: () d (b) b 05 Theorem 3 Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The survvl ucto o s gve by b 1 S 1 Φ b 1 b e 1 Φ b Proo: Let s cotuous rdom vrble wth dstrbuto ucto F( ) o the tervl [0, ) the the survvl ucto s deed by S t dt 1 F From the dstrbuto ucto o EMIG Theorem, the the survvl ucto o s gve by b 1 b S 1 Φ e 1 Φ b 1 b b 1 b 1 Φ e 1 Φ b 1 b Theorem 4 Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The hzrd rte ucto o c be wrtte s 3

4 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty h 3/ 1/ 1 b b 1 b e 1b b b 1 b 1 Φ e 1 Φ b 1 b Proo: Let s cotuous o-egtve rdom vrble wth the robblty desty ucto d survvl ucto the the hzrd rte ucto c be deed s h, S ( ) By usg Theorem 1 d Theorem 3, we hve S 3/ 1/ 1 b b 1 b b e b1 b h b 1 b 1 Φ e 1 Φ b 1 b Some Survvl uctos d hzrd rte uctos lots o the EMIG dstrbuto wth some rmeter vlues re dslyed Fgure 3 d Fgure 4 Fgure 4 Hzrd rte uctos o the EMIG dstrbuto or some vlues o rmeters: () d (b) b 05 3 Results 31 Theoretcl results 311 Sttstcl Proertes o the EMIG The chrcterstc ucto, cumults, momets, me d vrce o EMIG dstrbuto re studed ths secto Theorem 5 Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The chrcterstc ucto o c be wrtte the orm e 1 1bt 1 1bt t 1 bt 1 Proo: The chrcterstc ucto o rdom vrble s deed by t t E e The dstrbuto o s EMIG dstrbuto, the chrcterstc ucto tkes the orm Fgure 3 Survvl uctos o the EMIG dstrbuto or some vlues o rmeters: () d (b) b 05 3/ 1/ 1 b b t et 0 1 b 1 b e b d 4

5 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty 3/ 1 b 1 b e 1 b 0 t d b 1/ 1 b 1 b e t d 1b b 0 1/ b e 3/ 1 b/ e t d 1 b 0 e 1/ 1 b/ e t d b 1 b 0 From the Tble o tegrls, seres, d roducts by Grdshtey d Ryzhk 007, 369), the ormuls re tke rom the ollowg orm: 1/ q d q e / 1 e, q 0 where Re 0, Re q 0 The chrcterstc ucto c become 1/ b e t e 1 bt 1 b e b e 1 bt b 1 1 bt e 1 1bt 1 1bt 1 bt 1 Theorem 6 Let be rdom vrble o the EMIG dstrbuto wth rmeters d b The cumult geertg ucto o c be gve by 1/ 1 bt 1/ K t log e 1 1 bt 1 1 Proo: The cumult geertg ucto o rdom vrble s deed s K t log t K t e 1 1 bt 1 1bt log 1 bt 1 1 1bt log e 1 1 bt ( 1) 1bt 1/ 1 bt log e 1 1 bt 1 1 Recll tht Mclur seres, s deed s For 4, we hve 0! O 1!! 3! 4! Thus, the Mclur seres o d 1 1/ 1 1/, or 1, c be wrtte the ollowg orm; , / O / O Net, we cosder 1 bt 1/ d 1 bt 1/ term o 1 1/ d 1 1/ resectvely, the the cumult geertg ucto o becomes log t log 1 bt bt bt bt bt bt bt bt log bt bt bt bt, 8 bt bt bt bt 3 4 d usg the eso log 1 1 O, we obt 3 4 log t bt bt bt bt bt 3bt bt bt bt 4 1 bt bt bt bt bt bt bt

6 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty bt bt 1 bt 1 1 bt 3 1 1! 1 1! 3 bt ! bt ! log,! m From m1 K t t t Ot 1 the rst our cumults re gve s ollows: b, 3 1 b, b, b, d the rw momets re relted to the cumults by the ollowg ormul: E ( ) 1, E ( ), 1 E ( ) 3, E ( ) Substtutg the cumults the equto bove, we c d me d vrce re gve by 1 E b, Vr b 1 1 Some me d vrce lots o the EMIG dstrbuto wth some rmeter vlues re dslyed Fgure 5 Fgure 5 Me d Vrce o the EMIG dstrbuto or derece vlues o ( b, ) 31 Prmeters estmto The estmto o rmeters or the eteded mture verse Guss dstrbuto v the Mmum Lkelhood Estmto (MLE) rocedurethe lkelhood ucto o the dstrbuto wth rmeters d b s gve by L, b 3/ 1/ 1 b b 1b 1 1 b e b The log-lkelhood ucto c be wrtte s, b log L, b logb b log 1 b 1 b 1 1 log 3/ 1/ b b 1 6

7 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty log b b log 1 b 1 b log b b 1/ 3/ logb b log 1 b 1 3 log b b log log b 1 By tkg rst rtl dervtves o the loglkelhood ucto ech wth resect to d b, we obt the equtos; b 1 b, b b, b b b , b b b b b b b ˆ, ˆ b The MLE solutos o c be obted by equtg the bove equtos to zero d solvg the resultg equtos smulteously usg umercl rocedure wth the Newto-Rhso method R (R Develomet Core Tem, 015) 3 Numercl Result I ths secto, the EMIG dstrbuto s led o rel dt set whch s tke rom Gut d Kudu (009) whch rereset l emto mrks mthemtcs o the slow ce studets 003, The dt re gve Tble 1 Tble 1 The l emto mrks mthemtcs o the slow ce studets dstrbutos re show Tble Net, we comute Aderso-Drg (AD) test or goodess o t dstrbuto to these dt whch re show Tble 3 Tble MLE o the model rmeters or the mrks dt set Estmte Dstrbuto rmeters AIC EMIG MIG IG LBIG BS Dstrbuto Tble 3 Goodess o t test or the mrks dt set by usg Aderso-Drlg test Dstrbuto AD test Sttstc P-vlue EMIG MIG IG LBIG BS The P-vlue o Aderso-Drlg test s show tht the EMIG dstrbuto erorms better th MIG, IG, LBIG d BS dstrbutos d the vlue o AIC sttstc s show tht the EMIG dstrbuto s the best t or ths dt The tted robblty desty uctos d the observed hstogrms re gve Fgure 6 b Estmte rmeters AIC EMIG MIG IG LBIG BS b For ths dt, we hve tht me=589 d vrce= 345 We hve tted the EMIG, MIG, IG, LBIG d BS dstrbutos to ths dt set by usg mmum lkelhood estmto We obt the estmtes rmeters d AIC sttstcs or ll 7

8 Avlble ole t htt://wwwssstjscssructh Vol03No1 Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty Fgure 6 Hstogrm o the mrks dt 4 Cocluso I ths er, we hve reseted the EMIG dstrbuto whch s gve by ddg ew weght rmeter to the MIG dstrbuto The BS d LBIG dstrbuto re some secl cse o EMIG We mly studed the sttstcl roertes o EMIG dstrbuto such s d, desty ucto, survvl ucto, hzrd rte ucto, cumults, the rst our momets, me d vrce We dscuss the estmto o the rmeters by mmum lkelhood Flly, we comre the t o the EMIG dstrbuto wth MIG, IG, LBIG d the BS dstrbutos by usg mrks dt The AIC sttstcs dctes tht the EMIG s best t or rel dt Gut, RD, & Kudu, D (009) A ew clss o weghted eoetl dstrbutos, Sttstcs: A Jourl o Theoretcl d Aled Sttstcs, 43(6), , do:101080/ Gut, RC, & Kudu, D (011) Weghted verse Guss verstle le tme model Jourl o Aled Sttstcs, 38(1), , do: / Heze, N, & Klr, B (00) Goodess-o-t tests or the verse Guss dstrbuto bsed o the emrcl llce trsorm Als o the Isttute o Sttstcl Mthemtcs, 54(), , do:10103/a: Jorgese B, Seshdr, V, & Whtmore, GA (1991) O the mture o the verse Guss dstrbuto wth ts comlemetry recrocl Scdv Jourl o Sttstcs, 18(1), R Develomet Core Tem, (015) A lguge d evromet or sttstcl comutg R oudto or sttstcl comutg, Ve, Austr Vo Alve, WH, (1964) Relblty Egeerg New Jersey: retce-hll Ic Ackowledgemet The uthor would lke to thk the edtor d reeree or ther useul commets d suggestos whch cosderbly mroved ths rtcle Reereces Akm, O, & Gut, RC (199) A comrso o vrous estmtors o verse Guss dstrbuto Jourl o Sttstcl Comutto d Smulto,40,1-,71-81 do:101080/ Blkrsh, N, Lev, V, Shuez, A & E Cbrer, (009) Mture verse Guss dstrbutos d ts trsormtos, momets d lctos Sttstcs, 43(1), , do: / Grdshtey, IS, & Ryzhk, IM (007) Tble o tegrls, seres, d roducts (7th ed) MA, USA: Elsever Acdemc Press Gut, RC, & Akm, HO (1995) O the relblty studes o weghted verse Guss model Jourl o Sttstcl Plg d Ierece, 48 (1), 69-83, do: / (94)00148-O 8