Bayesian Nonlinear Regression Models based on Slash Skew-t Distribution

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1 Euopea Ole Joual of Natual ad Socal Sceces 05; Vol.4, No. Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet ISSN Bayesa Nolea Regesso Models based o Slash Skew-t Dstbuto Savash Pzadeh Nahooj, Rahma Faoosh *, Nade Nematollah 3 Depatmet of Statstcs, Scece ad Reseach Bach, Islamc Azad Uvesty, Teha, Ia; School of Mathematcs, Ia Uvesty of Scece ad Techology, Namak, Teha, Ia; 3 Depatmet of Statstcs, Allameh Tabataba' Uvesty, Teha, Ia. *E-mal: faoosh@ust.ac.. Abstact Ths pape cosdes the Bayesa aalyss fo estmatg the paametes of olea egesso model whe the eo tem has a slash skew-t dstbuto. Ths model s a asymmetc olea egesso model whch s sutable fo fttg the data sets wth heavy tal ad skewess. The popetes of ths model ae deved ad a heachcal epesetato of ths model based o the stochastc epesetato of slash skew-t dstbuto s gve. Ths epesetato allows us to use Makov Cha Mote Calo method to estmate the paametes of model. To compae ths model wth othe asymmetc olea egesso models, we use codtoal pedctve odate statstc ad devace fomato, expected Akake fomato ad expected Bayesa fomato cteos, ad show the pefomace of the poposed model by a smulato study. Also a applcato of the ew model to fttg a eal data set s dscussed. Keywods: Bayesa aalyss, Nolea egesso models, Slash skew-t dstbuto, Skew slash dstbuto, Skew-t dstbuto. Itoducto Nolea egesso model usually appled whe espose vaable s a olea fucto of explaatoy vaable. Oe of the fudametal assumptos of these models s the omalty of the eo tems. But ths assumpto does ot hold whe the data ude study have skewess ad/o heavy tal. Fo solvg ths poblem some authos use heavy tal dstbutos fo eo tems such as Studet-t, logstc ad expoetal powe famly whch ae symmetc ad have heavy tal tha the omal dstbuto. See fo example Cyseos ad Vaegas (008) ad Cyseos et al. (00) ths egad. Some authos used asymmetcal dstbuto such as skew-omal dstbuto fo eo tems to ovecome the skewess popety of the data sets, fo example see Cacho et al. (00) ad Xe et al. (009 a, b). Moteego et al. (009) have show that the estmato of the paametes of a olea egesso model wth skew-omal adom eo ca be sestve to outles. To solve the effect of outles, Gaay et al. (0) ad Lachos et al. (00) cosdeed the scale mxtue of skewomal dstbuto fo adom eos. They foud the ML estmates of the model paametes ad showed that these estmates dd ot sestve to outle obsevatos whe the dstbutos of adom eos ae skew-t o skew-slash dstbuto. Also, Cacho et al. (0) developed a Bayesa estmato of the paametes of a olea egesso model wth scale mxtue of skewomal dstbuto fo ts eo tem. Recetly Faoosh et al. (03) toduced a ew slash skew-ellptcal dstbuto whch cotas may of skew ad heavy tal dstbutos such as slash skew-t dstbuto. They showed that slash skew-t dstbuto s moe skewed ad heavy tal tha skew-omal ad skew-t dstbuto. I ths pape we use slash skew-t dstbuto as the dstbuto of the adom eo of a olea egesso model. We use the stochastc epesetato of slash skew-t to fd a heachcal epesetato of the poposed model ad fd the Bayes estmate of the model Opely accessble at 30

2 Savash Pzadeh Nahooj, Rahma Faoosh, Nade Nematollah paamete by employg a Makov Cha Mote Calo (MCMC) method. Also by usg a smulato study, we show the pefomace of the poposed model to othe exstg models by applyg Codtoal Pedctve Odate (CPO) statstcs, Devace Ifomato Cteo (DIC), Expected Akake Ifomato Cteo (EAIC) ad Expected Bayesa Ifomato Cteo (EBIC). Ths pape has bee aaged as follows. I Secto, we wll befly toduce slash skew-t dstbuto ad state ts mpotat popetes. I Secto 3 we gve a ough dea of olea egesso model wth adom eos that have slash skew-t dstbuto. I Secto 4, we wll descbe Bayesa method to estmate model paametes ad toduce some measues of model selecto whch we wll use to compae the models. I Secto 5, fo compag the poposed model to othe exstg models, we coduct a smulato study ad apply a eal example. A dscusso s gve Secto 6. Slash skew-t dstbuto I ths secto we dscuss some popetes of slash skew-t dstbuto whch has bee toduced by Faoosh et al. (03). A adom vaable Y has slash skew-t (SLST) dstbuto f t ca be wtte as X Y, () q U whee s the locato paamete, s a scale paamete, q 0 s the tal paamete ad X s a skew-t adom vaable wth locato 0, scale, skewess, degee of feedom ( X ~ ST(0,,, ) ) ad U s a ufom adom vaable ( U ~ U (0,)), X ad U ae depedet. Thus, the pobablty desty fucto (pdf) of Y s gve by q y q q ( ) u g( u) F ( ( )) q 0 g uh y du y y f ( y) () Y q g(0) y ( q ) ( ) ( ) t ( ) t whee ht () ad gt () ( ) s the desty geeato fucto of t () Studet-t dstbuto ad F g (.) s cumulatve dstbuto fucto (cdf) of Studet-t dstbuto. We deote t by Y ~ SLST (,,,, q). Now, we demostate the subsequet esults of Slash skew-t dstbuto. Fst, we peset the stochastc epesetato of SLST. Ths epesetato wll be employed to smulate adom vaabley. The dcated epesetato wll also be employed to tasact Gbbs sample algothm fo heachcal epesetato of adom vaabley. Fo a poof of the followg theoems see Faoosh et al. (03). Theoem. Let Y ~ SLST(,,, q, ), the Y has the followg stochastc epesetato q q 0 ( ) q, Y U V T U V T T U V T (3) Opely accessble at 3

3 Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet whee,, ( ), V ~, U ~ U (0,), q T U V T0 ad T 0 ad T ae depedet stadad omal adom vaables. Theoem. Let Y ~ SLST(,,,, q) ) If q,, the E( Y) c ; q ) If q,, the Va( Y ) ( c ) ; q whee ( ) q adc q ( ). The slash skew-t olea egesso model I ths secto, we assume that the adom eo of a olea egesso model s dstbuted as SLST dstbuto. So, ou olea egesso s gve by Y ( β, x ),,, (4) whee Y ae esposes, (.) s a jectve ad twce cotuously dffeetable fucto wth espect to the paamete vecto β (,, ) T p, x s a vecto of explaatoy vaable values ad ( ) q the adom eos ~ SLST ( c,,,, q) fo q, ad c, whch q ( ) coespods to egesso model whee the eo dstbuto has mea zeo. Fom Theoem., the followg esult ca be deduced q EY ( ) ( β, x), VaY ( ) ( c), q,. q So, we have Y ~ SLST( ( β, x ) c,,, q, ) fo,,. T T Fom () ad (4) the log-lkelhood fucto foθ (,,, q, ), gve the obseved sample y ( y,, y ) T s gve by ( ) log qqlog log log log( ) log( S ) y q ( ) u q 0 whee S u Fg( uh( y)) du y. I ths pape we wll use the Bayesa method to estmate model paametes. Ths method has bee bult based o MCMC algothms that would acheve posteo feece fo the paametes. Opely accessble at 3 (5)

4 Savash Pzadeh Nahooj, Rahma Faoosh, Nade Nematollah Bayesa model paamete estmato I ths secto we use Bayesa aalyss to estmate the paametes of the olea egesso model wth SLST adom eo (SLST-NLM). We use MCMC techques. Fom theoem. we ca obta the followg heachcal epesetato whch helps us to wte BUGS cods smply. q β x q, ~ (, ) (, ), Y U u, V v, T t ~ N( (, ) t, u v ), T U u V v TN c u v c (6) U ~ U(0,), V ~ gamma(, ), whee TN ( a, b ) (, s) deote the tucated omal dstbuto N ( a, b ) o ( s, ). Let y ( y,, y ) T, x ( x,, x ) T, t ( t,, t ) T, u ( u,, u ) T ad v ( v,, v ) T, so the complete lkelhood fucto of θ based o ( y,x,t,u ) s gve by q β x q t c u v c U u L( θ y, x, t, u, v) [ ( y ; (, ) t, u v ) ( v, ) ( ;, )I(, ) ( ;0,) ] To coduct a Bayesa aalyss, we eed to deteme the po dstbuto fo all ukow paametes,,, ad q. Because thee s o pevous fomato about ukow paametes, we cosde cojugate po dstbutos to these paametes. Fo the sake of avodg mpope posteo, we select pope pos wth kow hype-paametes. Theefoe, we choose a omal po dstbuto fo the elemets of wth pdf gve by ( j ). Fo paamete, omal po dstbuto wth mea ad vaace has bee selected. The vese Gamma po dstbuto s cosdeed fo,.e., IGamma(, ). Fo degees of feedom paamete ad tal paamete q we cosde tucated expoetal po dstbutos o the teval (, ) whch ae gve by exp( ) I(, ), q exp( ) I(, ). These tucato pots wee chose to pesuade that the vaace of y be fte. We assume that po dstbutos ae depedet. Theefoe, the complete po ca be wtte as follows p ( θ ) ( j;, ) ( ;, ) (, ) ( q) ( ). j j (7) j By mxg the lkelhood fucto ad the po dstbuto (7), the jot posteo ca be acheved as follows Opely accessble at 33

5 Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet q q β x ( θ, tuv,, y, x) [ ( y ; (, ) t, u v ) ( t ; c, u v ) ( v, ) I( c, ) U( u;0,) ] ( θ). Fo feg ad estmatg the paamete of the above dstbuto, MCMC method such as Gbbs sample ca be used. The full codtoal dstbutos whch eed the Gbbs sample ca be wtte as follows whee q T T T β,,,, q, y, u,v TN ( c, u v M )I( c, ), =,,; whee ( (, ) ) T y β x c, M T, β,, q,, y, u,v, t~ N(, ), q u vt, q β,,, q, y, u,v, t ~ IGamma ( ), ( u ( (, ) ) ) v y β x c q whee uvt( (, )) y β x, q p β x,, q,, yu,v,, t ( ; μ,d ) ( (, );( yt), u v ) T whee μβ (,, ), D dagoal(,, ) p p. Fo geeatg samples fom we use Metopols-Hastg algothm as follows a) We choose tal value fo β0 at the fst step b) At the ( )thstep espectvely. The * β ad U ae geeated fom ( ) β s gve by * * ( ) ( ) β f U ( β, β ) β ( ) β othewse N β S ad U (0,) ( ) ( p, ) * q * ( ) ( ) ( ) ( ) ( ) * ( ) p( β ; μ ) ( (, );( ), ) β,d β x y t u v q ( ) ( ) ( ) ( ) ( ) ( ) ( ) p β μ β,d β x y t u v ( β, β ). ( ; ) ( (, );( ), ) dstbutos, The matx S s selected afte geeatg ad tal MCMC cha whch would help to acheve a good covegece. Smla techque used to update ad. Fo completg Gbbs sample scheme, we eed the full codtoal posteo dstbutos of u, v, ad q. Fo each elemet ofu, the pdf s: Opely accessble at 34

6 Savash Pzadeh Nahooj, Rahma Faoosh, Nade Nematollah q q uv ( y ( β, x) t) (u β,,, q,, y, v, t) u exp ( ( t c) ) (0, fo,,. Fo each elemet of v, the pdf s: v ( (, ) ) q y β x t (v,,,,,,, ) exp ( β qy u t v u ( t c) ) (8) (9) fo,,. So we ca suppose that v β,,, q,, y, ut, has Gamma(, w ), whee v q ( y ( β, x) t) w u ( ( t c) ) fo,,. The codtoal posteo desty of s: ( β,,, q, y, u, v, t) ( ) ( ( )) (0) exp ( v l ) (, ). v The codtoal posteo desty of q s: ( q,,,,,,, ) u q q exp (, ), u vz q β y u v t () ( y ( β, x) t) whee z ( ( t c) ). Note that (8), (9), (0), () does ot have a closed fom, but a Metopols-Hastg algothm ca be embedded the MCMC scheme to obta daws ofu, v, ad q. Fally, sce (, ) has a oe to oe coespodece to (, ), the ad ca be calculate ad. Model selecto method Fo compag seveal models ad fdg the best model that fts to the gve data, thee ae seveal cteos fo selectg the best model. I ths secto we toduce some of these cteos to compae SLST-NLM to othe exstg models. The fst cteo s based o Codtoal Pedctve Odate (CPO) statstc whch s oe of the applcable model compaso ctea ad ecommeded by Gelfad et al. (99). Let be the ( ) full data ad deote the data wthout obsevato. I SLST-NLM model fo a obseved data, we have q y q q ( ) g( y ) u g( u) F ( ( )) q 0 g uh y du y, Opely accessble at 35

7 Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet whee ( ) t gt () ( ) () ( ) ( ) s the desty geeato fucto of Studet-t dstbuto, F g (.) s cdf of Studet-t dstbuto ad (, β x) c. The posteo desty of gve ( ) ( ) s deoted by ( ). Fo the obsevato, the CPO s gve by ( ) ( ) CPO g( y ) ( ) d d,,,. g( y ) I SLST model we could ot fd a closed fom fo CPO statstcs. Theefoe, a Mote Calo estmate of CPO ca be obtaed by usg a MCMC sample fom the posteo dstbuto ( ). Let, be a sample of sze Q of ( ) afte the bu. A Mote Calo estmate of, Q CPO (Che et al., 000) s gve by Q CPO. Q t g( y t) Now a statstc based o CPO 's s B log( CPO ). Sce lage value of CPO show a bette ft of the model, the lage value of B mply a bette ft of the model too. We use othe ctea, cludg the Devace Ifomato Cteo (DIC) toduce by Spegelhat et al. (00), the Expected Akake Ifomato Cteo (EAIC) (Books, 00) ad Expected Bayesa Ifomato Cteo (EBIC) (Cal ad Lous, 00). These ctea ca by appoxmated by the MCMC output as follows: estmate of DIC s DIC D D, whee D( ) log[ g( y )], Q D( t ) ad D ca be estmated by D t Q Q Q Q Q Q D D,,,,. t t t t t t t t q t t Q Q Q Q Q The EAIC ad EBIC ca be estmated by meas of EAIC D #( l) ad EBIC D #( l)log( ), whee #( l ) s the umbe of model paamete. Smulato ad eal data aalyses I ths secto, we vestgate popetes of SLST-NLM model by a smulato study ad the pefomace of ths model wth othe models such as olea egesso model based o skew-t adom eo (ST-NLM) ad olea egesso model based o skew-slash adom eo (SSL- NLM) toduced by Cacho et al. (0). Also a applcato to a eal data set s dscussed. Fequetst popetes At fst by usg smulated SLST-NLM data we study the fequtst popetes of the paamete estmates. I ths smulato we study the behavo of Bayesa estmates, based o fequetst Mea Squaed Eo (MSE) ad the fequetst Mea. We mplemet the smulato study by the followg logstc model: Y,,,50, exp( 3x ) (3) whee the vaable x has a age betwee to 50. Fo compag ou poposed model wth othe models, we use model paametes that have bee appled by Cacho et al. (0), thus we cosde the values, 30, 5, 3 0.7,, 4, q 3ad 4. The eo tems ae Opely accessble at 36

8 Savash Pzadeh Nahooj, Rahma Faoosh, Nade Nematollah depedet wth SLST( c,,,, q). I ths smulato we cosde the umbe of teatos equal to 00 ad the ftted olea egesso model wth thee dffeet dstbuto fo adom eos. These adom eos ae skew-t, skew slash ad slash skew-t. Fo mplemet the Gbbs sample the followg depedet po has bee cosdeed. Fo,, 3, (0,000) N, N(0,000), IG(0.0,0.0), ~ exp(0.)i(, ) ad q ~exp(0.)i(, ). Fo each geeated data sets we smulate two cha of sze fo each paamete, fo bug, we exclude the fst 0000 teatos. Also fo deceasg autocoelato, we cosde th sze 0, hece the effectve sample sze ae Fo each sample the posteo mea of the paamete ad the DIC, EAIC, BIC ad B ae calculated. The esult of smulato study fo ST, SSL ad SLST adom eo dstbutos wth estmated MSE ad Mea ae gve Table. I ths table the tue values have bee put to the 00 paethess, MC Mea has bee calculated accodg to ˆ 00 j kj ad MC MSE whch has bee 00 detemed by ˆ ( ) 00 j kj s the empcal mea squaed eo. I the dcated fomula ˆk k s the estmated paamete of tue paamete k. The values Table dcated that the MC MSE fo model paametes ae smalle SLST model tha ST ad SSL models. Thus SLST-NLM would be bette ftted athe tha ST-NLM ad SSL-NLM fo osymmetcal data. Ths statemet has bee appoved by the values Table. I ths table the athmetc aveage of the measuemets fo compag the models (DIC, EAIC, EBIC, B) ae gve. These measuemets would be smalle SLST-NLM tha the metoed models. Thus, ths patcula model has bette ftted tha the competto models. Table Mote Calo esults based o 00 smulated samples Paamete ST-NLM SSL-NLM SLST-NLM MC Mea MC MSE MC Mea MC MSE MC Mea MC MSE (30) (5) (0.7) () (4) q (3) ( 4) Table Compaso betwee ST-NLM, SSL-NLM ad SLST-NLM fttg by usg dffeet Bayesa ctea Model Cteo MC DIC MC EAIC MC EBIC MC B ST SSL SLST Opely accessble at 37

9 Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet Real Data We use a Bayesa aalyss of the ultasoc calbato data, whch has bee descbed L et al. (007) ad Lachos et. al. (0), to vestgate ou method. These data ae geeated fom Natoal sttutes of Stadads ad Techology (NIST) study by Chwut (979) volvg ultasoc calbato, whee the espose vaable y ad the pedcto vaable s metal dstace x. The data cossts of 4 obsevatos ad s avalable feely the R package Nlsms. We cosde exp(.. x ) d Y, SLST ( c,,, q, ). (4) 3x We wll use the ST, SSL ad SLST dstbutos to make a compaso. We cosde the followg depedet pos to apply the Gbbs sample algothm: Fo,, 3, (0,000) N, N(0,000), IG(0.0,0.0), ~ exp(0.)i(, ) fo skew-t model ad q ~ exp(0.)i(, ) fo SSL model ad SLST model. By cosdeg the above pos, we geeate two paallel depedet us of the Gbbs sample cha wth sze 00,000 fo each paamete, fo bug, we exclude the fst 0000 teatos. Also to avod coelato poblem we cosde th sze 0, hece, the effectve sample sze ae Fo each sample the posteo mea of the paamete ad the DIC, EAIC, BIC ad B ae calculated. Posteo mea ad stadad devato of paamete ude the dstbutos ST, SSL ad SLST have bee demostated Table 3. Also, Table 4 the values of model selecto ctea ae gve. Fom ths table we see that the SLST-NLM has a bette ft to ths data tha ST-NLM ad SSL-NLM. Table 3 Ultasoc calbato data set. Summay esult fom the posteo dstbuto, Mea ad Stadad Devato (SD) fo paamete ude ST, SSL ad SLST dstbutos paamete ST-NLM SSL-NLM SLST-NLM Mea SD Mea SD Mea SD e e e q Table 4 Ultasoc calbato data set. Compaso betwee ST-NLM, SSL-NLM ad SLST- NLM by usg dffeet Bayesa ctea. Cteo ST-NLM SSL-NLM SLST-NLM B DIC EAIC EBIC Opely accessble at 38

10 Savash Pzadeh Nahooj, Rahma Faoosh, Nade Nematollah Cocluso I ths pape, we suggest a ew olea egesso model based o a asymmetc dstbuto of adom eo of the model. We employ a MCMC method to obta the Bayes estmate of the model paametes. To show the pefomace of the ew model to the exstg models, we pefom a smulato study. By usg B, DIC, EAIC ad EBIC ctea, we see that SLST-NLM has bette ft tha ST- NLM ad SSL-NLM fo smulated data ad a eal data set. Refeeces Books, S.P. (00). Dscusso o the pape by Spegelhalte, Best, Cal, ad va de Lde. Joual of the Royal Statstcal Socety. Sees B. 64 (3): Cacho, V.C., Lachos, V.H., & Otega, E.M.M. (00). A olea egesso model wth skewomal eos. Statstcal Papes. 5: Cacho, V. C., Dey, D. K., Lachos, V. H., & Adade, M. G. (0). Bayesa olea egesso models wth scale mxtues of skew-omal dstbutos: Estmato ad case fluece dagostcs. Computatoal Statstcs ad Data Aalyss. 55: Cal, B. P., & Lous, T. A. (00). Bayes ad Empcal Bayes Methods fo Data Aalyss, d ed. New Yok, Chapma & Hall. Che M. H., Shao Q. M., & Ibahm, J. (000). Mote Calo Methods Bayesa Computato. New Yok, Spge-Velag. Chwut D. (979). Ultasoc Refeece Block Study. Avalable fom: Cyseos, F. J. A., & Vaegas, L. H. (008). Resduals ad the statstcal popetes symmetcal olea models. Statstcs ad Pobablty Lettes. 78: Cyseos, F., Codeo, G. M., & Cyseos, A. (00). Coected maxmum lkelhood estmatos heteoscedastc symmetc olea models. Joual of Statstcal Computato ad Smulato. 80: Faoosh, R., Nematollah, N., Rahamae. Z., & Hajajab, A. (03). A famly of skew-slash dstbutos ad estmato of ts paametes va a EM Algothm. Joual of the Iaa Statstcal Socety. ():7-9. Gaay, A.M, Lachos, V.H., & Abato-Valle, C. A. (0). Nolea egesso models based o scale mxtues of skew-omal dstbutos. Joual of the Koea Statstcal Socety. 40:5-4. Gelfad, A.E, Dey, D. K, & Chag, H. (99). Model detemato usg pedctve dstbutos wth mplemetato va samplg-based methods I: Bayesa Statstcs. New Yok, Oxfod Uv Pess. Lachos, V.H., Ghosh, P., & Aellao-Valle, R.B. (00). Lkelhood based feece fo skewomal depedet lea mxed models. Statstca Sca. 0: Lachos, V.H., Badyopadhyay, D., & Gaay, A.M. (0) Heteoscedastc olea egesso model based o scale mxtues of skew-omal dstbutos. Statstcs ad Pobablty Lettes. 8:08-7. L, T. I., Lee, J. C., & Hseh, W. J. (007). Robust mxtue modelg usg the skew-t dstbuto. Statstcs ad Computg. 7:8-9. Moteego, L. C., Lachos, V., & Bolfae, H. (009). Local fluece aalyss of skew-omal lea mxed models. Commucatos Statstcs. Theoy ad Methods. 38: Opely accessble at 39

11 Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet Spegelhalte, D. J., Best, N. G., Cal, B. P., & Va De Lde, A. (00). Bayesa measues of model complexty ad ft. Joual of the Royal Statstcal Socety. Sees B. 64(4): Xe, F., L, J., & We, B. (009a). Dagostcs fo skew-omal olea egesso models wth AR./ eos. Computatoal Statstcs ad Data Aalyss. 53(): Xe, F. C, We, B. C., &L, J. G., (009b) Homogeety dagostcs fo skew-omal olea egesso models. Statstcs ad Pobablty Lettes. 79:8-87. Opely accessble at 40