β (cf Khan, 2006). In this model, p independent

Transcription

1 Proc. ICCS-3, Bogor, Idoes December 8-4 Vol. Testg the Equlty of the Two Itercets for the Prllel Regresso Model Bud Prtko d Shhjh Kh Dertmet of Mthemtcs d Nturl Scece Jederl Soedrm Uversty, Purwokerto, Jw Tegh, Idoes brtkto@gml.com School of Agrculturl, Comuttol d Evrometl Sceces Itertol Cetre for Aled Clmte Sceces Uversty of Souther Queesld, Toowoomb, Austrl khs@usq.edu.u ABSTRACT Testg the equlty of the two tercets of two rllel regresso models s cosdered whe the sloes re sucted to be equl. For three dfferet sceros o the vlues of the sloe rmeters, mely () ukow (ucfed), () kow (cfed), d () sucted, we derve the urestrcted (), restrcted () d retest (T) tests for testg the tercet rmeters. The test sttstcs, ther smlg dstrbutos, d ower fuctos of the tests re obted. Comrso of ower fuctos d szes of the tests re rovded. Keywords d hrs: Ler regresso; tercet d sloe rmeters; re-test test; o-smle ror formto; d ower fucto. Mthemtcs Subject Clssfcto: Prmry 6F3 d Secodry 6J5. Itroducto Two ler regresso les re rllel f the two sloes re equl. A rllelsm roblem c be descrbed s cl c of two relted regresso les o the sme deedet d deedet vrbles tht come from two dfferet ctegores of the resodets. If the deedet dt ts come from two rdom smles ( ), rerchers ofte wsh to model the regresso les for les grous tht re rllel (.e. the sloes of the two regresso les re equl) or whether the les hve the sme tercet. To test the rllelsm of the two regresso equtos, mely y θ + β x + e d y θ + β x + e, j,,...,, j j j j j j for the two dt ts: y [ y, y ] d [, ] y y,..., y, x x,..., x, x H : β β t ( β β )/ S, ( β β ) where β d β re estmte of the sloes β d x x x where x,..., x. We u rorte twosmle t test for testg (rllelsm). Ths t sttstc s gve s β rectvely, d y y,..., y S β s estmte of ( β ) the stdrd error of the estmted dfferece betwee sloes (Klebum, 8,. 3). The rllelsm of the two regresso equtos bove c be exresd s sgle model of mtrx form, tht s, y XΦ + e, where Φ [,,, ], X [ X, X ] wth X [,,,] d X [ ] d e [, ] θ θ β β x,,, x e e. The mtrx form of the tercet d sloe rmeters c be wrtte, rectvely, s θ [, ] d [, ] θ θ β β β (cf Kh, 6). I ths model, deedet ICCS-3, 8- Dec. 4, Bogor, Idoes.

2 bvrte smles re cosdered such tht yj N( θ + βx j, σ ) for,..., d j,...,. The rmeters θ ( θ,..., θ ) β ( β,..., β ) re the tercet d sloe vectors of the les. See Kh (3, 6, d 8) for detls o rllel regresso models d lys. To exl the mortce of testg the equlty of the tercets (rllelsm) whe the equlty of sloes s ucert, we cosder the geerl form of the PRM of t of ( > ) smle regresso models s Y θ + β x + e,,,...,, d j,,...,, (.) j j where Y ( Y,..., Y ) s vector of obrvble rdom vrbles, (,...,) s - tule of s, x j ( x,..., x ) s vector of deedet vrbles, θ d β re ukow tercet d sloe, rectvely, d e ( e,..., e ) s the vector of errors whch re mutully deedet d detclly dstrbuted s orml vrble, tht s, e I where N(, σ ) s the detty mtrx of order. Equto (.) reret ler models wth dfferet tercet d sloe rmeters. If β... β β, θ s the there re rllel smle ler models f ( θ,..., θ ) ( β,..., β ) re dfferet. Here, the rmeters θ d β re the tercet d sloe vectors of the les. Bcroft (944) troduced the de of retestg NSPI to remove ucertty. The outcome of the retestg o the ucert NSPI s ud the hyothess testg to mrove the erformce of the sttstcl test (Kh d Sleh, ; Sleh, 6, ; Yuus d Kh, ). The sucted vlue of the sloes my be () ukow or ucfed f NSPI s ot vlble, () kow or cfed f the exct vlue s vlble from NSPI, d () ucert f the sucted vlue s usure. For the three dfferet sceros, three dfferet of sttstcl tests, mely the () urestrcted test (), () restrcted test () d () re-test test (T) re defed. I the re of estmto wth NSPI there hs bee lot of work, otbly Bcroft (944, 964), Hd d Bcroft (968), d Judge d Bock (978) troduced relmry test estmto of rmeters to estmte the rmeters of model wth ucert ror formto. Kh (3, 8), Kh d Sleh (997,, 5, 8), Kh et l. (), Kh d Hoque (3), Sleh (6) d Yuus () covered vrous work the re of mroved estmto usg NSPI, but there s very lmted umber of studes o the testg of rmeters the rece of ucert NSPI. Although Tmur (965), Sleh d Se (978, 98), Yuus d Kh (7,, b), d Yuus () ud the NSPI for testg hyothes usg ormetrc methods, the roblem hs ot bee ddresd the rmetrc cotext. The study tests the equlty of the tercets for whe the equlty of sloes s sucted. We test the tercet vector θ ( θ,..., θ ) whe t s ucert f the sloe rmeters re equl (rllel). We the cosder the three dfferet sceros of the sloe rmeters, d defe three dfferet tests: for the, let φ be the test fucto d T be the test sttstc for testg H : θ θ H : θ > θ whe ( β,..., β ) gst for the, let gst β s ucfed, φ be the test fucto d H : θ > θ whe β for the T, let β (fxed vector), T φ be the test fucto d I H : θ θ T be the test sttstc for testg H : θ θ T T be the test sttstc for testg gst H : θ > θ followg re-test () o the sloe rmeters. For the, let φ be the test fucto for testg H : β β ( sucted costt) gst H : β > β (to ICCS-3, 8- Dec. 4, Bogor, Idoes.

3 3 remove ucertty). If the H s rejected the, the the s ud to test the tercet, otherw the s ud to test H. Thus, the T deeds o the whch s choce betwee the d. The urestrcted mxmum lkelhood estmtor or lest squre estmtor of tercet d sloe vectors, θ ( θ,..., θ ) d β ( β,..., β ), re gve s ( xy ) ( )[ ] x y θ Y T β d β, (.) where θ ( θ,..., θ ), β ( β,..., β ) T Dg(,..., ),, x x xx ( )[ ] x, d θ Y βx for,..,. Furthermore, the lkelhood rto (LR) test sttstcs for testg H : θ θ gst H : θ > θ s gve by θ HD H θ F, (.3) ( ) s e where H I D, D Dg(,.., ),, Q ( ) xx x d ( ) S ( )( ) e Y θ βx Y θ βx (Sleh, 6,. 4-5). Uder H, F follows cetrl F dstrbuto wth (, ) degrees of freedom (d.f.), d uder H t follows ocetrl F dstrbuto wth (, ) degrees of freedom d ocetrlty rmeter /, where ( θ θ) ( θ θ) (.4) σ d D HD H. Whe the sloe ( β ) s equl to β (cfed), the restrcted mle of tercet d sloe vectors re gve s ˆ ˆ β θ d k k θ + D TH β β (.5) The followg cto rovdes the rood tests. Secto 3 derves the dstrbuto of the test sttstcs. The ower fucto of the tests re obted Secto 4. A llustrtve exmle s gve Secto 5. The comrso of the ower of the tests d cocludg remrks re rovded Sectos 6 d 7. The Three Tests To test the equlty of the tercets whe the equlty of sloes s sucted, we cosder three dfferet sceros of the sloes. The test sttstcs of the, d T re the defed s follows. For β ucfed, the test sttstc of the s gve by where θ HD Hθ ( ) s T, (.) e e s ( ) ( Y θ βx )( Y θ βx ). ICCS-3, 8- Dec. 4, Bogor, Idoes.

4 4 Uder The T follows cetrl F dstrbuto wth (, ) degrees of freedom. H, t follows ocetrl F dstrbuto wth (, ) degrees of freedom d ocetrlty rmeter /. Uder orml model we hve where θ θ D TD N σ D N + TDT β d N Dg(,..., ). β,, β β TD D Whe the sloe s cfed to be β (fxed vector), the test sttstc of the s gve by ( ˆ θ HD Hˆ θ) + ( β HD H β), (.) T where (.3) ( ) ( ) ( ˆ ˆ ) ( ˆ ˆ ) d ˆ r Y x Y x s θ β θ β β β The T follows cetrl F dstrbuto wth (, ) degrees of freedom. Uder t follows ocetrl F dstrbuto wth (, ) degrees of freedom d ocetrlty rmeter /. Ag, ote tht ˆ * * θ θ TH β D D N, σ, ˆ β * β D D T Tβ D N + d D T. β where * * Whe the vlue of the sloe s sucted to be β but usure, re-test o the sloe s requred before testg the tercet. For the relmry test () of H : β β gst H : β (.4) β>, the test sttstc uder the ull hyothess s defed s T β HD H β ( ), (.5) whch follows cetrl F dstrbuto wth (, ) degrees of freedom. Uder follows ocetrl F dstrbuto wth (, ) degrees of freedom d ocetrlty H, H, t rmeter /. Ag, ote tht * θ β ( β β) / N, σ, (.6) β ˆ β H β HD * where β D β (Sleh, 6,. 73). Let us choo ostve umber α ( < α <, for,, 3) d rel vlue v, v, v3 j j j F ( v be umertor d.f. d v be deomtor d.f.) such tht ( > F α θ θ ) α, (.7),, ICCS-3, 8- Dec. 4, Bogor, Idoes.

5 5 ( > F α θ θ ) α, (.8),, ( > F α β β ) α. (.9),, 3 3 Now the test fucto for testg H : θ θ gst H : θ > θ s defed by, f ( T Fc, T > Fb) or ( T > Fc, T > F); F (.), otherw, F F F F d F F where,,, b,, c 3,,. 3 Dstrbuto of Test Sttstcs To derve the ower fucto of the, d T, the smlg dstrbuto of the test sttstcs rood Secto re requred. For the ower fucto of the T the jot dstrbuto of ( T, T ) d ( T, T ) s estl. Let { N } be quece of ltertve hyothes defed s λ λ N :( θ θ, β β ), λ, (3.) where λ s vector of fxed rel umbers d θ s the true vlue of the tercet. Uder vlue of ) H the vlue of (θ θ ) s equl zero. N the (θ θ s greter th zero d uder Followg Yuus d Kh (b) d equto (.), we defe the test sttstc of the whe β s ucfed, uder N, s ( θ θ) HD H ( θ θ) T T. (3.) ( ) The T follows ocetrl F dstrbuto wth ocetrlty rmeter whch s fucto of ( θ θ ) d (, ) degrees of freedom, uder N. From equto (.3) uder N, ( θ θ ) > d ( β β ) >, the test sttstc of the becomes ( ) ( ) ( ) ( ) θ θ HD H θ θ + β β HD H β β T T ( ) sr (3.3) The T lso follows ocetrl F dstrbuto wth ocetrlty rmeter whch s fucto of ( θ θ ) d (, ) degrees of freedom, uder N. Smlrly, from the equto (.5) the test sttstc of the s gve by ( ) ( ) β β HD H β β T3 T (3.4) ( ) Uder H, the T 3 follows ocetrl F dstrbuto wth ocetrlty rmeter whch s fucto of ( β β) d (, ) d.f. From equtos (.), (.3) d (.5) the T d T re correlted, d the T d T re ucorrelted. The jot dstrbuto of the T d T, tht s, ( T, T ) (3.5) ICCS-3, 8- Dec. 4, Bogor, Idoes.

6 6 s correlted bvrte F dstrbuto wth (, ) degrees of freedom. The robblty desty fucto (df) d cumultve dstrbuto fucto (cdf) of the correlted bvrte F dstrbuto s foud Krshh (964), Amos d Bulgre (97) d El-Bssouy d Joes (9). Lter, Johso et l. (995,. 35) descrbed reltosh of the bvrte F dstrbuto wth the bvrte bet dstrbuto. Ths s due to the df of the bvrte F dstrbuto hs smlr form wth the df of bet dstrbuto of the cod kd. Followg El-Bssouy d Joes (9), the covrce d correlto betwee the T d T re the gve s Cov( T, T 3 ) ff ( f )( f )( f 4) ρ +, d ( 4 4 ) ( ) ( 4) dd ( f 4) ( + )( + )( 4) T T3 f d f d f ( + )( 4). ( 3 ) ( 4) (3.6) (3.7) Note the bove exressos degrees of freedom for the T d d d d f f re the rorte T rectvely. 4 The Power d Sze of Tests The ower fucto of the, d T re derved below. From equto (.) d (3.), (.3) d (3.3), d (.5) d (3.4), the ower fucto of the, d T re gve, rectvely, s: the ower of the ( λ ) ( > F N ) α,, ( F kδ ), (4.) where d α,, λ Dλ d k. ( ) s e the ower of the ( λ) ( > F N ) α,, ( λ HD Hλ ) + ( λ HD H λ ) P T Fα,, ( ) s r ( Fα ),, k( δ + δ), (4.) d λ D λ d k. where ( ) s r The ower fucto of the s ICCS-3, 8- Dec. 4, Bogor, Idoes.

7 7 ( λ) ( > F K ) α 3,, λ HD Hλ P T ( F kδ ). (4.3) the ower of the T where d (, ) whch 3 Fα 3,, ( ) 3 α3,, T λ < Fα3,,, T > Fα,, ( ) ( ) ( α,,, > α,, ) + F T F r 3 ( π ) π + d ( b, ), (4.4) r b s bvrte F robblty tegrls, d t s defed s b b f ( F, F ) df df, (4.5) d (, b) f ( F, F ) df df r λ HD Hλ F F k 3,, 3,, ( ) δ, d ( θ θ ) HD H( θ θ ) b F F k,,,, ( ) α α δ b The f ( F, F ) df df equto (4.5) s the cdf of the correlted bvrte ocetrl F (BNCF) dstrbuto of the d. From equto (4.4), t s cler tht the cdf of the BNCF dstrbuto volved the exresso of the ower fucto of the T. Usg equto (4.7), we u t the clculto of the ower fucto of the T. R codes re wrtte, d the R ckge s ud for comuttos of the ower d sze d grhcl lyss. Furthermore, the sze of the, d T re gve rectvely s: the sze of the α > Fα,, H θ θ,, α > Fα,, H θ θ ( : ) ( F α ), (4.8) the sze of the ( : ) ( F kδ ), (4.9) α,, The sze of the s gve by α ( λ ) ( > Fα ) 3,, H ( 3 F α 3,, ). (4.) the sze of the T T ( H, T > d ) (, ) H + > T > h H ( ( > F3,, )) ( > F,, ) + d r( h, ), (4.) where h F. α,,. ICCS-3, 8- Dec. 4, Bogor, Idoes.

8 8 5 Power Comrso by Smulto To comre the tests grhclly we coducted smultos usg the R ckge. For 3, ech of three deedet vrbles ( x,,,3, j,..., ) re geerted from the uform j dstrbuto betwee d. The errors ( e,,, 3) re geerted from the orml dstrbuto wth µ d σ. I ech c rdom vrtes were geerted. The deedet vrble ( y j ) s determed by y j θ + βx j + e for θ 3 d β. Smlrly, defe y j θ + βx j + e for θ 3.6 d β ; y, 3 j θ3 + β3x3 j + e3 for θ 3 4 d β 3, rectvely. For the comutto of the ower fucto of the tests (, d T) we t α α α3 α.5. The grhs for the ower fucto of the three tests re roduced usg the formuls equtos (4.), (4.) d (4.4). The grhs for the sze of the three tests re roduced usg the formuls equtos (4.8), (4.9) d (4.). The grhs of the ower d sze of the tests re reted the Fgures d. ICCS-3, 8- Dec. 4, Bogor, Idoes.

9 9 6 Comrso d Cocluso The form of the ower curve of the Fgure s cocve, strtg from very smll vlue of er zero (whe δ s lso er ), t roches s δ grows lrger. The ower of the cres rdly s the vlue of δ becomes lrger. The she of the ower curve of the s lso cocve for ll vlues of δ d δ. The ower of the cres s the vlues of δ d/or δ cre (e Fgures () d (), d equto (4.)). The ower of the T (e Fgure ) cres s the vlues of δ cre. Moreover, the ower of the T s lwys lrger th tht of the d for the vlues of δ roud.7 to.5. The sze of the does ot deed o δ. It s costt d rems uchged for ll vlues of δ d δ. The sze of the cres s the vlue of δ cres. Moreover, the sze of the s lwys lrger th tht of the, but ot for T for the smller vlues of the δ (ot fr from ). The sze of the T s clor to tht of the for lrger vlues of δ. The dfferece (or g) betwee the sze of the d T cres sgfctly s the vlue of δ d ρ cres. The sze of the s α.5 for ll vlues of δ d δ. For ll vlues of δ d δ, the T sze of the s lrger th tht of the, α > α. For ll the vlues of ρ, α α. ICCS-3, 8- Dec. 4, Bogor, Idoes.

10 Bd o the bove lys, the ower of the s lwys hgher th tht of the for ll vlues of δ d δ. Also, the ower of the T s lwys lrger th tht of the for ll vlues δ (e the curves for tervl vlues of δ.7 < <.5), δ d ρ. The sze of the s smller th tht of the d T for ll δ. The ower of the T s hgher th tht of the d teds to be lower th tht of the. The sze of the T s less th tht of the but hgher th tht of the. Referece [] Amos, D. E. d Bulgre, W. G. (97). Comutto of multvrte F dstrbuto. Jourl of Mthemtcs of Comutto, 6, [] Bcroft, T. A. (944). O bs estmto due to the u of the relmry tests of sgfcce. Als of Mthemtcl Sttstcs, 5, 9-4. [3] Bcroft, T. A. (964). Alyss d ferece for comletely cfed models volvg the u of the relmry test(s) of sgfcce. Bometrcs, (3), [4] El-Bssouy, A. H. d Joes, M. C. (9). A bvrte F dstrbuto wth mrgls o rbtrry umertor d deomtor degrees of freedom, d relted bvrte bet d t dstrbutos. Sttstcl Methods d Alctos, 8(4), [5] H, C. P. d Bcroft, T. A. (968). O oolg mes whe vrce s ukow. Jourl of Amerc Sttstcl Assocto, 63, [6] Johso, N. L., Kotz, S. d Blkrsh, N. (995). Cotuous uvrte dstrbutos, Vol., d Edto. Joh Wley d Sos, Ic., New York. [7] Judge, G. G. d Bock, M. E. (978). The Sttstcl Imlctos of Pre-test d Ste-rule Estmtors Ecooetrcs. North-Holld, New York. [8] Kh, S. (3). Estmto of the Prmeters of two Prllel Regresso Les Uder Ucert Pror Iformto. Bometrcl Jourl, 44, [] Kh, S. (5). Estmto of rmeters of the multvrte regresso model wth ucert ror formto d Studet-t errors. Jourl of Sttstcl Rerch, 39(), [9] Kh, S. (6). Shrkge estmto of the sloe rmeters of two rllel regresso les uder ucert ror formto. Jourl of Model Asssted d Alctos,, [] Kh, S. (8). Shrkge estmtors of tercet rmeters of two smle regresso models wth sucted equl sloes. Commuctos Sttstcs - Theory d Methods, 37, [] Kh, S. d Sleh, A. K. Md. E. (997). Shrkge re-test estmtor of the tercet rmeter for regresso model wth multvrte Studet-t errors. Bometrcl Jourl, 39, -7. [] Kh, S. d Sleh, A. K. Md. E. (). O the comrso of the re-test d shrkge estmtors for the uvrte orml me. Sttstcl Pers, 4(4), [3] Kh, S., Hoque, Z. d Sleh, A. K. Md. E. (). Imroved estmto of the sloe rmeter for ler regresso model wth orml errors d ucert ror formto. Jourl of Sttstcl Rerch, 3(), 5-7. [4] Kh, S. d Hoque, Z. (3). Prelmry test estmtors for the multvrte orml me bd o the modfed W, LR d LM tests. Jourl of Sttstcl Rerch, Vol 37, [5] Kh, S. d Sleh, A. K. Md. E. (5). Estmto of tercet rmeter for ler regresso wth ucert o-smle ror formto. Sttstcl Pers. 46, [6] Kh, S. d Sleh, A. K. Md. E. (8). Estmto of sloe for ler regresso model wth ucert ror formto d Studet-t error. Commuctos Sttstcs-Theory d Methods, 37(6), [7] Klebum, D. G., Kuer, L. L., Nzm, A. d Muller, K. E. (8). Aled regresso lyss d other multvrble methods. Duxbury, USA. [8] Krshh, P. R. (964). O the smulteous ov d mov tests. Prt of PhD thess, Uversty of Mesot. [9] Sleh, A. K. Md. E. (6). Theory of relmry test d Ste-tye estmto wth lctos. Joh Wley d Sos, Ic., New Jery. [] Sleh, A. K. Md. E. d Se, P. K. (978). Normetrc estmto of locto rmeter fter relmry test o regresso. Als of Sttstcs, 6, ICCS-3, 8- Dec. 4, Bogor, Idoes.

11 [] Sleh, A. K. Md. E. d Se, P. K. (98). Shrkge lest squres estm-to geerl multvrte ler model. Procedgs of the Ffth Po Symosum o Mthemtcl Sttstcs, [] Schuurm, F. J., Krshh, P. R. d Chttodhyy, A. K. (975). Tble for multvrte F dstrbuto. The Id Jourl of Sttstcs, 37, [3] Tmur, R. (965). Normetrc fereces wth relmry test. Bull. Mth. Stt., [4] Yuus, R. M. (). Icresg ower of M-test through re-testg. Uublshed PhD Thess, Uversty of Souther Queesld, Austrl. [5] Yuus, R. M. d Kh, S. (7). Test for tercet fter re-testg o sloe - robust method. I: 9th Islmc Coutres Coferece o Sttstcl Sceces (ICCS-IX): Sttstcs the Cotemorry World - Theores, Methods d Alctos. [6] Yuus, R. M. d Kh, S. (). Icresg ower of the test through re-test - robust method. Commuctos Sttstcs-Theory d Methods, 4, [7] Yuus, R. M. d Kh, S. (b). M-tests for multvrte regresso model. Jourl of Normtrc Sttstcs, 3, -8. ICCS-3, 8- Dec. 4, Bogor, Idoes.