Functional Weak Laws for the Weighted Mean Losses or Gains and Applications

Transcription

1 Applied Mahemaic Publihed Olie May 5 i SciRe hp://wwwcirporg/joural/am hp://dxdoiorg/436/am56579 Fucioal Wea Law for he Weighed Mea Loe or Gai ad Applicaio Gae Samb Lo Serige Touba Sall 3 Pape Djiby Mergae LERSTAD Uiverié Gao Berger de Sai-Loui Sai-Loui Séégal LSTA Uiverié Pierre e Marie Curie Pari Frace 3 Ecole Normale Supérieure Daar Séégal pdmergae@ufraorg Received February 5; acceped 4 May 5; publihed 7 May 5 Copyrigh 5 by auhor ad Scieific Reearch Publihig Ic Thi wor i liceed uder he Creaive Commo Aribuio Ieraioal Licee (CC BY hp://creaivecommoorg/licee/by/4/ Abrac I hi paper we how ha may ri meaure ariig i Acuarial Sciece Fiace Medicie Welfare aalyi ec are gahered i clae of Weighed Mea Lo or Gai (WMLG aiic Some of hem are Upper Threhold Baed (UTH or Lower Threhold Baed (LTH Thee aiic may be ime-depede whe he cee i moiored i he ime ad deped o pecific fucio w ad d Thi paper provide ime-depede ad uiformly fucioal wea aympoic law ha allow emporal ad paial udie of he ri a well a compario amog aiic i erm of depedece ad muual ifluece The reul are paricularized for uual aiic lie he Kawai ad Shorroc oe ha are maily ued i welfare aalyi Daa-drive applicaio baed o peudo-pael daa are provided Keyword Empirical Proce Time Depede Proce Wea Theory Ri Meaure Povery Idex Lo Fucio Ecoomic Welfare Iroducio ad Moivaio I may iuaio ad may area we face he double problem of eimaig he ri of lyig i ome mared zoe ad a he ame ime he co aociaed wih i To fix idea we may be iereed i eimaig he immuocompromied paie umber Q ad he ize of he e of ifeced people i ome populaio A he ame ime we ow ha he everiy of he ifecio i meaured by he viral load Y expreed i RNA copie per millilier of blood plama The co of reaeme for example a coure of chemoherapy heavily deped o he viral load If oe ha o rea all he paie here i a co o pay for each reame How o cie hi paper: Lo GS Sall ST ad Mergae PD (5 Fucioal Wea Law for he Weighed Mea Loe or Gai ad Applicaio Applied Mahemaic hp://dxdoiorg/436/am56579

2 G S Lo e al d Y Facig hee wo problem a he ame ime comparig wo differe populaio Qd Y raher ha o which i commoly called he HIV/AIDS adul prevalece rae o wha i baed ieraioal compario I order o mae a worable aiic coider a ample of idividual = { } draw for ad meaure he viral load Y for each j A geeral comparaive aiic hould be of he form which i a co fucio or moiorig he evoluio of he global iuaio hould be baed o he couple ( j j ( j d Y Sice compario over he ime are baed o hi idex oe will be iereed i puig more or le emphai o he more ifeced or o i erm of viral load Thi i achieved by affecig a weigh ρ ( j o j a a moooe fucio of he ra R j of Y j i he ample For a icreaig ρ i i paid more aeio o le ifeced while he corary hold for a decreaig oe Thi lead o aiic lie J( d ρ = ρ( Rj d( Yj ( I i alo ow ha he viral load i deecable oly above a hrehold of value Z = 4 RNA copie per millilier of blood plama We hu have ad j j Y Z J ( d ρ = ρ( R d( Y j j Yj Z We may decide o cocerae o he very expaive chemoherapy coure due o fiacial preure I ha cae we chage he hrehold o Z > Z accordig o he available budge Such aiic are alo ued i iurace heory Suppoe ha oe iurace compay receive claim { Y Y } We may fix a hrehold Z uch ha ay claim greaer ha Z i ee a cauig a lo ( Yj Z for he compay I he become iereig o eimae he umber of poible claim over Z Y Z j Q = ( j ( ad o chooe a diorio fucio of he idividual lo ( Z Y j ; hece ( i raformed here io J Zd R Y jdy Z j j j ( ρ = ρ( = ρ ( Yj Z j Q where Y Y Y Y J d ρ may be ee a a ri meaure I poor courie a idividual i coidered a a poor oe whe hi icome Y below ome hrehold Z called povery lie Ad he are he order aiic baed o { } I hi cae Q = ( j (3 Y Z i he oal umber of poor people i he ample while Q i he poor headcou Uually he co fucio Y Z Y Z J d ρ here deped o he relaive povery gap ( j ( j = I hi field followig Lo [] may be called a Geeral Povery Idex (GPI The ame form may alo be ued i medical ciece whe dealig wih viamie (ay viamie D deficiecy I hi cae J ( Zdρ i ued a a geeral meaure of viamie deficiecy o evaluae he mea co of viamie upply a a reame We ee from he lie above ha ( i a very geeral aiic which wor i variou field wih loe or gai depede o he meaig of he co fucio c We are eiled o ame i a a Weighed Mea Lo or Gai (WMLG aiic or radom meaure or idex I may ae a pecific ame depedig o he paricular field where i operae I he lo (rep gai cae we imply deoe i WML (rep WMG Whe we have ime-depede daa over he ime [ T ] wih coiuou obervaio 848

3 ({ Y Y } T we are led o a ime-depede WMLG aiic i he form J( ρ = ρ( Rj d( Yj j G S Lo e al I he cae where i baed o he hrehold Z ; he laer hould eveually deped o he ime ad become Z = Z Alo i a paial aalyi i would be poible o have a paricular hrehold for ay area The choice of d ad ρ deped of he pecific role played by ( Bu a e of axiom which are deirable or madaory o be fulfilled for a welfare or a ri meaure i uually adoped For ri meaure uch axiome alogide a axiomaic foudaio are o be foud i Arzer e al [] For povery aalyi a large ad deep review of he axiomaic approach due o Se [3] i available i Zheg [4] Fially aig io accou variou form of ( i he lieraure he followig form of hrehold-baed weighed mea lo eem o be a geeral oe or he followig depedig o wheher we hadle lo (wih ( Q ( 3 4 (( Q A J ( Z ω d = w µ + µ Q µ j+ µ d Z Y Z (4 j B ( Q ( 3 4 (( A J Z d w Q j d Z Y Z (5 ( ω = j B µ + µ µ + µ j Q Q defied i (3 or gai (wih Q w( j B Q = Q defied i ( ad where From a mahemaical poi of view he aympoic behavior of he wo form radically differ alhough he wriig eem ymerical The reao i ha for he fir he radom variable ued i (4 are bouded ad he aympoic hadlig i much eaier A for (5 we hould face heavy ail problem ad furher complicaio may arie Thi paper i aimed a offerig a full fucioal wea heory accordig o he mo rece eig of uch heorie a aed i [5] Paricularly we are iereed here i he ime-depede iveigaio of (4 ad ex he fucioal wea heory i d ad w We call he fir cla of aiic Upper Threhold Baed Weighed Mea Lo or Gai (UTB WMLG oe ad he oher are amed Lower Threhold Baed Weighed Mea Lo idice (LTB WMLG Thi paper i oly cocered wih he fir cla of aiic The oher will be objec of furher udie Coider for a while ha w ad d are fixed a well a he ime We oice ha aympoic reul of J ( Z ω d are available for pecific form i Welfare heory or i Acuarial Sciece For example Lo [] proved ha Z ( ω ( ω = ( G d d = G wg ( ZY d( ( ZY z ( Y Z J Z d J Z d w Z y z y z y = wmgl where wmgl may be called he Exac UTB WMLG For iace he weigh wg ( Z y = ( y i relaed o he Shorroc [6] ad Tho [7] aiic wg ( Z y = ( y Z i he Kawai weigh (ee [8] ha iclude he Se [3] oe correpodig o = For wg ( Z y = we ge he oweighed mea loe or gai To be able o bae aiical e of uch reul we may be iereed i fidig he aympoic law of ( ( ω ( ω J d J d a However we ill eed o hadle logiidual daa where he ri iuaio i aalyed over a coiuou period of ime [ T ] I hi cae we are faced wih coiuou daa i he form of { Y < < T} ad ome modificaio i eeded i he defiiio of idice o ae hi io accou We are he led o coider he 849

4 G S Lo e al ime-depede ad UTB WMLG aiic defied by ( B( Q AQ Z Q Z Yj ( µ µ µ 3 µ 4 Z J = w + Q j+ d (6 wih T ad T R Iead of aalyig uch UTB WMLG for ome pecific fucio w or d or a a fixed poi i may be more valuable o have a oce a uiform wea heory o w d ad T Such a reul will provide idividual e ad eable paial ad emporal compario of he ri meaure A well ice all he meaure are expreed i he ame Gauia field we have joi aympoic diribuio of he differe idice hemelve Thi paper i aimed a elig he uiform wea covergece of uch aiic which i he aympoic heory C T of real coiuou fucio defied o of he ime-depede povery meaure (6 i he pace ([ ] [ ] T Fir aemp were reaed for he pecial cae of ime-depede oweighed mea lo or gai (MLG meaure i [9] ad i [] for oradomly WMLG aiic ha i WMLG aiic for which he weigh i oradom lie he Shorroc oe i deal wih Now we arge o give here he mo geeral reul o he ime-depede UTB-WMLG aiic Two poeial applicaio area here are viamie deficiecy ri meaure ad povery meaure I i he aural o coider a hrehold depedig o he ime Bu we uppoe ha i lie i ome fiie ierval < Z Z Z < + A impora applicaio i he aiical eimaio of he Relaive Mea Lo Variaio (RMLV from ime o defied a follow RJ = J J J ( i a povery meaure oe of he Milleium Developme Goal (MDG 5% Our reul below acle hi iue We will eed a umber of hypohee oward a adequae frame for our udy Thee hypohee may appear evere ad umerou a fir igh bu mo of hem are aural ad eay o ge We fir eed he followig hape codiio for he WMLG meaure hemelve The leer S i he hypohee ame refer o hape codiio by cofidece ierval where J i halvig of exreme povery from = o ime = 5 Thi mea ha we arge o have RJ ( (HS There exi fucio h( pq of ( pq N cuv ( ad π ( uv of ( ( of [ T] uch ha a + up max ( ( ( µ µ µ 3 µ 4 P [ T] j Q where uv idepede A Q h Q w + Q j+ c Q j = o o P deoe he covergece o zero i ouer probabiliy (ee [5] (HS up max w( j h ( Q Q j op T j Q π = [ ] 3 (HS3 There exi a fucio cuv ( of ( uv ( idepede of [ T] up ( µ µ µ 3 µ 4 [ T] j Q uch ha a + max A Q B Q w + Q j+ c Q j = o P We will require oher aumpio depedig o he regulariy of he fucio c ad π The leer R i hee hypohee ame refer o Regulariy codiio (HR The bivariae fucio c ad π have equi-coiuou parial differeial o ( βξ ( where β ad ξ are wo real umber o be defied laer o c (HR For a fixed x he fucio y ( xy ad y ( xy y π are moooe y (HR3 There exi T H > ad H < + uch ha for [ ] ( H < Hc = c G Z G y y dg y < H 85

5 G S Lo e al ad ( ( H < Hπ = π G Z G y e y dg y < H Our fial achieveme i ha whe puig J Hc Hπ law of { ( J J T} ad o decribe he limiig Gauia proce { T} eable he aiical uiform eimae of J ( = J J ( by J( J J( = we are able o ge he uiform aympoic Thi = by ierval cofidece We alo paricularize he reul for he o-impora Kawai cla of WMLG aiic of which he Se oe i a member The reul ha have direcly bee derived for he Shorroc cae are redicovered here Our Reul Our reul will rely o he repreeaio of Theorem [] which i ur will eed he followig aumpio (HL There exi β > ad < ξ < uch ha T (HL The ubcla = π : Z x T x Z iuou fucio i a up < β < if G Z < G Z < ξ < T { [ ]} of l C( [ T] -Gliveco-Caelli cla ha i a Y G Z G Z ao p [ T] up where for ay [ T] ad y R i= he e of real bouded ad co- G y = ( Y y A a remider R z aop a + mea ( z i ouer probabiliy ha i : here exi a equece of meaurable radom variable u uch ha for ay z u ad u a + f x x x l C T Fially le u deoe = where ([ ] (HL3 For ay [ T] [ T] G i ricly icreaig ad he fucio G are uiformly coiuou i (HL4 d i bouded by oe ad i differeiable wih derivaive fucio d bouded by M : d d M J = H H ad Theorem Suppoe ha (HS-(HS (HR-(HR3 ad (HL-(HL4 hold Pu Defie c Kc = ( G ( ( Z G d x (7 π Kπ = (( G ( ( d Z e G x K = H K H H K (8 π c c π π ( ( ( ( π g = cg Z G f f c g = G Z G f e f + K e f ν ν π c y = ( G Z G f y ( f ( y y c π y = ( G Z G f y e ( f ( y y π g = H g H H g π c c π π c π (9 85

6 G S Lo e al ad ν = Hπ νc Hc Hπ νπ The we have uiformly i [ T] ( J J α ( g β( ν op wih ad α he followig repreeaio a = + + ( { ( } = j j g g Y g Y { ( } ( β( ν = G Yj G Yj ν Yj ( Suppoe ha (HS3 (HR-(HR3 ad (HL-(HL4 hold The ( hold wih c c ad c K = K g = g v = ν ( Thi heorem expree our udied ime-depede aiic a he um of a fucioal empirical proce ad β ν i aympoically a iegral of he V baed o G Y G Y (where V ( i he empirical quaile he ochaic proce ( I will be ee for a fixed ime ha quaile proce ( ( fucio ad he of empirical proce α ( = ( I ( G Y j ( J J Thee fac mae eay he hadlig of i he moder empirical proce eig a aed i [5] We ill eed a horough udy of ( ad i coecio wih α while he compuaio of he variace ad covariace fucio Thi i doe eparaely i [] o avoid leghy paper Now we ue hee ool o give fir geeral law for he WMLG aiic below ad he for he Kawai cla of idice i Secio ad for he Shorroc-Tho idice i Secio 3 We fiih by a pecial udy of he abolue ad he relaive povery chage i Secio 4 While we deal wih he geeral idex ad we ue he oucome of Theorem we adop he followig wriig: α j j ( g = W W where Wj g Yj Wj ( i place of he Yj ( for he geeral cae Ad we uppoe ha Wj m for each [ T] I paricular cae we will ur bac o hypohee o he Y = The we are eiled o expre he hypohee (HT ad (HT below o he admi a deiy probabiliy j for eablihig (HT ad (HT3 ad ubequely recover he reul I he equel r i a fixed poiive real umber uch ha r o are performed whe ad < < Ad from ow he limi ad he P T for ome coa K W W( K + r (HT For (HT For T for ome coa K r W W K + I order o defie our la aumpio we eed he followig fucio: wih by coveio h hu dg ( u = y v g ν ν ν x d G x ν y d G y d G uv x u ( ν = ν d d c x G x G u x u = for a fucio h Se (HT3 If here i a uiveral coa K 3 uch ha for ay δ > for large eough value of { } ( 3 + r δ c ν g ν ν + Gν Gν Gν K3 (3 85

7 G S Lo e al We are ow able o give our geeral mai reul Theorem Aume he codiio of Theorem hold ad ha (HT-(HT3 are aified The he ochaic { } proce ( J J T coverge i ([ T ] { T} wih covariace fucio ( ( g g ( ν ν ( ( g ν wih ad wih ad g ad l o a ceered Gauia proce Γ =Γ +Γ +Γ +Γ 3 ( g g ( g( x η ( g( y η( d G( xy Γ = ( ν ν g( ν ν ( Gν ( Gν Γ = ( ( ν ν = ν d ν d d g x G x x G x G u v x u x v ( gν κ( gν κ( g ν Γ 3 = + ( g = g( u ( x dg( x d G uv κ ν ν ν are give i Theorem ad x v η = g ( y dg ( y Proof We have o do hree hig Fir we how ha ( J J i aympoically igh Nex we have o prove ha i coverge i fiie diribuio Ad fially we hould compue he covariace fucio We will oly ech he fir ad he ecod a wih he appropriae ciaio The ecod will be properly adreed Sice he aumpio of Theorem hold we have he repreeaio ( Pu N α ( g β( ν Fir (HT ad (HT yield for each j [ ] ad hece by repeaed ue of for ome coa K = + (4 for ome coa K r W W + W W K + j j j j c -iequaliy (ha i for ay couple ( α ( g α ( g K ab of calar a b ( a b + r + + (5 We remid agai ha r i ricly le ha oherwie fucio aifyig 5 are coa Here ad i he equel K i a geeric coa eveually aig differe value from oe formula o aoher Nex we fid i [] ha ( β ( ( ν β ν where ( i { } ( c( ν g( ν ν ( Gν( Gν ( Gν ( K + + (6 K i bouded uiformly i ad So by combiig (4 (5 (6 ad (HT3 ad by he c -iequaliy we ge for ome K ha for ay δ > for large eough value of + r 4 Thu { N [ T] } i [5] To fiih he proof we have o eablih ha fiie-diribuio of N δ N N K (7 i aympoically igh by Lemma i [9] which i a adapaio of Example coverge o hoe of ome Gauia igh proce For impliciy ae we do i i he wo dimeioal cae for 853

8 G S Lo e al ( N N( Coider N an bn( ad X Y where he ( j j = + Sill for impiciy ae le u e ( = + ( ν N g X G X G X X j j j j ( = + ( ν N g Y G Y G Y Y j j j j ( j j Y Y ad for he G i he empirical fucio baed o X X a idepede obervaio of ( XY G (rep (rep Y Y Pu ( ( ( ( G x y = P X x Y y G x = G x + G y = G + y Now le for each ε (rep ε be he quaile procee baed repecively o G( X G( X G( X (rep G( Y G( Y G( Y I i o hard o ee ha ad ( ( j ( j ν ( j = ε ν ( d + P G X G X X G o ( ( j ( j ν ( j = ε ν d + P G Y G Y Y G o Now le α ad α be he empirical procee baed repecively o G( X G( X G( X o G( Y G( Y G( Y We have (ee [3] p 584 ha αi ( εi ( op ( which give ε( = ( ε ( ε = ( α ( α + op uiformly i ( ( Now le u coider he fucioal empirical proce ( N = G X G Y ha i j j j α j j ( F = f ( N f ( N where f a real fucio defied o ( uch ha i proce baed o he ( X Y defied for h : R R We have i i ad = + uiformly i α baed o he f N < Fially le he focioal empirical ( H = { h( X Y h( X Y } j j j j α ( f ( h = f for hf ( xy = f( G( x G( y We have by he claical reul of empirical proce ha { ( h h } coverge o a Gauia proce { G( h h } wheever i a Doer cla I follow ha { ( C C } coverge o a Gauia proce { G( C C } wheever i a Vapi-Cervoei cla Bu = ] ] ] ] ( R i VC-cla of idex o greaer ha (ee [5] for VC-clae ue o em- { xy x y } pirical procee Thu puig hxy = ] x] ] y] we have i ( xy = ( h Gh = ( xy xy xy l R where i a igh Gauia proce uch ha ( h ( ( h d ( h h = h xy h xy Gxy 854

9 G S Lo e al Furher for f [ ] [ ] ad for h ( xy h xy = h ( xy = [ ] [ ] ( G ( x G ( y = ( ( xy h xy = = f + G f G R G + + ( ( f h α = α = G ( + ( = h xy α G Now by uig he Sorohod-Wichura-Dudley Theorem we are eiled o uppoe ha we are o a probabiliy pace uch ha Now ice he fucio i N = an + N i equal o ( h ( h up xy xy P ( xy R ν are bouded ad puig h( xy = g( x ad h( xy g( y { } = { ( ν ( ( ν } d G + + G P a h + b h + a h G + b h G + o Oe eaily prove ha { } { ( ν ( ( ν ( G } + + G a h + b h + a h G + b h G d i a Gauia radom variable ice he ecod erm i a Riema iegral which i a limi of fiie liear combiaio of Gauia radom variable Thu N i aympoically Gauia We are able o do he ame for a arbirary fiie-diribuio ( N N( The compuaio of he limiig Gauia proce require heavy calculaio doe i [] The proof ed wih providig he covariace fucio Γ of α ( g Γ of β ( ν ad he covariace Γ 3 fucio bewee hem Special Cae Sice he reul are aed i a more geeral form ad may appear very ophiicaed i eem eceary o how how hey wor for commo cae We apply our reul o wo ey example i Welfare aalyi: he cla of Kawai ad Shorroc aiic Thee wo example are paricularly iereig ice hey pu he emphai o he le deprived idividual wihi he whole populaio (wih weigh j+ for Shorroc aiic or wihi he mared idividual (wih weigh Q j+ for Kawai cla of aiic icludig e meaure I boh cae aig he weigh a he power may lead o more accuracy i he aiical eimaio The Kawai Cae We are ow applyig he geeral reul o he Kawai WMLG aiic of parameer defied by Q j Z Q Z Y J = ( Q j d Q + j The way we are uig here i o be repeaed for ay paricular idex For iace he reul i [9] ad [] may be redicovered i hi way I hi pecific cae we ur he hypohee (HT ad (HT o W j o he Y j a follow Suppoe he d f G ( x = ( Yj x admi a derivaive m ( x Pu G ( uv = ( X ux ( v Iroduce: (H For T for ome coa K Z( Z K + r (H There exi a poiive fucio m uch ha for T u R < r < ( + r Z d m um u mu ad mu u= K< 855

10 G S Lo e al ad ad up x ( Z Z (H For T for ome coa K u m x ( = M < r up G uu G u K + ( ( r G Z G Z Z K + (H3 For T for ome coa K r We chec i he Kawai cae ha he repreeaio of Theorem hold wih ( ( = ( ( xy cxy x y o ha Nex ad he where For ad y π = ad he x Y Y K + ( Hπ = G Z + ( Z c = d H G Z G y y G y Z = c π = ( + d hq ( J H H G y G Z y G y ( d ( Z c π K = G Z G y y G y K = G Z + { } ( K = + G Z r + r ( Z = d r G Z G y y G y ( ( g = + G f G Z f + K ( f Z Z = + Z y ν ( y = ( + G ( Z G ( Z ( G ( Z G ( y ( + r G ( y ( f ( Z we will ge he repreeaio wih ( = + P J J N o N = α ( g + β( ν Theorem 3 Le (HL (HL3 (HL4 (H-(H3 hold The ( ([ T ] Proof We begi o remar ha (H3 eure ha { J J T} { ( G Z G Z T} coverge i l o a ceered Gauia proce wih covariace fucio Γ give i Theorem i aympocially igh ad hece (HL I i he eough o how ha (HT ad (HT hold from (H (H (H ad (H3 856

11 G S Lo e al Bu hi follow from rouie calculaio ha we oly ech here We place hee calculaio i he appedix 3 The Shorroc-Tho-Lie Cae We apply our reul o he Shorroc-Tho WMLG aiic meaure defied by Z Y j d j Z Q J = j+ = Thi i he Tho idex Oe obai he Shorroc oe by replacig by ( here ha repreeaio of Theorem hold i he imple cae correpodig o (HS3 wih cxy ( ( x y I hi cae π i uele The Z c d J = H = G y y G y K = Kc = ν ( y = νc( y = ( y Here agai { ( J J T} g ( y = ( G ( y ad ν ( y = νc( y = ( y + We alo chec = ha he ame aympoic behaviour decribed i Theorem 3 wih uder he ame hypohee (HL-(HL4 ad (H-(H3 4 Eimaio of he WLMG Saiic Variaio Alhough hey are very expeive o collec logiudial daa are highly preferred for adequae eimae of he abolue idex variaio J( = J( J which i he exac meaure of WMLG chage bewee he RJ = J J J Their repecive period ad ad he aociae relaive WMLG variaio ( aural eimaor are of coure J( = J( J ad RJ( ( J( J J viou reul yield he follow Theorem 4 Uder he aumpiom of Theorem or Theorem ( ( ( ( Γ4 ( J J where Γ 4 ( =Γ ( +Γ( Γ ( ad ( RJ ( RJ( ( Γ5 ( where wih ( a ( a ( aa ( Γ = Γ + Γ + Γ 5 ( a = + RJ J = Our pre- a = J The proof i raighforward We alo migh coider he covergece of ( J ( J( Gauia proce ( = ( i l ([ T ] Ayway for fixed ad ( J ( J( coverge o he Gauia radom variable ( = ( uiy Theorem wih 4 ( ( RJ ( RJ ( = a ( + a + o P o he by he coi- Γ a variace Alo by uig he Sorohod-Wichura-Dudley Theorem we have A impora applicaio of he ecod par of hi heorem i relaed o checig he achieveme of pecific goal Oe may wihi a aioal or regioal raegy whih o have ome deprivaio limied o ome exe For example he UN ha aiged a umber of goal amed Milleium Developme Goal (MDG o i member We are cocered here by oe of hem Ideed i i whihed o halve he exreme povery i he world 857

12 G S Lo e al i year = 5 arig from year = Whe he WMLG aiic i a povery meaure we may ue RJ ( ad chec wheher i i le ha 5 Ad a ( α -cofidece ierval IR ( α baed o hee reul i where J ( α 5 ( ( ( RJ Γ5 u α RJ + Γ5 u α J α J α ( u α = α Thi MDG will be repored achieved a he 95% level if he umber 4 Daadrive Applicaio ad Variace Compuaio We apply our reul i Ecoomic ad Welfare aalyi Epecially we coider he houehold urvey i Seegal i (ESAM II ad i 6 (EPS from which we coruc peudo-pael daa ad apply our reul 4 Variace Compuaio for Seegalee Daa We apply our reul o Seegalee daa We do o really have logiudial daa So we have coruced peudo-pael daa of ize = 6 from wo urvey: ESAM II coduced from o ad EPS from 5 o 6 We ge wo erie X ad X We pree below he value of Γ I ( deoed here Γ J ( deoed here ( ad Γ ( deoed here ( 3 Whe corucig peudo-pael daa we ge mall ize lie = 6 here We ue hee ize o compue he aympoic variace i our reul wih oparameric mehod I real coex we hould ue high ize comparable o hoe of he real daabae ha i aroud e houad lie i he Seegalee cae Neverhele we bac o medium ize for iace = 696 which give very accurae cofidece ierval a how i Table Before we pree he oucome le u ay ome word o he pacage We provide differe R crip file a hp://wwwufraorg/lerad/reource/allmerglozip The uer hould already have hi daa i wo file daax ad daax The fir crip file amed afer gamma_mergloda provide he value of ( ad ( 3 for he FGT meaure for α = ad for he ix iequaliy meaure ued here The ecod crip file amed gamma_mergloda perform he ame for he Shorroc meaure Fially gamma_merglo3da cocer he awai meaure Ule he uer upload ew daax ad daax file he oucome hould he ame a hoe preeed i Appedix 4 Aalyi Fir of all we fid ha a a aympoical level all our iequaliy meaure ad povery idice ued here have decreaed Whe ipecig he aympoic variace we ee ha for he povery idex he FGT ad he Kawai clae repecively for α = α = ad = ad = have he miimum variace pecially for α = ad = Thi advocae for he ue of he Kawai ad he FGT meaure for povery reducio evaluaio Table Variaio of he povery idice Idex J J ( Γ 4 ( CI J 95% ( SHOR 346 KAK ( 895 KAK ( FGT ( FGT ( FGT ( [ ] 973 [ ] 78 [ ] [ ] 999 [ ] 8383 [ ] 858

13 G S Lo e al 5 Cocluio We obaied aympoic law of he UTB WMLG aiic wih i mid amog oher arge he uiform eimaio of he variaio J( ad he relaive variaio RJ ( The reul are oly illuraed wih imple daadrive applicaio o icome daabae i Seegal Thi ope large daadrive applicaio i whole ecoomic area I he heoriical had he Lower Threhold Baed weighed mea lo or gai aiic i o be udied i accordace wih heavy ail codiio ad o be applied i Iurace ad HIV/VIH field Referece [] Lo GS (3 The Geeralized Povery Idex Far Ea Joural of Theoreical Saiic 4 - hp://pphmjcom/joural/fjhm [] Arzer P Delbae F Eber JM ad Heah D (999 Cohere Meaure of Ri Mahemaical Fiace hp://dxdoiorg// [3] Se AK (976 Povery: A Ordial Approach o Meaureme Ecoomerica hp://dxdoiorg/37/978 [4] Zheg B (997 Aggregae Povery Meaure Joural of Ecoomic Survey 3-6 hp://dxdoiorg// [5] va der Vaar AW ad Weller JA (996 Wea Covergece ad Empirical Procee Wih Applicaio o Saiic Spriger New Yor [6] Shorroc AF (995 Reviiig he Se Povery Idex Ecoomerica hp://dxdoiorg/37/778 [7] Tho D (979 O Meaurig Povery Review of Icome ad Wealh hp://dxdoiorg//j b7x [8] Kawai N (98 O a Cla of Povery Meaure Ecoomerica hp://dxdoiorg/37/96 [9] Sall ST ad Lo GS (9 Uiform Wea Covergece of he Time-Depede Povery Meaure for Coiuou Logiudial Daa Brazilia Joural of Probabiliy ad Saiic hp://dxdoiorg/4/8-bjps [] Lo GS ad Sall ST (9 Uiform Wea Covergece of No Radomly Weighed Povery Meaure for Logiudial Daa 57h ISI Seio [] Lo GS ad Sall ST ( Aympoic Repreeaio Theorem for Povery Idice Afria Saiia [] Lo GS ( A Simple Noe o Some Empirical Sochaic Proce a a Tool i Uiform L-Saiic Wea Law Afria Saiia [3] Shorac GR ad Weller JA (986 Empirical Procee wih Applicaio o Saiic Wiley-Ierciece New Yor 859

14 G S Lo e al Appedix Pu wih ad We have fir o prove ha for i = = ( = + W g Y W W = W K ( Y Z ( ( = ( + W G Y G Z Y i r W W K + Baed o he expreio of K( ad o he fac ha r ( ad bouded for [ T] i uffice o prove ha for ad Thi would help o coclude wih he Le u eablih (4 We have ( ( G Z G Z ( ( i ( + r G u for Z u Z are uiformly G Z G Z K (8 ( + r r r K (9 ( ( K +r ( Y Z Y Z c -iequaliy ha W W K +r ( G Z G Z + G Z G Z ( ( ( Z Z mz G Z + G Z G Z B where Z( lie bewee Z( ad Z( ad B( lie bewee G ( Z( ad ( he ge ice Now we how (5 { } ( ( β ζ + ( ( G Z G Z K Z Z G Z G Z K + r Z Z( d r r = u m u u u m u du Z( Z Z( Z( Z Z Z Z G Z We u m u u m u + u m u du+ u m u d u i uiformly bouded we have by (H ad (H Z ( d Z + d r u m Z Z u u+ Z Z u m u u K (3 ( 86

15 G S Lo e al Furher Z( Z Z Z d + ( u m ( u ( u m ( u ( u ( u m ( u u ( u m ( u m ( u du (4 ad ice ( x d ( Z u Z = we ge ha u u m u d u Z Z B ud (( Z u Z( m( u du (5 Z( Z Z( Z where Z( lie bewee Z( ad From (4-(6 we coclude ha ad for > ad wih ad Z The Z ( Z ( ( d r u u m u u Kβ Z Z Z K + (6 ( + r r r K d Z Z( Z = d + Z Z r G y G Z u m u u G y G Z u m u u d Z Z( Z( = d + Z Z r G y G Z u m u u G y G Z u m u u Z Z Z( Z( Z Z( le ha M β Z Z( Now r r ( ad ( d G y G Z u m u u ( d G y G Z u m u u i le ha A+ B wih ( Z Z( A= G y G Z G y G Z u m u du Z By (H A i le ha ξ for Z Z( T B = G y G Z u m u u m u du ( + r ( ad Z m u du ( + r B K by (4 (5 ad (6 The ( + r r r K which prove (5 Le u fially prove (6 We have by (H for a fixed z ( r G z G zz + G z G zz K + Y z Y z for ome coa K The by he c -iequaliy wih + ( ( Y Z ( Y( Z( ( Y Z ( Y( Z ( Y( Z ( Y( Z( ( ( ( ( G Z G Z Z + G Z G Z Z K + Y Z Y Z r 86

16 G S Lo e al ad ( ( ( = GZ + G Z G Z Z K + Y Z Y Z ad he (6 hold By puig ogeher (4 (5 ad (6 ad by repeaedly uig he c -iequaliy we arrive a ( Now we have o eablih ha Pu W W K +r (7 W = + A B Y ( Z ( wih A = G ( Y G ( Z ( ad wih ad ad B = d Z Y Z ( We have by readily chec ha β β β ( ( A A Y Y + M Z Z + G Z G Z The by (H-(H3 ad he c -iequaliy Nex The by (H ( r A A K + Y Z Z( r ( ( ( Z Z Y Z B = d Z Y Z + = B + B = ( + ( B B B Y Z Z( ( B = d Z Y Z ( Z Z( Y Z = ( B d Z Y Z ( ( r B = G G Z Z K + Nex by puig C( Z ( Z ( Y ( Z ( = B K +r ( ( ( ( ( Z Z( Z Y Z Y B B = C d D where D( lie bewee ( ( Z Y Z ad ( Z( Y( Z( By imilar mehod we ge We fially ge ( ( B B Z + ZZ Z Z + Z Y Y ( ( B B K +r By combiig all ha precede we ge (7 which ogeher wih ( eablihe by he r c iequaliy W W K + (8 86

17 G S Lo e al ad Now we have o prove ha r W W K + We oly ech hi ecod par Le u coider i W i = We have Z W = K m u du Z Z Z u W = ( + ( G ( d u G Z d m u u By (4(5 ad he decompoiio of ( wih Furhermore ued i (5 we have K K K +r Z Z( Z Z( Z Z( m ( u d u d d d ( d m u u = Z Z( m u u Z Z( m u u m u m u u + We he ge The Now The Z ( m ( u d u Z d r m u u M Z Z Z K + + W W K +r Z W = S u du Z Z u S( u = ( + ( G ( u G ( Z d m ( u Z Z( W W = S u d u S u du Sice S( u i uiformly bouded we have Z Z( Z Z( Z Z( Z Z( ( = S u d u S u d u+ Su ( S u d u Z Z( r S( u u S( u u K + Z Z( Z Z( d d Moreover oe eaily how by he (H-(H3 wih imilar echique ued whe hadlig r Thu ( ( S u S u K +r W W K +r ha 863