Study on Bayes Semiparametric Regression

Transcription

1 Joural of Avace Apple Mathematc, Vol., No. 4, October Stuy o Baye Semparametrc Regreo Abulhue Saber AL-Mouel a Ameera Jaber Mohae Mathematc Departmet College of Eucato for Pure Scece, AL-Barah Uverty, Iraq Emal: ameera.jaber@yahoo.com Abtract. I th paper, Bayea approach bae o Markov cha Mote Carlo (MCMC to (fully Semparametrc regreo problem ecrbe a a mxe moel ug a coveet coecto betwee pealze ple a mxe moel. We vetgate the ferece o the moel coeffcet uer ome coto o the pror, a well a tuyg ome properte of the poteror trbuto a etfyg the aalytc form of the Baye factor. Keywor: Semparametrc regreo, pealze ple, mxe moel, baye approach, pror trbuto, poteror trbuto, baye factor. Itroucto Semparametrc regreo moel have bee vetgate by may reearcher. Lek (999 preete the Bayea ferece of a emparametrc regreo moel ug a Fourer repreetato[6]. Nato ( tue emparametrc regreo wth multplcatve ajutmet[]. Alo Brezger et al. ( vetgate a aalyze Bayea emparametrc moel[]. Ruppert et al. (3 trouce emparametrc regreo moel bae o pealze regreo ple a mxe moel[]. at a MA (4 tue locally effcet emparametrc etmator for fuctoal meauremet error moel by efg after projectg the core vector wth repect to the parameter o to the uace taget pace for the oparametrc cotoal trbuto of X gve Z, where Z the prector varable meaure precely[4]. I (7, Yua a DE Goojer preete emparametrc regreo wth kerel error moel[6]. Alo (7, Jee a Maheu tue bayea emparametrc tochatc volatlty moelg[3]. Cho, Lee a Roy (8 vetgate the large ample property of the Baye factor for tetg the parametrc ull moel agat the Semparametrc alteratve moel[]. Wa (9 preete emparametrc regreo a graphcal moel[5]. armaratram ( propoe a robut etmato metho emparametrc regreo moel for pealze regreo ple that ca be ue the preece of outler the repoe varable, a tue a robut vero of the moel electo crtero AIC, Akak' formato crtero for regreo moel where S- a MM- etmator are ue for etmato[3]. Pele ( tue Bayea emparametrc regreo that coere a Bayea etmato of retrcte cotoal momet moel wth lear regreo a a partcular example[]. Mohae a Abulhua vetgate Bayea emparametrc regreo bae o pealze ple[7-9]. h paper came to he lght o the emparametrc regreo moel whch ha two part, the parametrc (frt part aume to be lear fucto of p-meoal covarate a the oparametrc (eco part aume to be a mooth pealze ple, a well a the error term whch ha ormal trbuto wth mea zero a varace. We ca repreet emparametrc regreo moel a a mxe moel by ug a coveet coecto betwee pealze ple a mxe moel. I th paper, Bayea approach bae o Markov cha Mote Carlo (MCMC to (fully Semparametrc regreo problem ecrbe a a mxe moel ug a coveet coecto betwee pealze ple a mxe moel. We vetgate the ferece o the moel coeffcet uer ome coto o the pror, a well a tuyg ome properte of the poteror trbuto a etfyg the aalytc form of the Baye factor.

2 98 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 Decrpto of the Problem a the Pror Dtrbuto Coer the moel: where p j = p j j p+, j = y = x + gx ( +, =,,..., ( x the parametrc part whch aume to be lear fucto of p-meoal j j covarate, gx ( the oparametrc part a the uoberve error,,..., are kow to be... p+, ormal wth mea zero a covarace I wth ukow. By ug pealze ple of egree q for the moel ( we get: p q K j q y = x + x + u ( x k +, =,,..., ( j j p+ j p+, k p+, k + j= j= k= where k,..., k are er kot a < k <... < k < b. By ug a coveet coecto betwee K K pealze ple a mxe moel, the moel ( rewrtte a follow: Y = X + Zu + (3 where Y ha legth, X a ( p + q + eg matrx of pure polyomal compoet of the ple, Z a K eg matrx of ple trucate fucto, a ( p + q + -vector of parameter of pure polyomal compoet of the ple, u a K - vector wth ple trucate fucto, a the vector of error term ha legth, ~ N(, I. Aume that u a are epeet a the pror trbuto o u, π ( u N(, I, the u pror trbuto o the parameter vector, π ( N(, I, a we wll aume that the pror trbuto o, π ( vere gamma IG (,, alo we aume that ~ IG (,, where u u u the hyperparameter,,, that eterme the pror a mut be choe by the tattca. 3 Poteror Dtrbuto u u From the moel (3 we have Y θ,, ~ NC ( θ, I (4 u where, C = [ X Z ] a θ = [ u ]. he, the lkelhoo fucto LY ( θ,, u LY ( θ, /, exp ( Y C θ ( I ( Y C θ (5 { } u he, the poteror trbuto of the vector of coeffcet θ a the error varace a u are π ( θ Y,, LY ( θ,, π ( θ u u (6 π ( θ,, exp { θ θ } Y ( Y C ( I ( Y C π ( θ u a π ( / Y, θ, exp ( Y C θ ( I ( Y C θ π ( (7 { } { } u / u u π Y, θ exp ( Y Cθ ( I ( Y C θ π (8 From (6 we ca ee θ Y, N µ, (9 where θ Y,, u (,,,, u θ Y θ Y u u { }{ [ ] } µ = Λ I + ΛCC C (

3 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 99 θ Y,, u { [ ] } { } = Λ Λ C I + ΛCC C Λ ( a I = Λ=, I u ( the I Λ = I u ( Now, by pectral ecompoto we obta C C = PDP [4], where D = ag(,..., the matrx of egevalue a P the orthogoal matrx of egevector. hu I I + Λ CC= P + I D P ( I ( where, = a γ = u. he, the cotoal ety of Y gve, a γ ca be wrtte a: where = (,..., = PY. my (, γ, = I / ( π et I + D γ I ( p q I Y exp Y P I + D P I ( = / / / ( π [ + ] [ + γ ] = = + exp = + = + p q heorem. he jot poteror ety of, γ gve Y / ( b/ / / γ e π( γ, Y γ + ( + ( + ( a b/ ( a + bγ = = = + = + p q (3 Proof. Sce ~ Gamma (,, γ ~ Fba (, [5] = π(, γ Y my (, γ, (, f, (,, f γ b a ( f,, / / = ( + ( + γ / ( π = = Γ(

4 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 = ( π b/ a/ ( b/ γ b a exp + + exp ( a b/ = + = p q + (, ba ( a + bγ ( + ( exp Γ( / e b/ a/ b a ( b/ γ ( ( Γ ( (, ba ( a + bγ ( a+ b/ / / ( + ( + γ / ( π = = ( + exp ( + / = + = p q + e b/ a/ b a ( b/ γ ( / = ( π ( ( a+ b/ ( Γ ( (, ba ( a + bγ / / ( + ( + γ / ( π = = exp γ = + = + = + = + γ = + = + p q ( + + / ( + + / ( + + / ( b/ / / γ e γ + ( + ( + ( a b/ / ( a + bγ ( π = = exp = + = p q + p q = + = p+ q + γ + = + = + p q [( + + /] ( + + / ( b/ / / γ e Γ (( / + + γ + ( ( ( a b/ ( a + bγ = =

5 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 = + = + p q = + = + ( + + / ( b/ / / γ e π( γ, Y + + γ + ( ( ( a b/ ( a + bγ = = γ ( + + / heorem. he poteror mea a covarace matrx of θ are I E( θ Y = Λ P E I + D Y C I ( a = = θ + p q + Var( Y = E Λ + + Y Λ C P E + + = + = p q + I I + D Y PC Λ+ γ I ( p q I E Λ C P I + D I ( (4 (5 Proof. { }{ } E( θ Y = µ = Λ I + Λ θ Y CC CY I = Λ P I + D P CY I ( I = Λ ( P I + D P CY I ( P the orthogoal matrx of egevector, the P = P a ( P = P. herefore I I p q E( θ Y = Λ P I + D P CY I ( I p q (

6 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 I = Λ P E I + D Y C I ( I where the expectato E I + D Y take wth repect to π I ( ( γ, Y. By followg the ame way we ca prove the varace of ( θ Y 4 Moel Checkg a Baye Factor We woul lke to chooe betwee a Bayea pealze ple emparametrc regreo moel a a mxe moel a a Bayea pealze ple emparametrc regreo moel wth kow coeffcet by ug Baye factor for two hypothee veru p q K j q H : y = x + x + u ( x k + j j j+ p+, k p+, k + j= j= k= p q K j q H : y = x + x + u ( x k + j j j+ p+, k p+, k + j= j= k= or H : Y = X + Zu + veru (6 H : Y = X + Zu + where a u are kow. We compute the Baye factor, B, of H relatve to H for tetg problem (6 a follow my ( H B ( Y = (7 my ( H where my ( H the margal ety of Y uer moel H, =,. We have: my ( H = f Y, u, π ( ( = ( π ( exp ( exp = ( π ( exp = ( π / / ( + ( Y X Zu ( ( Y X Zu / ( + + / ( + ( ( Y X Zu ( ( Y X Zu + ( Y X Zu exp + + ( + ( Y X Zu / π Γ( + + ( ( = ( + ( Y X Zu + ex ( + + p ( Y X Zu

7 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 3 a herefore, + ( Y X Zu / = ( π ( + ( Y X Zu + + ( exp ( Y X Zu + + / ( Y X Zu Γ( = ( π Γ ( ( / / / my ( H,,, γ = ( π + + γ ( ( = = exp + = + = p q + my ( H = my ( H,, γπ, (, γ, γ / / ( + / = ( exp ( π γ + + ( ( ( = = + exp π ( γ, γ = + = p q + / / / ( = = = ( π ( + ( + γ π( γ, exp γ = + = p q + / π Γ( / / my ( H = ( Γ + ( + ( + γ π( γ, = = γ = + = + γ p q (8 (9 5 Smulato Reult I th ecto, we llutrate the effectvee of our methoology, we geerate obervato from the moel ( wth the followg regreo fucto: ( y = + 3x + exp{( x +.4 }, {( x +.} ( y = 3x + x. ( x +. 3 he obervato for x are geerate from uform trbuto o the terval [,]. he ample ze take are = 5, 5,, 5,.

8 4 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 he gooe of ft of the etmate moel quatfe by computg the crtero average mea quare error( AMSE a average mea abolute error ( AMAE are efe a: N AMSE = MSE( x, N = N AMAE = MAE( x, N where MSE a MAE are mea quare error a mea abolute error crtero repectvely. able ( preet ummary value of the ( AMSE a ( AMAE for the etmato metho. From th table we ca ee that the value of ( AMSE a ( AMAE whe ( = are maller tha ther value for the frt tet fucto, whch were (.5367 a (.644 repectvely. Whle the value of ( AMSE a ( AMAE are maller whe ( = for the eco tet fucto were (.63 a (.3437 repectvely. Fgure ( a (3 below how the umber of terato of Gbb ampler ue th paper, whch wa ( terato for the frt a eco tet fucto repectvely whe ( =. Whle fgure ( a (4 how the ety etmate bae o ( terato of =. a for the frt a eco tet fucto repectvely whe u ( able. Reult of the( AMSE a ( AMAE crtero for Bayea emparametrc regreo moel. = et fucto Sample ze AMSE AMAE y y he moel checkg approach bae o Baye factor ha bee tete o mulate example. hee Baye factor are gve table (. From th table, t ca be ee that the moel correpog to the frt tet fucto obta the larget Baye factor whe ( = 5 followe by that the eco tet fucto whe ( = 5, a the Baye factor favor H wth trog evece wth all ample ze for two tet fucto. able. Value of Baye factor et fucto Sample ze B ( y Evece Strogly favor H Strogly favor H -9 y.7653 Strogly favor H Strogly favor H Strogly favor H Strogly favor H Strogly favor H -8 y Strogly favor H Strogly favor H Strogly favor H

9 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 5 Fgure. ( terato of the Gbb ampler for the frt tet fucto whe ( =. Fgure. he ety etmate bae o ( terato of ( =. a u for the frt tet fucto whe Fgure 3. ( terato of the Gbb ampler for the eco tet fucto whe ( =.

10 6 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 Fgure 4. he ety etmate bae o ( terato of ( =. a u for the eco tet fucto whe 6 Cocluo he cocluo obtae throughout th paper are a follow: ( he jot poteror ety of, γ gve Y ( b/ / / γ e π( γ, Y + + γ + ( ( ( a b/ ( a + bγ = = + + = + = + p q ( + + / ( he margal ety of Y uer moel H, =, : a / = π Γ Γ( ( + + m( Y H ( ( ( Y X Zu, / / / = Γ + ( = = my ( H ( π ( ( γ π( γ, Γ γ = + = + γ p q (3 I the mulato reult, we coclue the followg: (a he value of ( AMSE a ( AMAE whe ( = are maller tha ther value for the frt tet fucto, whch were (.5367 a (.644 repectvely. (b he value of ( AMSE a ( AMAE are maller whe ( = for the eco tet fucto were (.63 a (.3437 repectvely.

11 Joural of Avace Apple Mathematc, Vol., No. 4, October 7 7 (c he moel correpog to the frt tet fucto obta the larget Baye factor whe ( = 5 followe by that the eco tet fucto whe ( = 5. ( he Baye factor favor H wth trog evece wth all ample ze for two tet fucto. Ackowlegmet. We thak the etor a referee for provg crtcal commet whch have brought gfcat mprovemet to our preetato. Referece. Brezger, Aerea, Keb, homa a Lag, Stefa, "Aalyzg Bayea emparametrc regreo moel", Work hop AG-Baye, Uvertät Müche,.. Cho,., Lee, J. a Roy, A., "A ote o the Baye Factor a Semparametrc regreo moel", Joural of Multvarate Aaly, 36-37, Jee, M. J. a Maheu, J. M.,"Bayea Semparametrc tochatc volatlty moelg, he Rm Ceter for Ecoomc Aaly, Italy, Joo, R. A. a Wcher, D. W. "Apple Multvarate Stattcal Aaly" Pretce Hall, Eglewoo Clff, New Jerey 763, Jo, W., "Bayea a frequett regreo metho, Sprger New York Heelberg Dorrecht Loo, (3. 6. Lek, P. J., "Bayea ferece for Semparametrc regreo ug a Fourer repreetato ", J.R. Statt. Soc. Ser. B6, part 4, Mohae, A. J. a Abulhua, A. M., "A ote o Baye Semparametrc regreo ", Mathematcal heory a Moelg ISSN (Paper4-584 ISSN (Ole5-5 Vol.3, No., Mohae, A. J. a Abulhua, A. M., "Bayea Semparametrc regreo wth fuzzy et ", Iteratoal Joural of Pure a Apple Reearch Egeerg a echology IJPRE, Vol. (3:-8 IJPRE, Reearch artcle ISSN: 39-57X, Mohae, A. J. a Abulhua, A. M., "Fuzzy et a pealze ple Bayea Semparametrc regreo ", LAP LAMBER Acaemc Publhg, ISBN: , 4.. Nato, K., "Semparametrc regreo wth multplcatve ajutmet, commucato tattc, theory a metho", ,.. Pele, J., "Bayea Semparametrc Regreo" Ittute for Avace Stue, Vea -39, (.. Ruppert, D., Wa, M.P. a Carroll, R. J. "Semparametrc regreo", Cambrge uverty pre, (3. 3. emaratram, K., Robut etmato a moel electo Semparametrc regreo moel", Proefchrft voorgerage tot het behale va e gra va Doctor e oegepate Ecoomche Wetechappe oor, (. 4. at, A. A. a MA, Y., "locally effcet Semparametrc etmator for fuctoal meauremet error moel ", Bometrka, 9, 4, pp , Wa, M.P. "Semparametrc regreo a graphcal moel". Aut, N. Z. J. Stat., 5 (, Yua, A. a DE Goojer, J.G. "Semparametrc regreo wth kerel error moel", Boar of the Fouato of the Scaava Joural of tattc, Publhe by Blackwell Publhg Lt, 7.