2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 07 Fuzzy approach to Sem-parametrc sample selecton moel L. MUHMD SFIIH *..BSH KMIL 2 M. T. BU OSMN Mathematcs Department Faculty of Scence Technology Unversty Malaysa Terengganu 200 Kuala Terengganu Terengganu MLYSI. 2 School of Dstance Learnng Unverst Sans Malaysa 800 USM Penang MLYSI. bstract: The sample selecton moel stue n the contet of sem-parametrc methos. Wth the efcency of the parametrc moel such as nconsstent estmators etc the sem-parametrc estmaton methos prove the best alternatve to hle ths efcency. Sem-parametrc of a sample selecton moel s an econometrc moel has been foun nterestng applcaton n emprcal stues. The ssue of uncertanty ambguty stll become are major problem complcate n the moelng of sem-parametrc sample selecton moel as well as ts parametrc. In ths stuy we wll focus n the contet of fuzzy concept as a hybr to the sem-parametrc sample selecton moel. The best approach of accountng for uncertanty ambguty s to take avantage of the tools prove by the theory of fuzzy sets. It seems partcularly approprate for moelng vague concepts. Fuzzy sets theory ts propertes through the concept of fuzzy number prove an eal framework n orer to solve the problem of uncertanty ata. In ths paper we ntrouce a fuzzy membershp functon for solvng uncertanty ata of a sem-parametrc sample selecton moel. Key-Wors:- uncertanty sem-parametrc sample selecton moel cr ata fuzzy number membershp functon. Introucton The sample selecton moel stue n the contet of sem-parametrc methos. Wth the efcency of the parametrc moel such as nconsstent estmators etc the sem-parametrc estmaton methos prove an alternatve to hle ths efcency. The stuy of sem-parametrc econometrcs of the sample selecton moels has receve conserable attenton from statstcans as well as econometrcans n the late of 2 st century (see Schafgans 996. The terme sem-parametrc use as a hybr moel for the selecton moels whch o not nvolve parametrc forms on error strbutons; hence only the regresson functon part of the moel of nterest s use. Conseraton base on two perectves frstly; no restrcton of estmaton of the parameters of nterest for the strbuton functon of the error terms seconly; restrctng the functonal form of heteroskeastcty to le n a fnte-mensonal parametrc famly (Schafgans996. Cosslett (990 consere sem-parametrc estmaton of two-stage metho smlar to Heckman (976 for the bvarate normal case where the frst stage consste of sem-parametrc estmaton of bnary selecton moel the secon stage consste of estmatng the regresson equaton. Ichmura Lee (990 propose an etenson of applcablty of a sem-parametrc approach. It was proven that all moels can be represente n the contet of multple ne frameworks (Stoker 986 shown that t can be estmate by the sem-
2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 08 parametrc least squares metho f entfcaton contons are met (see also Klen Spay (99 Gerfn (996 Martns (200 Khan Powell (200. Frankly eakng the prevous stuy n ths area concentrates on sample selecton moel use parametrc sem-parametrc or nonparametrc approaches. More ecfcally none of these researchers put efforts nto stues that analyze semparametrc sample selecton moels n the contet of fuzzy envronment lke fuzzy sets fuzzy logc or fuzzy sets systems (M.Safh (2007. The purpose ths paper s to ntrouce a membershp functon of a sample selecton moel n whch hstorcal ata contans some uncertanty. Wth ths proves an eal framework to eal wth problems n whch there oes not est a efnte crteron for scoverng what elements belongs or o not belongs to a gven set. Fuzzy set efnes by a fuzzy sets n a unverse of scourse U s characterze by a membershp functon enote by the functon maps all elements of U that take the values n the nterval [0] that s : [0] (aeh 965. The concept of fuzzy sets by aeh s etene from the cr sets that s the two-value evaluaton of 0 or {0 } to the nfnte number of values from 0 to [0 ]. (see Terano et.al. 994. 2 Representaton of uncertanty other wor representng partal belongng or egree of membershp are use. The trangular membershp functon s use. These represente as a ecal form as: ( c f [ c n] ( n c f n ( ( f [ n ] ( n 0 otherwse From that functon the -cuts of a trangular fuzzy number can be efne as a set of close ntervals as [( n c + c( n + n] (0] (2 the graph of a typcal membershp functon s llustrate n Fgure µ ( Generally fuzzy number represents an appromaton of some value whch s n the ntervals ( l ( l ( l ( l terms [ c ] c for l 0.. n s gven by the - cuts at the -levels µ l wth μl μl + μ μ0 0 μ n usually prove a better job set to compare the correonng cr values. s wely practce use each -cuts of fuzzy set are close relate wth nterval of real numbers of fuzzy numbers for all (0] base on the coeffcent ( : f then f < then 0 whch s the cr set epens on. Closely relate wth a fuzzy number s the concept of membershp functon. In ths concept the element of a real contnuous number n the nterval [0] or n 0 c n Fgure : trangular fuzzy number For the membershp functon µ ( the assumptons are as follows: ( monotoncally ncreasng functon for membershp functon ( wth 0 lm for n ( monotoncally ecreasng functon for membershp functon µ ( wth µ lm 0 for n
2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 09 The -cuts representaton of fuzzy number Before gong eeper nto fuzzy moelng of PSSM an overvew some efntons are presente (Yen et.at. (999 Chen Wang (999 use n ths stuy s relate to the estence fuzzy set theory ntrouce by aehs(965. The efntons propertes are as follows: Defnton : the fuzzy functon s efne by f : Y ; Y f ( where ; s a cr set; 2 s a fuzzy set Y s the cooman of assocate wth the fuzzy set Defnton 2: Let F(R be calle a fuzzy number f: est R such that µ 2 for any [0] [ µ a] s a close nterval wth F(R represents all fuzzy sets R s the set of real numbers. Defnton : efne a fuzzy number on R to be a trangular fuzzy number f ts membershp functon : R [0] s equal to ( l ( m l ( u ( u m 0 f f f [ l m] m [ m u] otherwse where l m u s a moel value wth l u be a lower upper boun of the support of reectvely. Then the trangular fuzzy number enote by ( l m u. The support of s the set elements { R l < m < u}. non-fuzzy number by conventon occurre when l m u. Theorem : The values of estmator coeffcents of the partcpaton structural equatons for fuzzy ata converge to the values of estmator coeffcents of the partcpaton structural equatons for nonfuzzy ata reectvely whenever the value of cut ten to from below. Proof. From the centro metho that followe to get the cr value the fuzzy number for all observaton of w as W c ( Lb( w + w + Ub( w when there s no cut. The lower boun upper boun for each observaton referre to by the efnton above. Snce we follow the trangular membershp functon Lb s followe see Fgure 4.2 then ( ( w ( B ( Ub where Lb Ub cut ( w ( ( w Lb( w ( w( Lb( w + ( w Ub( w ( w ( Ub( w + Lb ( w w Ub ( w Fgure 2: Membershp functon -cut pplyng the cut nto the trangular membershp functon the fuzzy number s obtane that epens on the gven value of the cut over the range 0 s as follow: B
2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 0 W c( Lbw ( + Lbw ( ( ( w Lbw ( + w + Ubw ( + ( w Ubw ( + w + Ubw ( (. When approaches from below then Lb( w ( w Ub( w ( w. Further w + w + w obtane s W c( w W c ( w. The last equaton states that when approaches from below then Wc ( w. Smlarly for all observatons z c ( c ( z reectvely as tens to from below. Ths mples that the values of estmator coeffcents of the partcpaton structural equatons for fuzzy ata converge to the values of estmator coeffcents of the partcpaton structural equatons for non-fuzzy ata reectvely whenever the value of cut ten to from below. Defnton 4: LR-type fuzzy number enote as Y wth functons L( Y f(( ( YC Y β R( Y f 2 (( ( Y YC. Y consst the lower boun γ ( Y L center Y upper boun Y ( C ( U ( L U mn satsfyng L Y R( Y 0( L( YC R( YC ( ma. The sze of Y s YU YL where mn ma can be any preetermne levels. 4 Development of Fuzzy Semparametrc of Sample Selecton Moels Before constructng a fuzzy SPSSM frst the sample selecton moel purpose by Heckman (976 consere. In SPSSM t s assume that the strbutonal assumpton of ( ε u s weaker than the strbutonal assumpton of the parametrc of sample selecton moel. Then the sample selecton moel s now calle a sem-parametrc of sample selecton moel (SPSSM. In the evelopment of SPSSM moelng usng fuzzy concept as a evelopment of fuzzy PSSM the basc confguraton of fuzzy moelng.e. nvolve fuzzfcaton fuzzy envronment efuzzfcaton (see M.Safh 2007. For fuzzfcaton stage an element of real-value nput varables converte n the unverse of scourse nto value of membershp fuzzy set. t ths approach a trangular fuzzy number use over all observatons. The - cut metho wth ncrement value of 0.2 starte wth 0 up to 0.8. Ths s then apple to the trangular membershp functon to get a lower upper boun for each observatons ( w z whch s efne as w ( wl wm wu ( l m u z ( z z z l m u ( In orer to solve the moel n whch occurs uncertantes fuzzy envronment such as fuzzy sets fuzzy number are more sutable as the processng of the fuzzfe nput parameters. Snce t s assume that some orgnal ata contans uncertanty uner the vagueness of the orgnal ata the ata wll then be consere as fuzzy ata. That means each observaton consere has are varaton values. The upper boun lower boun of the observaton are commonly chosen epenng on the each ata structure eperence of the researchers. For a large sze of observaton the upper boun lower boun of each observaton are qute ffcult to be obtane.
2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 Base on the fuzzy number a fuzzy SPSSM s bult wth the form as: z * The terms z z w ' f * c γ + ε * ' β + u... N 0 otherwse... N w z ε > 0 numbers wth the membershp functons µ µ ε u (4 u are fuzzy µ W µ µ reectvely. Snce the strbutonal assumpton for the SPSSM s weak then for the analyss of the fuzzy SPSSM t s assume that the strbutonal assumpton s weak. To fn an estmate for γ β of the fuzzy parametrc of sample selecton moel one ea s to efuzzfy the fuzzy observatons W. That means convertng ths trangular fuzzy membershp real-value nto a sngle (cr value (or a vector of values that n the same sense s the best representatve of the fuzzy sets that wll actually be apple. Centro metho or the center of gravty metho s use.e. computes the outputs of the cr value as the center of area uner the curve. Let W c centro metho for c be the efuzzfe values of c W reectvely. The calculaton of the W c reectvely va the followng formula: c wμw ( w w W c ( W + W l + W m μw ( w w μ c ( + l + m μ u u c zμ z ( z z c ( + l + m μ z ( z z u (5 Then the cr values for the fuzzy observaton are calculate followng the centro formula as state above. To estmate γ β of SPSSM approach applyng the proceure as n Powell then the parameter s estmate for the fuzzy semparametrc sample selecton moel (fuzzy SPSSM. Before gettng a real value for the fuzzy SPSSM coeffcent estmate frst the coeffcent estmate values of γ β are use as a shaow of reflecton to the real one. The value of γ β are then apple to the parameters of the parametrc moel to get a real value for the fuzzy SPSSM coeffcent estmate of γ β σ. The Powell ε u SPSSM proceure s then eecute usng the plore software. Eecutng the Powell (Powell 987 proceure by plore takes the ata as nput from the outcome equaton ( y where may not contan a vector of ones. The vector contanng the ' estmate for the frst-step ne β the bwth vector h where h s the threshol parameter k that s use for estmatng the ntercept coeffcent from the frst element. The bwth h from the secon element s use for estmatng the slope coeffcents. For fuzzy PSSM follows the above proceure then another set of cr values W s obtane. pple the c c c - cut values on the trangular membershp functon of the fuzzy observatons W wth the orgnal observaton fuzzy ata wthout - cut fuzzy ata wth - cut to estmate the parameters of the fuzzy SPSSM. The same proceure above s applyng. The parameters of the fuzzy SPSSM are estmate.
2th WSES Int. Conf. on PPLIED MTHEMTICS Caro Egypt December 29-2007 2 5 Concluson Bascally moelng take as an mportant part n estmatng the parameters of economc problems. Dfferng from other system esgn the moel tself s generate by a mathematcal functon. In ths paper a escrpton of the evelopment of the FSPSSM has been presente. For hlng the uncertanty whch s nvolve n the orgnal ata fuzzy number together wth membershp functon takes an mportant part erve from epert knowlege. References: Chen T. Wang. M.J.J. (999. Forecastng methos usng fuzzy concepts Fuzzy sets systems.05. p. 9-52. Cosslett S. (99. Semparametrc estmaton of a regresson moels wth sample selectvty n W.. Barnett J. Powell G.E. Tauchen (es Nonparametrc semparametrc estmaton methos n Econometrcs Statstcs Cambrge Unversty Press p 75-98. Gerfn M. (996. Parametrc semparametrc estmaton of the bnary reonse moel of labor market partcpaton. Journal of pple Econometrcs.. p. 2-9. Heckman J.J. (976. The common structure of statstcal moels of truncaton sample selecton lmte epenent varables a smple estmaton for such moels nnals of Economc Socal Measurement 5 p. 475-492. Ichmura H Lee L..F. (990. Semparametrc least square estmaton of multple ne moels: Sngle equaton estmaton n W.. Barnett J. Powell G.E. Tauchen (es Nonparametrc semparametrc estmaton methos n Econometrcs Statstcs Cambrge Unversty Press. Klen R. SpayR. (99. n effcent semparametrc estmator of the bnary reonse moel Econometrca 6 2 p.87-42. L. M. Safh Lola (2007. Fuzzy Sem-parametrc of a Sample Selecton Moels. Ph.D. ssertaton. Unversty Scence of Malaysa. Penang. Malaysa. Martn M.F.O. (200. Parametrc semparametrc estmaton of sample selecton moels: n emprcal applcaton to the female labor force n Portugal. Journal of pple Econometrcs 6 p. 2-9. Powell J.L. (987. Semparametrc estmaton of bvarate latent varable moels. Socal Systems Research Insttute. Unversty of Wsconsn- Mason Workng paper No.8704. Schafgans M. (996. Semparametrc estmaton of a sample selecton moel: estmaton of the ntercept; theory applcatons Ph.D. ssertaton Yale Unversty New Haven US. Stoker T.M. (986. Consstent estmaton of scale coeffcents Econometrca 54. p. 46-8. Terano T. sa K. Sugeno M. (994. pple fuzzy systems Cambrge. P Professonal. Yen K.K. Ghoshray. S. Rog. G. (999. lnear regresson moel usng trangular fuzzy number coeffcents. Fuzzy sets systems. 06. p. 67-77. aeh L.. (965. Fuzzy Sets systems. In: Fo J. e. System Theory. Brooklyn New York. Polytechnc Press. Khan S. Powell J. L. (200. Two-step estmaton of semparametrc censore regresson moels. Journal of Econometrcs. 0. p. 7-0.