Asymptotic Behaviors of the Lorenz Curve for Left Truncated and Dependent Data

Transcription

1 Joural of Sceces, Islamc Reublc of Ira 23(2): 7-77 (22) Uversty of Tehra, ISSN 6-4 htt://jsceces.ut.ac.r Asymtotc Behavors of the Lorez Curve for Left Trucated ad Deedet Data M. Bolbola Ghalbaf,,* V. akoor, 2 ad.a. Azaroosh 2 Deartmet of Statstcs, School of Mathematcal Sceces ad Comuter, akm Sabzevar Uvercty, Sabzevar, Islamc Reublc of Ira 2 Deartmet of Statstcs, School of Mathematcal Sceces, erdows Uversty of Mashhad, Mashhad, Islamc Reublc of Ira Receved: 6 October 2 / Revsed: 29 Arl 22 / Acceted: 3 Jue 22 Abstract The urose of ths aer s to rovde some asymtotc results for oarametrc estmator of the Lorez curve ad Lorez rocess for the case whch data are assumed to be strog mxg subject to radom left trucato. rst, we show that oarametrc estmator of the Lorez curve s uformly strogly cosstet for the assocated Lorez curve. Also, a strog Gaussa aroxmato for the assocated Lorez rocess s establshed uder arorate assumtos. Usg ths strog Gaussa aroxmato, a law of the terated logarthm for the Lorez rocess s also derved. Keywords: Law of the terated logarthm; Lorez curve; Strog Gaussa aroxmato; Strog mxg; Trucated data Itroducto Petra [25] ad Gastwrth [8] deedetly troduced the Lorez curve corresodg to a oegatve radom varable (rv) X wth a dstrbuto fucto (df), quatle fucto Q ad fte mea E( X ) = as: t L ( t ) : = Q ( s ) ds, t. I ecoometrcs, wth X reresetg come, Lt () gves the fracto of total come that the holders of the lowest t th fracto of come ossesses. Most of the measures of come equalty are derved from the Lorez curve. A mortat examle s the G dex assocated wth defed by G where [ u L ] u du : = = 2( CL ), udu ( CL ) = L ( u ) du s the cumulatve Lorez curve corresodg to. Ths s a rato of the area betwee the Lorez curve ad the 45 le to the area uder the 45 le. The umerator s usually called the area of cocetrato. Kedall ad Stuart [2] showed that ths s equvalet to a rato of a measure of dserso to the mea. I geeral, these otos are useful for measurg cocetrato ad equalty dstrbutos of resources, ad sze dstrbutos. or a lst of alcatos dfferet areas, we refer the * Corresodg author, Tel.: +98(57) 438, ax: +98(5)882865, E-mal: m_bolbola@yahoo.com 7

2 Vol. 23 No. 2 Srg 22 Bolbola Ghalbaf et al. J. Sc. I. R. Ira readers to Cso rgo ad Ztks [9]. To estmate the Lorez curve, oe ca use the Lorez statstc L ( y ) defed by where y L( y ) = Q,, u du y s the samle mea ad Q ( y ) s the emrcal quatle fucto costructed from deedet ad detcally dstrbuted (..d.) samle take from. Golde [9] roved the uform cosstecy of L to L ad derved the weak covergece of the Lorez rocess [ ] l(): t = L() t L(), t t to a Gaussa rocess uder sutable codtos. Cso rgo et al. [6] gave a ufed treatmet of strog ad weak aroxmatos of the Lorez ad other related rocesses. I artcular, they establshed a strog varace rcle for the Lorez rocess, by whch Rao ad Zhao [26] derved oe of ther two versos of the law of the terated logarthm (LIL) for the Lorez rocess. Dfferet versos of the LIL uder weaker assumtos are also obtaed by Cso rgo ad Ztks ([9], []). I Cso rgo ad Ztks [], cofdece bads for the Lorez curve that are based o weghted aroxmatos of the Lorez rocess are costructed. Cso rgo et al. [7], obtaed weak aroxmatos for Lorez curves uder radom rght cesorsh. Strog Gaussa aroxmatos for the Lorez rocess whe data are subject to radom rght cesorsh ad left trucato was establshed by Tse [27], he s also derved a fuctoal LIL for the Lorez rocess. owever, most ecoomc stuatos, the basc sequece of observatos may ot be deedet. It s more realstc to assume some form of deedece amog the data are observed. Cso rgo ad Yu [8], obtaed weak aroxmatos for Lorez curves ad ts verse uder the assumto of mxg deedece. Glveko-Catell-tye asymtotc behavor of the emrcal geeralzed Lorez curves based o radom varables formg a statoary ergodc sequece wth determstc ose were cosdered by Davydov ad Ztks[2]. Davydov ad Ztks [3] establshed a large samle asymtotc theory for the emrcal geeralzed Lorez curves whe observatos are statoary ad ether short-rage or log-rage deedet. Strog laws for the geeralzed absolute Lorez curves whe data are statoary ad ergodc sequeces establshed by elmers ad Ztks [2]. Based o the geeralzed Lorez curves Davydov et al. [4] roosed a statstcal dex for measurg the fluctuatos of a stochastc rocess. They develoed some of the asymtotc theory of the statstcal dex the case where the stochastc rocess s a Gaussa rocess wth statoary cremets ad a cely behaved correlato fucto. The uform strog covergece rate of the Lorez curve estmator uder strog mxg hyothess s obtaed by akoor et al. [7]. They also establshed a strog Gaussa aroxmato for the Lorez rocess, by whch they derved a fuctoal LIL for the Lorez rocess, uder the assumto of strog mxg. The couterart of these results for the cesored deedet model was establshed by Bolbola et al. [2]. The urose of ths aer s to rovde some asymtotc results for Lorez rocess l (t) for the case whch data are assumed to be strog mxg subject to radom left trucato. Cosder a sequece of rv's X, X 2,..., X wth commo ukow absolutely cotuous df ad fte mea. These rv's are regarded as the lfetmes of the tems uder study whch may ot be mutually deedet. Amog the dfferet forms whch comlete data aear, rght cesorg ad left trucato are two commo oes. Left trucato may occur f the tme org of the lfetme recedes the tme org of the study. Oly subjects that fal after the start of the study are beg observed, otherwse they are left trucated. Ths meas that some subjects are samled, whle others are eglected. Ths model arses varous felds, e.g., astroomy, ecoomy ad medcal studes (see, e.g. [28]). Let T, T 2,..., T be a sequece of..d. radom varables wth cotuous df G, they are also assumed to be deedet of the rv's X 's. I the left trucato model, ( X, T ) s observed oly whe X T. Let ( X, T),...,( X, T ) be a samle whch oe observes (.e., X T ), ad γ : =Ρ( T X ) >, where Ρ s the absolute robablty (related to the N - samle). Note that tself s a rv ad that γ ca be estmated by (although ths estmator caot be N calculated sce N s ukow). Assume, wthout loss of geeralty, that X ad T are oegatve radom varables, =,..., N. or ay df L deotes the left ad rght edots of ts suort by al = f{ x: Lx > } ad bl = su{ x: Lx < }, resectvely. The uder the curret model, as dscussed by Woodroofe [28], we assume that ag a ad bg b. Defe C(x)= Ρ(T x X T X ) (.) - = P (T x X )= γ G(x)(-(x)), 72

3 Asymtotc Behavors of the Lorez Curve for Left Trucated ad Deedet Data where P(.) = Ρ (. ) s the codtoal robablty (related to the -samle) ad cosder ts emrcal estmate - (.2) C (x)= I(T x X ), = where I(.) s the dcator fucto. The the roductlmt (PL) estmator ˆ of s gve by (.3) ˆ (x)=-. X x C ( X ) The cumulatve hazard fucto Λ ( x ) s defed by x d ( u ) (.4) Λ ( x ) =. u Let ( x) =Ρ( X x T X ) (.5) x = P ( X x ) = γ G ( u ) d ( u ), be the df of the observed lfetmes. Its emrcal estmator s gve by = ( ). = x I X x O the other had, the df of the observed T 's s gve by G ( x) =Ρ( T x T X ) ad s estmated by γ = P ( T x ) = G ( x u ) d ( u ), = ( ). = G x IT x It the follows from (.) ad (.2) that C( x) = G ( x) ( x), (.6) C ( x) = G ( x) ( x). ally (.), (.4) ad (.5) gve x d ( u ) ( x ). Cu Λ = ece, a atural estmator of Λ s gve by x ˆ d ( u ) I ( X x ) Λ ( x ) = =, C ( u ) C ( X ) = whch s the usual so-called Nelso-Aale estmator of Λ. Moreover, Λ ˆ s the cumulatve hazard fucto of the PL estmator ˆ defed (.3). The quatle fucto Q ad ts emrcal couterart Q are defed by Q()=f{x R; (x) } ad (.7) Q ()=f{x R; ˆ (x) } Suose that < <. We defed the Lorez curve corresodg to rv X as: L : = Q ( s ) ds,, where = Q ( s ) ds. Therefore the atural estmator for the Lorez curve L ( ) s L: = Q( s ) ds,, where = Q ( s ) ds. The ma ams of ths aer are to derve strog uform cosstecy of the Lorez statstc ad strog Gaussa aroxmato for Lorez rocess, for the case whch data are assumed to be deedet subject to radom left trucato. As a result of our strog Gaussa aroxmato, we obta a fuctoal LIL for the Lorez rocess. I ths aer we cosder the strog mxg deedece, whch amouts to a form of asymtotc deedece betwee the ast ad the future as show by ts defto. Defto. Let { X, } deote a sequece of radom varables. Gve a ostve teger m, set α( m ) = (.8) k su PA ( B) PAPB ; A I, B I k { k+ m} k where I deote the σ -feld of evets geerated by { X j ; j k}. The sequece s sad to be strog mxg ( α -mxg) f the mxg coeffcet α( m ) as m. Amog varous mxg codtos used the lterature, strog mxg s reasoably weak ad has may ractcal alcatos (see, e.g. [6], [4] or [5] for more detals). I artcular, Masry ad Tjosthem [24] 73

4 Vol. 23 No. 2 Srg 22 Bolbola Ghalbaf et al. J. Sc. I. R. Ira roved that, both ARC rocesses ad olear addtve AR models wth exogeous varables, whch are artcularly oular face ad ecoometrcs, are statoary ad strog mxg. Now we troduce our ma assumto that s used to state our results gathered below for easy referece. A. { X, } s a sequece of statoary strog + ν (log ) mxg rv's wth mxg coeffcet α = Oe for some ν >. I the ext Secto, we reset our ma results. Results I ths secto we frst derve strog uform cosstecy of the Lorez statstc ad strog Gaussa aroxmato for Lorez rocess, for the case whch data are assumed to be deedet subject to radom left trucato ad fally as a result of our strog Gaussa aroxmato, we obta a fuctoal LIL for the Lorez rocess. Theorem below roves the uform strog cosstecy wth rate of the estmator L. Theorem. Let < <. Uder Assumto A, assumg that = f s bouded away from zero o [Q( )δ, Q( )+δ], for some δ >. The log log su L L = O (2.) Proof. A elemetary comutato shows that, L ( ) L ( ) = Q ( s ) Q ( s ) ds (2.2) L. It s easy to see that, (2.3) = Q s Q s ds. Now, by usg (2.2), (2.3) ad Lemma 3 of [23], we obta the results. or costruct strog Gaussa aroxmato we frst troduce the followg Gaussa rocess, whch lays a mortat role to reset our strog aroxmato. * g s = I X s s, j, Let (2.4) j j ( s, s ) cov ( g( s), g( s )) Γ = j = 2 ( g( s) g j ( s )) g( s ) g j ( s) + cov, + cov,. Defe, for t b, two arameter mea zero Gaussa rocess (2.5) B ( t ) = + 2 K t, / K u, /, : dc u, t C t C u where { K( st, ), st, b} s a Kefer rocess Theorem 3 of [5] wth covarace fucto ( tt ss ) ( tt ) ( ss ) * Γ = Γ,,, m,,, ad Γ ( ss, ) gve by (2.4). We ow restate below a strog aroxmato by Bolbola et al. [3] for the ormed PL-quatle rocess ρ ( u): = f( Qu )[ Qu Q( u)] by a two arameter Gaussa rocess at the rate O((log ) λ ), for some λ >. The statemets are codtoal o the observed samle sze. Theorem 2. (Bolbola et al. [3]) Let < <. Uder Assumto A, assume that s Lschtz cotuous ad that s twce cotuously dfferetable o [Q( )δ, Q( )+δ], for some δ >, such that f s bouded away from zero, the there exsts a two arameter mea zero Gaussa rocess B(t,u) for t,u, such that, ( ) B ( Q ) su ρ, (( log ) ) = O λ, for some λ > We wll gve strog Gaussa aroxmato of the Lorez rocess over restrcted terval [, ] for fxed < <. I the full model, Lagberg et al. [22] defe the total tme o test trasform curve corresodg to a cotuous dstrbuto o [, ),, for [,] as = ( ) y dq y ( ) Q ( ) Q ( y ) dy Q = +, =. Obvously, (): = lm ( ) =. or the our model, we modfy the defto of as = Q (2.6) + Q y dy,,. [ ] 74

5 Asymtotc Behavors of the Lorez Curve for Left Trucated ad Deedet Data As ad, Q ( y ) dy =. We ca regard ( ) as a surrogate for the fte mea μ. A atural estmator for s = ( ) Q [ ] + Q y dy,,. I the ext theorem, we costruct a two arameter mea zero Gaussa rocess that strogly uformly aroxmate the emrcal rocess (). Theorem 3. Let < <. Uder Assumto A, assume that s Lschtz cotuous ad that s twce cotuously dfferetable o [Q( )δ, Q( )+δ], for some δ >, such that f s bouded away from zero. The there exsts a two arameter mea zero Gaussa rocess B(t,u) for t,u, such that, almost surely, (2.7) su for some λ >. ( ) ( ) ( ) (, ) f ( Q ( y )) y B Q y dy ( y ) B ( Q ( y ), ) f ( Q ( y )) (( log ) ), L dy = O λ Proof. See the Aedx. The ext theorem gves a fuctoal LIL for the Lorez rocess. We work o the robablty sace of Theorem 3. Let D[a,b] Dab [, ] be the sace of fuctos o [a,b] that are rght cotuous ad have left lmts ad B s the ut ball the reroduce kerel lbert sace (Γ * ). Theorem 4. Suose that codtos of Theorem 3 are satsfed. O a rch eough robablty sace, l (.) / 2log log s almost surly relatvely comact D[, ] wth resect to the suremum orm ad ts set of lmt ots s h( y ) ( ) u G = g h : g h ( u ) = dy f Q y where h( y ) ( Q( y )),,, L u dy u h f ( x ) u + dc x g B 2 C g u = h :[, ] R, h( u) = C u g x :. Proof. Theorem 4 follows at oce from (2.7) ad Theorem A []. Aedx I establshg Theorem 3, we were aded by some deas foud [27], but frst we start wth the followg lemmas whch s ecessary for achevg the establshmet of the our results. Lemma. Suose the codtos of Theorem 2 are satsfed. We have, - - log log lm su ()- () =O Proof. By Lemma 3 of [25], we have, [ ] su ()- () su ( -) Q ()-Q() su Q (y)-q(y) dy log log = O Next, defe the ormed total tme o test emrcal rocess t ( ) by - - t = ()- (), [, ]. Lemma 2 characterze the asymtotc lmt of t ( ). Lemma 2. Suose the codtos of Theorem 2 are satsfed. The there exsts a two arameter mea zero Gaussa rocess Btu (, ) for tu,, such that, ( -y)b(q(y), ) su t ()- dy f(q(y)) 75

6 Vol. 23 No. 2 Srg 22 Bolbola Ghalbaf et al. J. Sc. I. R. Ira 2 (-) B(Q(), ) -λ + = O ((log ) ), f(q()) Proof. Proof of ths lemma ca be doe usg smlar augmet of Lemma 3.2 [27], we therefore omt the roof. Next, we defe the scaled total tme o test trasform, ts statstc ad assocated emrcal rocess corresodg to. ad (3.) W ():=, W ():=, w ():= [ W ()-W () ] for [, ]. The followg lemmas gve the strog uform cosstecy of W () ad strog Gaussa aroxmato of the scaled total tme o test emrcal rocess resectvely. Lemma 3. Suose that codtos of Theorem 2 are satsfed. We have, log log su W ()-W () = O Proof. By tragular equalty ad Lemma, the left had sde s bouded by su su ( ) ( ) su su ( ) - log log = O Lemma 4. Suose that codtos of Theorem 2 are satsfed. The there exsts a two arameter mea zero Gaussa rocess Btu (, ) for tu,, such that, su w ()- - 2 (-y)b(q(y), ) (-) B(Q(), ) dy + for some λ >. + f(q(y)) ( ) f(q()) ( -y)b(q(y), ) dy f(q(y)) -λ = O (log ), Proof. Proof ca be doe alog the les of Lemma 3.5 of [27], we therefore omt the roof. Proof of Theorem 3. By Defto of the Lorez curve corresodg to the our model ad by usg (2.6) ad (3.) we have (-y)q(y) (3.2) W (y) = + L( y). Q(u)du We have also (-y)q (y) (3.3) W (y) = + L( y), y [, ]. Q (u)du Substtutg (3.2) ad (3.3) Lemma 4, we obta the result. Ackowledgemets The authors would lke to scerely thak aoymous referees for the careful readg of the mauscrt Refereces. Berkes, I. ad Phl, W. A almost sure varace rcle for the emrcal dstrbuto fucto of mxg radom varables. Z. Wahrschelchketstheore Verw. Gebete. 4: 5-37 (977). 2. Bolbola Ghalbaf, M., akoor, V. ad Azaroosh,.A. Asymtotc behavors of the Lorez curve for cesored data uder strog mxg. Commucatos Statstcs - Theory ad Methods. 4: (2). 3. Bolbola Ghalbaf, M., akoor, V. ad Azaroosh,.A. Strog Gaussa aroxmatos of roduct-lmt ad quatle rocesses for trucated data uder strog mxg. Statstcs ad Probablty Letters. 8: (2). 4. Ca, Z. Kerel desty ad hazard rate estmato for cesored deedet data. Joural of Multvarate Aalyss. 76

7 Asymtotc Behavors of the Lorez Curve for Left Trucated ad Deedet Data 67: (998). 5. Ca, Z. Estmatg a dstrbuto fucto for cesored tmes seres data. Joural of Multvarate Aalyss. 78: (2). 6. Csörgő, M., Csörgő, S. ad orváth, L. A asymtotc theory for emrcal relablty ad cocetrato rocesses. Lecture Notes Statstcs, Srger Berl edelberg NewYork Vol. 33, (986). 7. Csörgő, M., Csörgő, S. ad orváth, L. Estmato of total tme o test trasforms ad Lorez curves uder radom cesorsh. Statstcs. 8: (987). 8. Csörgő, M. ad Yu,. Weak aroxmatos for emrcal Lorez curves ad ther Golde verses of statoary observatos. Adv. Al. Prob. 3: (999). 9. Csörgő, M. ad Ztks, R. Strasse s LIL for the Lorez curve. Joural of Multvarate Aalyss. 59: -2 (996a).. Csörgő, M. ad Ztks, R. Cofdece bads for the Lorez curve ad Golde curves. I A volume hoor of Samuel Kotz. NewYork: Wley (996b).. Csörgő, M. ad Ztks, R. O the rate of strog cosstecy of Lorez curves. Statstcs ad Probablty Letters. 34: 3-2 (997). 2. Davydov, Y. ad Ztks, R. Covergece of geeralzed Lorez curves based o statoary ergodc radom sequeces wth determstc ose. Statstcs ad Probablty Letters. 59: (22). 3. Davydov, Y. ad Ztks, R. Geeralzed Lorez curves ad covexfcatos of stochastc rocesses. J. Al. Probab. Vol. 4, No. 4: (23). 4. Davydov, Y., Khoshevsa, D., Shc, Z. ad Ztks, R. Covex rearragemets, geeralzed Lorez curves, ad correlated Gaussa data. Joural of Statstcal Plag ad Iferece. 37: (27). 5. Dhomogsa, S. A ote o the almost sure aroxmato of the emrcal rocess of weakly deedet radom varables. Yokohama Math. J. 32: 3-2 (984). 6. Doukha, P. Mxg: Proertes ad examles. Lecture Notes Statstcs. 85, Srger-Verlag, New York (994). 7. akoor, V. ad Nakhae Rad, N. Asymtotc behavors of the Lorez curve uder strog mxg. Pak. J. Stat. I Press (29). 8. Gastwrth, J.L. A geeral defto of the Lorez curve. Ecoometrca. 39: (97). 9. Golde, C.M. Covergece theorems for emrcal Lorez curve ad ther verses. Advaces Aled Probablty. 9: (977). 2. elmers, R. ad Ztks, R. Strog laws for geeralzed absolute Lorez curves whe data are statoary ad ergodc sequeces. Proc. Amer. Math. Soc. 33: (25). 2. Kedall, M.G. ad Stuart, A. The advaced theory of statstcs I. (2d. ed.) Charles Grffe ad Comay, Lodo (963). 22. Lagberg, N.A., Leo, R.V. ad Proscha,. Characterzato of oarametrc classes of lfe dstrbutos. Aals of Probablty. 8: 63-7 (98). 23. Lemda, M., Ould-Saïd, E. ad Poul, N. Strog reresetato of the quatle fucto for left trucated ad deedet data. Mathematcal Methods of statstcs. Vol. 4, No. 3: (25). 24. Masry, E. ad Tjosthem, D. Noarametrc estmato ad detfcato of olear ARC tme seres: Strog covergece ad asymtotc ormalty. Eco. Theor. : (995). 25. Petra, G. Delle relazo fra dc d varabltá, ote I e II. Att del Reale Isttuto Veeto d Sceze, Lettere ed Art. 74: (95). 26. Rao, C.R. ad Zhao, L.C. Strasse s law of the terated logarthm for the Lorez curves. Joural of Multvarate Aalyss. 54: (995). 27. Tse, S.M. Lorez curve for trucated ad cesored data. AISM. 58: (26). 28. Woodroofe, M. Estmatg a dstrbuto fucto wth trucated data. A. Statst. 3: (985). 77