Strassen's LIL for the Lorenz Curve

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1 joural of multivariate aalysis 59, 112 (1996) article o. 5 Strasse's LIL for the Lorez Curve Miklo s Cso rgo * ad Ric$ ardas Zitikis - Carleto Uiversity, Ottawa, Otario, Caada We prove Strasse's law of the iterated logarithm for the Lorez process assumig that the uderlyig distributio fuctio F ad its iverse F &1 are cotiuous, ad the momet EX 2+= is fiite for some =>. Previous work i this area is based o assumig the existece of the desity f :=F$ combied with further assumptios o F ad f. Beig based oly o cotiuity ad momet assumptios, our method of proof is differet from that used previously by others, ad is maily based o a limit theorem for the (geeral) itegrated empirical differece process. The obtaied result covers all those we are aware of o the LIL problem i this area Academic Press, Ic. 1. INTRODUCTION AND THE MAIN RESULT Let X be a o-egative radom variable with distributio fuctio F. We assume throughout that the mea + :=EX is fiite ad positive. The Lorez curve correspodig to the radom variable X, deoted by L F,is defied (cf. Gastwirth, 1971) by the formula t [ L F (t) := 1 + t F &1 (s) ds, t1, where F &1 deotes the left-cotiuous iverse of F. I ecoometrics it is customary to iterpret L F (t) as the proportio of total amout of ``wealth'' that is owed by the least fortuate t_1 percet of a ``populatio.'' For some details o the variety of situatios where estimatig the curve L F is of importace, we may refer, for example, to: Received April 3, 1995; revised February AMS 1991 subject classificatios: 6F15, 6K1. Key words ad phrases: Lorez curve, Lorez process, Strasse's law of the iterated logarithm, Vervaat process, itegrated empirical differece process, empirical process, quatile process, relative compactess, mea residual life process, total time o test fuctio, Lorez process of order &, Shao process, redudacy process. * Research supported by a NSERC Caada grat at Carleto Uiversity, Ottawa. - Work doe while the author was a Caada Iteratioal Fellow at Carleto Uiversity, Ottawa; o leave from the Istitute of Mathematics ad Iformatics, Vilius X Copyright 1996 by Academic Press, Ic. All rights of reproductio i ay form reserved.

2 2 CSO RGO AND ZITIKIS (a) measuremets of icome ad wealthatkiso (197), Dasgupta, Se, ad Starrett (1973), Gastwirth (1971, 1972, 1988a, 1988b), Hart (1971, 1975), Se (1973, 1974, 1976); (b) school segregatioalker (1965); (c) atitrust ad idustrial cocetratiohart (1971, 1975); (d) oe perso oe vote casesgastwirth (1988a, 1988b); (e) fisherma's luckthompso (1976); (f) bibliographyleimkuhler (1967); (g) publishig productivity amog scietistsgoldie (1977); etc. Lorez curves have bee i use for more tha 9 years (cf. Lorez, 195). The empirical couterpart, deoted by L, to the Lorez curve L F is defied (cf. Gastwirth, 1971, 1972) as follows: Let X 1,..., X be idepedet copies of X, ad let F be the empirical (right-cotiuous) distributio fuctio based o X 1,..., X. If we ow deote the left-cotiuous iverse of F by F &1, the the empirical Lorez curve L is the fuctio t [ L (t) := 1 + t F &1 (s) ds, t1, where + stads for the empirical mea of the radom sample X 1,..., X. I this paper we study Strasse's law of the iterated logarithm for the empirical Lorez process l :=- [L &L F ]. Before formulatig our mai result, we eed to itroduce further otatios. Let H deote the so called Fikelstei class (cf. Fikelstei, 1971) cosistig of all absolutely cotiuous fuctios h: [,1]R such that h()==h(1) ad t [h$(s)]2 ds1. We use D[, 1] to deote the set of all left-cotiuous fuctios o [, 1] that have right-had limits at each poit. Let L be the set [l h : h # H], where l h (t) :=& 1 + F&1 (t) h b F(x) dx+ 1 + L F(t) h b F(x) dx. Throughout this paper ``wrt'' stads for ``with respect to,'' ad &}& deotes the sup-orm sup[ }(t): t#(,1)].

3 STRASSEN'S LIL FOR LORENZ CURVE 3 The mai aim of this ote is to prove the followig Strasse's law of the iterated logarithm for the empirical Lorez process l. Mai Theorem. Assume the two coditios: (A1)F ad F &1 are cotiuous, ad (A2) EX 2+= < o for some =>. The l - 2 log log L a.s. wrt &}& o D[, 1]. We proceed with some historical remarks cocerig rates of strog cosistecy for the empirical Lorez curve L. Assumig the fiiteess of the first momet +>, Gail ad Gastwirth (1978) obtaied poitwise strog cosistecy of L. Uder the same assumptio, Goldie (1977) proved strog cosistecy of the Lorez curve L uiformly over the iterval [, 1], that is to say, Goldie (1977) proved the GlivekoCatelli type result &L &L F &,, almost surely. Together with other processes of similar vei, ad assumig the existece of the desity f :=F$, as well as further assumptios o f, M. Cso rgo, S. Cso rgo, ad Horva th [CsCsH] (1986) established a strog ivariace priciple for the Lorez process l by which they easily cocluded the followig (right) rate of strog cosistecy (cf. Corollary 11.4 o page 96 therei) that ca be cosidered as the first LIL type result for l. Theorem 1.1 (CsCsH, 1986). Assume the two coditios: (B1) F is absolutely cotiuous ad the desity f is positive o the iterior of the support of F; (B2) for some :, ; #[, 3 ) 2 The, almost surely, with the costat J \1 :,1 ;+ lim sup t : (1&t) ; := sup <t<1 f(f &1 (t)) <. &l &- 2 log log 4 1 (F ) 4 1 (F ):= { 1 12 s(1&s) log log 1 df s(1&s)= &1 (s). It is easy to see that assumptio (B2) implies (A2). Cosequetly, Theorem 1.1 follows from our Mai Theorem.

4 4 CSO RGO AND ZITIKIS The first ``real'' law of the Iterated logarithm i Strasse's LIL form for the Lorez process l was proved i two versios by Rao ad Zhao (1995). We restate their two LIL results as the followig two theorems. Theorem 1.2 (Rao ad Zhao, 1995). Uder the assumptios (B1) ad (B2), the statemet of the Mai Theorem holds true. Theorem 1.3 (Rao ad Zhao, 1995). Assume the coditios: (C1) Fis twice differetiable o the iterior I of the support of F, ad the desity f is positive o I; (C2) (i) for some #>, sup F(x)(1&F )(x)) f$(x) f 2 (x)#, x # I (ii) for some * #(, 1 2 ) (iii) (1&F(x)) 12&* dx<, 1 1&$ F&1 (s) dx=(-$ log log (1$)), $. The the statemet of the Mai Theorem holds true. As was correctly idicated i Rao ad Zhao (1995), these two theorems (i.e., Theorems 1.2 ad 1.3) are, i geeral, differet results i that their coditios caot really be compared. Both of them are corollaries, however, to our Mai Theorem. I the case of Theorem 1.2 this is maily a cosequece of the implicatio (B2) O (A2), while i the case of Theorem 1.3 we have that (C2, ii) O (A2). We emphasize agai that our Mai Theorem does ot require absolute cotiuity of F, which is a requiremet i the so far kow LIL Theorems We coclude this sectio with several remarks. Remark 1.1. A careful ad tedious ispectio of the origial proof of the Mai Theorem i M. Cso rgo ad Zitikis (1995b) shows that assumptio (A2) could possibly be replaced by the followig: (A2)$ - 1&F(x) dx<, ad the result of the Mai Theorem would cotiue to hold true. Although this replacemet would lead to a somewhat stroger result due to (A2)O(A2)$O (EX 2 <), we retaied assumptio (A2) deliberately i the statemet ad the proof of the Mai Theorem sice it seems to us that,

5 STRASSEN'S LIL FOR LORENZ CURVE 5 just as for the approximatio i probability of the process l by appropriate Gaussia processes (cf. Theorem 11.2 of CsCsH, 1986), the optimal assumptio for the validity of the Mai Theorem is EX 2 <, that is to say, assumptio (A2) with ==. Thus, chagig assumptio (A2) to (A2)$ ca be somewhat misleadig. O the other had, all our attempts to replace (A2) by EX 2 < i the Mai Theorem have failed so far. Remark 1.2. At this stage it is ot clear to us what oe should do i order to relax (or get rid of) assumig cotiuity of the distributio fuctio F ad its iverse F &1. The otios preseted i Remark 3 of Major ad Rejto (1988) might tur out to be helpful i fidig a solutio of this problem. Remark 1.3. result From the Mai Theorem we easily coclude the followig lim sup &l &- 2 log log =4 (F ) :=sup[&l h &: h # H] a.s. It is easy to check that 4 (F )<4 1 (F). Therefore, havig proved Strasse's LIL for the Lorez process, we may ow derive a.s. better cofidece bads tha those obtaiable from Theorem 1.1 (cf. M. Cso rgo ad Zitikis, 1996, for more details o the subject). Remark 1.4. The ideas preseted i, ad the method of proof of, M. Cso rgo ad Zitikis, 1994a, ca be used to prove weighted approximatio results for the Lorez process, ad thus costruct cofidece bads for the (theoretical) Lorez curve L F as well (cf. M. Cso rgo ad Zitikis, 1996). These goals ca also be achieved idirectly by usig results of M. Cso rgo ad Zitikis (1995a) ad the represetatio of Chadra ad Sigpurwalla (1978) (cf. Eq. (6) o page 776 of Shorack ad Weller, 1986, for a coveiet referece) L F (F(x))=1&(1&F)(x))[M F (x)+x]+, (1.1) where x [ M F (x):=e[x&x X>x] is the mea residual life fuctio. O the other had, represetatio (1.1) ca be used to derive Strasse's LIL for mea residual life processes via utilizig ow the Mai Theorem here established for the Lorez process. Furthermore, based o these observatios, ad o havig T F (t)=+l F (t)+(1&t) F &1 (t), (1.2) aother result from Chadra ad Sigpurwalla (1978) (cf. Eq. (7) o page 776 of Shorack ad Weller, 1986, for a coveiet referece) cocerig the total time o test fuctio T F, oe ca also have Strasse's LIL, weak

6 6 CSO RGO AND ZITIKIS approximatio results, cofidece bads, etc., for the total time o test fuctio T F as well. I additio to these otios cocerig T F, we ote also that, with emphasis put o Strasse's LIL, the same remarks are applicable to the Lorez process of order &, the Shao process, as well as to the empirical redudacy process ad some others of similar vei. For defiitios ad a first uified treatmet of strog ad weak approximatios of all these processes, we refer to CsCsH (1986) ad to Shorack ad Weller (1986) for further related results ad discussios. I this regard we also ote results by M. Cso rgo ad Horva th (1989), where cofidece bads with prescribed cofidece levels are costructed for the quatile fuctio F &1 without assumig the existece of the desity fuctio f, which is ot assumed i this paper either. 2. PROOF OF THE MAIN THEOREM A elemetary computatio shows that L (t)&l F (t)= 1 + t = 1 + t & L(t) + 1 (s)&f &1 (s)] ds& + &+ L(t) + (s)&f &1 (s)] ds where the last equality holds true because of + &+= 1 (s)&f &1 (s)] ds, (2.1) (s)&f &1 (s)] ds. Havig represetatio (2.1), oe's atural icliatio is to make use of the theory ad assumptios of geeral empirical quatile processes as i M. Cso rgo ad Re ve sz (1978, 1981), ad this is, i fact, the very route Rao ad Zhao (1995) took i provig their Theorem 1 (# Theorem 1.3 i the preset paper). Aother ivitig way is to use the strog approximatio of the process l give i Theorem 11.3 of CsCsH (1986). Ideed, Rao ad Zhao (1995) based their Theorem 2 (# Theorem 1.2 above) o the latter strog ivariace priciple. I retrospect, however, quatile methods i this cotext appear to have bee somewhat misleadig i that, roughly speakig, itegrals of quatile processes are almost equal to itegrals of their correspodig empirical processes. Cosequetly the, i geeral, less restrictive theory ad methods of (weighted) empirical processes ca be

7 STRASSEN'S LIL FOR LORENZ CURVE 7 used istead, ad this, i tur, should result i stroger results tha those obtaiable via a direct use of quatile processes ad methods. As a prelimiary support of these claims, we state the followig easy-to-prove equality: 1 (s)&f &1 (s)] ds=& [F (x)&f(x)]dx. The problem of showig that the ``remaider'' term V (t) i the ``expasio'' t (s)&f &1 (s)] ds=& F&1(t) [F (x)&f(x)]dx+v (t) (2.2) is small for t # (, 1) is a slightly more difficult task. Specifically, it is show i Sectio 3 that, uder the assumptios of the Mai Theorem, the statemet - log log &V &=o(1),, (2.3) holds true almost surely. I this sectio we take (2.3) for grated. We ote i passig that, as idicated above, V (1)=. I geeral, however, for t # (, 1), the quatity V (t) is ot equal to. This reders the result (2.3) o-trivial. Assumig the for the time beig (2.3), we ow proceed with the proof of the Mai Theorem. A elemetary calculatio o the right-had side of (2.1) yields the represetatio L (t)&l F (t)=& 1 + F&1 (t) _ where Q (t) deotes the remaider term: [F (x)&f(x)]dx+ L(t) + [F (x)&f(x)]dx+q (t) (2.4) + &+ + + F &1(t) [F (x)&f(x)]dx& + &+ + + L(t) _ [F (x)&f(x)]dx+ 1 + V (t). Now, by the classical law of the iterated logarithm, we have that - [+ &+]-2 log log [&_, _] a.s. wrt } o R, (2.5)

8 8 CSO RGO AND ZITIKIS where _ 2 :=Var X. Furthermore, usig assumptio (A2), together with Corollary 2 o page 771 of James (1975), we get that the itegral f (x)&f(x) dx coverges almost surely to as. Therefore, the statemet that - log log &Q &=o(1),, (2.7) holds true almost surely is equivalet to the claim (2.3) (to be proved i Sectio 3). O the other had, statemet (2.7) i combiatio with represetatio (2.4) implies that the Mai Theorem amouts to Strasse's (LIL) for the process t [ & 1 + F &1 (t) _ [F (x)&f(x)]dx+ L(t) + [F (x)&f(x)]dx, t1, (2.8) which we ow proceed to prove. To this ed we eed some additioal otatio. Let D[, ) deote the set of all bouded ad right-cotiuous fuctios o [, ) that have left-had limits at each poit. Furthermore, fix a (small) $>, let q $ be the fuctio t [ [t(1&t)] (12)&$, ad let $ be the mappig from (D[, ), &}&) ito (D[, 1], &}&) defied by $ (v)(t) := 1 + F&1 (t) v(x) q $ b F(x) dx& L(t) + v(x) q $ b F(x) dx for fuctios v # D[, ). If we take $> sufficietly small (depedig o => that appears i assumptio (A2)), the we have that the itegral q $ bf(x)dx is fiite. Thus, for small $>, the mappig $ is cotiuous. (2.9) Sice the process W := 2 log log F &F q $ b F is a elemet of D[, ), we have by Theorem o page 77 of James (1975) (cf. also Theorem 1 o page 517 of Shorack ad Weller, 1986) that where W H $ a.s. wrt &}& o D[, ), (2.1) H $ :=[hbfq $ b F: h # H].

9 STRASSEN'S LIL FOR LORENZ CURVE 9 Furthermore, by (2.9), (2.1), ad the `` mappig theorem'' (cf. Theorem 5 o page 78 of Shorack ad Weller, 1986), we get $ (W ) $ (H $ ) a.s. wrt &}& o D[, 1]. (2.11) The simple observatio $ (H $ )=[l h : h # H]#L completes the proof of Strasse's LIL for the process (2.8). This, i tur, yields the Mai Theorem as well. K 3. THE VERVAAT PROCESS AND POLONIK'S PROOF OF (2.3) The ``expasio'' i (2.2) actually defies the process V (t):= t (s)&f &1 (s)] ds+ F&1 (t) [F (x)&f(x)]dx, t1, whose (, 1)-uiform versio is kow i the literature as the itegrated empirical differece process or, briefly the Vervaat processes (cf., for example, Shorack ad Weller, 1986). The process V (i the (, 1)-uiform case) was itroduced ad ivestigated by Vervaat (1972) (cf. also Sectio 2 i Chapter 15 of Shorack ad Weller, 1986). I particular, Vervaat (1972) proved Strasse's law of the iterated logarithm for the process V i the (, 1)-uiform case, which easily implies that the statemet - log log &V &=O(- &1 log log ),, (3.1) holds true almost surely (compare (3.1) with (2.3)). I Sectio 2 we faced the crucial (for this paper) problem of showig that statemet (2.3) holds true uder the coditios of the Mai Theorem. The followig very beautiful ad elegat proof of this fact is due to Wolfgag Poloik. Proof of (2.3) (Due to W. Poloik). It follows from elemetary geometrical cosideratios (see, for example, Fig. 1 o p. 585 ad Eq. (a) o page 594 of Shorack ad Weller, 1986) that V (t)= F&1 (t) F &1 (t) [F (x)&t] dx F bf &1 (t)&t F &1 (t)&f &1 (t). (3.2) Dividig ad multiplyig the right had side of (3.2) by the weight fuctio q $ (t) :=[t(1&t)] (12)&$ with some (small) $>, ad the takig the

10 1 CSO RGO AND ZITIKIS supremum, we get from (3.2) that statemet (2.3) is a elemetary cosequece of the followig two facts: (1) Corollary 2 o page 771 of James (1975); ad (2) Theorem 3 o page 51 of Maso (1982) (cf. also Exercise 5 o page 651 of Shorack ad Weller, 1986) which implies, i particular, that &q $ &F &1 ]&=o(1),, holds true almost surely for some small $> depedig o =>. Remark 3.1. A careful ispectio of the boud (3.2) shows that the right-had side of it reflects the true asymptotic behavior of the Vervaat process V (t). This fact therefore suggests that the a.s. o(1) rate of covergece i (2.3) ca hardly be, i geeral, icreased without postulatig more smoothess tha the mere cotiuity of the fuctios F ad F &1. This, i tur, suggests the followig quite itriguig K Ope Problem. does the result Uder what coditios o the distributio fuctio F - log log &V &=o(c ) or O(c ),, hold true almost surely (or i probability) for a fixed sequece c, =1, 2,...? Let us ote i cocludig that Vervaat (1982) ad result (2.3) are special solutios of the Ope Problem. ACKNOWLEDGMENTS Sicere thaks are due to Joseph L. Gastwirth for his ecouragemet ad suggestios durig the preparatio of this paper. A editor ad a referee made several very valuable remarks ad observatios that eabled us to make a substatial revisio of the paper. I particular, Wolfgag Poloik acquaited us with the beautiful, short, ad elegat proof give above for the crucial (for this papers) result (2.3). Our origial proof took almost six pages to preset it. REFERENCES [1] Alker, H. R., J. (1965). Mathematics ad Politics. MacMilla Co., New York. [2] Atkiso, A. B. (197). O the measuremet of iequality. J. Ecoom. Theory [3] Chadra, M., ad Sigpurwalla, N. D. (1978). The Gii idex, the Lorez curve, ad the total time o test trasforms. Upublished techical report. George Washigto Uiversity. [4] Cso rgo, M. (1983). Quatile Processes with Statistical Applicatios. SIAM, Philadelphia.

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