Journal of Statistical Theory and Applications, Vol. 16, No. 1 (March 2017) On Some New Results of Poverty Orderings and Their Applications

Transcription

1 Jorna of Statistica Theory Appications Vo 6 No (arch On Some New Rests of Poverty Orderings Their Appications ervat ahdy * epartment of Statistics athematics Insrance Benha niversity Egypt drmervatmahdy@fcombeeg Received 6 Jne 4 Accepted September 6 Abstract The paper proposes to derive some new poverty indices which depend on aging casses e aso give some properties of it show the connection between economic measre new poverty measres these based on the concept of reversed resia incomes In addition the characterization of Pareto distribtion based on new poverty fnctions is obtained rthermore the stochastic orderings of new poverty indices are stdied their properties In addition the weighted poverty gap indices stochastic dominance which invove the concept of inactivity incomes its featres are stdied eywords: Poverty gap the severity of poverty poverty ordering weighed fnctions orenz crve the reversed proportiona faire rate athematics Sbject Cassification: 6E5 Introction Let be the rom variabe epresses the income of indivias with a density fnction f ( a distribtion fnction ( ( a reversed hazard rate ~ r ( f ( / ( (R a poverty ine where R are considered sch that ( represents the proportion of the poor The poverty is then qantified in term of the proportion of the poor peope their income ineqaity (see Atinson [] oster Shorrocs [3] Beznce et a [4] The difference between the poverty ine indivia income y is defined as the poverty gap i P i where Pi y The traditiona poverty gap inde is introced by Sen [5] as the percentage of indivias i beow Amost the poverty papers consider the poverty gap inde as i ℵ ( m is the set of poor indivias P / ℵ i it depends on niform distribtion In many cases the distribtion of income is not niform distribtion There are some reated aging casses which reated to r~ ( sch as mean reversed hazard ifetime It r~ fnction its reated fnctions are ess intitive fnctions shod be noted that the * Te: / fa: Copyright 7 the Athors Pbished by Atantis Press This is an open access artice nder the CC B-NC icense ( 38

2 Jorna of Statistica Theory Appications Vo 6 No (arch The first goa of this stdy is to introce new poverty indices which depend on distribtion fnction of income in genera cases ie the distribtion of income may be not niform sch as eponentia or Pareto distribtion so on oreover these indices consider as right trncated distribtion In addition the second goa of this paper is spread ot the sefness of r~ ( by reating it to new poverty gap indices The stdy of the poverty measres (poverty gap the poverty severity based on the concept of reversed resia incomes are defined in Section e aso give some properties of it show the reationship between Lorenz crve new poverty measres these based on the concept of reversed resia incomes In addition the characterization of Pareto distribtion based on new poverty fnctions is obtained Section 3 consists of a throgh the stdy of the stochastic dominance of new poverty measres The ast section provides the weighted poverty gap fnction stochastic dominance which invove the concept of inactivity incomes its featres are stdied The Poverty easres nctions Let be the rom variabe epresses the income of indivias with an absotey continos distribtion fnction ( a density fnction f ( e consider ( for a poverty ine as a right trncated income distribtion at then we interesting to stdy the rom variabe: its distribtion fnction is given by r r ( ( ( { } ( ( for Then the mean of ( (the average income beow the poverty ine can define as foows: r ψ It is satisfies the foowing properties: ψ ( for a R im ψ ( E( ( E( sd ( s / ( R ( its distribtion fnction is given by The foowing definition is essentia for this wor: efinition : Let the rom variabe epresses the income of indivias with distribtion fnction ( a density fnction f ( Then the reversed proportiona faire rate (RP p~ ( is defined as foows: ~ p ( / for a R f ( The reversed proportiona faire rate fnction has been stdied by Boc et a [6] Chra Roy [7] inestein [8 9 ] pta et a [] Then by sing ( we can obtain the poverty gap (P inde as the foowing forma ( ψ ( / for a R (3 oreover from ( (3 we obtain: [ ~ / / p ( ( ] for a R where p~ ( is the reversed proportiona faire rate fnction at a poverty ine oreover the reationship between ψ ( ( is given by ψ ( [ ( ] 39

3 Jorna of Statistica Theory Appications Vo 6 No (arch ~ ( / p ( ( for a R ψ In the crrent investigation we stdy the properties of new poverty inde in term of aioms for a good inde of poverty Theorem : The poverty gap inde of (3 satisfying aioms: Normaization ( Ν Increasing in sbsistence income (ISI onotonicity ( Proof It is easy to proof that if a incomes have zero Therefore ( ( is satisfying N or proof that ( ~ p ( ( < since E ( < E( / < p~ ( is probabiity vae then ~ p ( ( < This means that ( is satisfying ISI hen we rection in an income with vae a beow the poverty gap inde can be shown as taes the vae nity it is means that is increasing in we shod be achieve the foowing reation ( d ( / ( a a / for a R ( ψ ( / ( a / ( ( a / since a are positive nmbers then a ( / ( / for a R it is compete proof that ( is satisfying To compare the distribtion of income of different contries at the same time we may be sed orenz crve that is introced by Lorenz [] it is aso defined by astwirth [3] as foowing definition: efinition : Let fnction ( eqations in namey be a rom variabe with cmative distribtion fnction ( f with a finite mean µ The Lorenz crve ( p a density L of is defined in terms of two parametric ( d ( t p L ( ( tf ( t dt (4 w µ where ( can interpret as the proportion of the poor beow the eve ( can be viewed as the proportiona share of the tota income of poor beow the eve It foows from (4 that the Lorenz crve is the first moment distribtion fnction of ( It may be noticed that both ( ( ies between zero one In addition the Lorenz crve being the pot of the points ( ( ( is represented in the nit sqare L ( p can be interpreted as the proportion of the tota weath possessed by the poorest p th fraction of the popation The Lorenz crve defined by (4 acqires the foowing properties: L( L( L ( p is continos stricty increasing on as ( p / ( / µ for a R L 4

4 Jorna of Statistica Theory Appications Vo 6 No (arch where ( p L p ( p ( d ( d ( inf y : ( y { } / L is twice differentiabe is stricty conve on ( as L ( p / / µ f ( ( w > where the inverse of the distribtion fnction can interpret as ( or be any rom variabe with distribtion fnction a finite mean µ the Lorenz crve L ( p is defined as p L ( p ( t dt for a p [ ] (5 w µ where ( t inf{ : ( t} is the eft continos inverse of (aso nown as the qantie fnction Thompson [4] has proved the foowing properties for the Lorenz crve defined by (5 e can se the foowing characterizations theorem for the Pareto distribtion (Pareto( by sing new measres Theorem : Sppose be a non-negative rom variabe Then we get The poverty gap fnction is κ κ α κ ( κ p If r ( f ( /( ( represents the hazard rate ( the reationship ( ( κ 3 If have the Lorenz crve L ( p the poverty gap ( (6 the poverty gap fnction therefore r ( (7 the finite mean µ then the reationship ( g ( Θ L ( / µ g ( p / (8 4e have the foowing reationship ( L ( p pl p p if ony if foows ( κα where L ( p L ( p / p i hod for a rea Pαreto with distribtion fnction as foows: κ ( ( α / α αnd κ > (9 Proof Since κ κ κ κ a κ α κ ψ ( for a R κ κ κ ( κ ( a ( κ p ence by (3 the reqired rest ( foows In addition when r ( hods ence we can cacate ( directy by sing (6 as in Eq (7 rthermore κ κ L ( µ κ/ µ td t dt α ( κ Then the reqired rest (3 foows Assme Eq (9 hods according to Arnod [5] we obtain: 4

5 Jorna of Statistica Theory Appications Vo 6 No (arch L L -/ κ ( s - ( - s ( s / s ( κ / κ( s / κ foraκ > ence (6 gives the needed rest (3 Net we consider a new measre of the poverty that taes the variations in the distribtion of wefare amongst the poor into accont It is defined as the severity of poverty Let be a rom variabe denote an income beow eve with cmative distribtion fnction ( In many economic probems there are attention to the foowing rom variabe: r ( w { } The severity of the poverty (SP of the poor peope is defined as ρ E By sing integration by parts we can concde that: r ( Φ ( y ( ddy [ ( Φ( ] for a R Φ where ( ( d / ( Now we can define new cass of the severity as foows: A rom variabe is said to have increasing (decreasing the poverty severity IPS (PS if ρ ( is increasing (decreasing in or eqivaenty IPS ( PS iff Φ 3 Stochastic ominance of The Poverty easres nctions Let have the distribtion fnctions the reversed proportiona faire rate (RP fnctions ~ r q ~ respectivey Then is said to be smaer than or eqa ( in RP order (denoted as if rp ( ~ ( for a R ~ r To compare the poverty gap fnction of two grop poor peope et be two nonnegative rom variabes (incomes of two poor grops having distribtion fnctions the poverty gap fnctions ( ( respectivey The foowing definition of the poverty gap ordering is essentia for this section: efinition 3: is said to be smaer than or eqa to in the poverty gap fnction ordering ( where is a poverty ine It can be written as ( ( for a if 4

6 Jorna of Statistica Theory Appications Vo 6 No (arch ( ( ( ( for a R There is important notice that reversed mean resia ifetime ordering is same of new poverty inde that is introced in above definition where if we sppose m ( / ( is reversed mean resia ifetime we get that ( m ( / Now by sing Theorem ( in Ahmad et a [6] Coroary (3 Theorems (3 33 in ayid Ahmad [7] we can concde that: (i If Z be three nonnegative rom variabes where Z is independent of both et Z have a density g If g is a og-concave that is Z has the decreasing the reversed proportiona faire rate property then we can get that Z Z (ii If Z Z aso is independent of Z is independent of Z then the foowing statements hods: (a If Z have og-concave densities then Z Z (b If Z have og-concave densities then Z Z (iii If Z Z are seqences of independent rom variabes with i Z i i Z have og-concave densities for a i then i n i n i Zi n (3 i According (3 we can say that the poverty gap order is cosed nder convotion (iv Sppose that ( ς be a rom variabe having distribtion fnction ς et Θ i be a rom variabe having distribtion i for i Let ( ς ς R be a famiy of a rom variabes independent of Θ Θ If ( ς ( ς for a ς ς then ( Θ ( Θ (3 According (3 we can say that is cosed nder mitre The ower higher bonds for mitre weights of two distribtions by sing RP ordered are stdied in the net rest Theorem 3: Sppose V be two rom variabes with distribtion fnctions ( ( respectivey Let Z is mitre of V with distribtion ( A( ( A ( A Then we have Z V if V rp Proof: If Aso et rp rp rp V then we have ( is increasing in (33 ( V Z denote the right endpoints of the spports of the corresponding rom variabes V Z ma Z ma then V A( ( A ( ( ( A A respectivey 43

7 Jorna of Statistica Theory Appications Vo 6 No (arch is increasing in ( ma ( Therefore by (33 we can get that Z The proof that Z V Z is simiar Net we define a new ordering of rom variabes which depend on the poverty severity fnction that is the poverty severity order Let be rom variabes with distribtion fnction ( ( SP fnctions ( ( ρ ρ respectivey Then is said to be smaer than in the poverty severity ordering if eqation (34 can eqivaenty be written as In addition Eq (35 is eqivaent 4 The eighted Poverty ap y y ( ddy / ( ( ddy / ( y ( ( ddy ddy is increasing in R y r ( r ( for a R rp for a R Among the ey cases or at east correates of poverty is hors wored ( according to Borjas [8] we can write the reationship between income variabe ( hors wored variabe ( as foows: ( 4ep( a n for a R If we need to tae into accont the hors wored variabe ring measre the poverty inde we can se this variabe as weighted fnction ( ( Let the observation of is recorded with ( Then is an observation on the weighted rom variabe with pdf fnction as foows: ( f ( f E( ( (4 4 ep( a n f ( for a R E 4ep( a n where ( ( > E α are parameters is poverty ine The rom variabe is caed the weighted version of its distribtion is: ( [ ( A ( ] E( 4ep( α n (4 where y ( r ( r dr y 4 ep( α n r ( r dr A y ( y r( y where ( ( Note that if ( > then A ( R Now we can be defined s [ ] s f d 4ep( α n f ( d B s E s s ( s (43 ( s Then (4 can be rewritten as the foowing ( s B ( s s E( ( s s (44 oreover if ps rp (34 (35 44

8 Jorna of Statistica Theory Appications Vo 6 No (arch [ ] s y f y dy 4ep( α s E s > s s ( s Reintrocing E( ( s s into (44 it foows E( ( s s B ( s ( s ( s ( s By sing ( then RP of Now sppose we se ( can rewrite as p s ( s ~ p ( s / B ~ s n f as the rate of increase in eqaity per pond sqared as income is transmitted throgh income eve from poorer to richer indivias (see for more detais i Crescenzo [9] Soon [] Barrett Saes [] Ichino et a [] Aso by sing ( (4 (44 we have the weight P fnction ( of ( as foows [ ] y y A y dy y B y dy (45 ( [ ( A ( ] ( B ( In the foowing eampes we istrate the weighted poverty gap fnctions for some important income distribtions Eampe 4 (The Pareto II distribtion The Pareto II distribtion is important for stdying the growth rate of income poor Loma [3] sggested pdf of the Pareto II distribtion as: g ( ( for a R In addition the cmative distribtion fnction of g ( is foows: ( ( for a R iven (4 after some agebraic cacations we obtain E( 4ep( α n ep( α n ( ep( α ( 4 ep( α n ( ( A ( ( 4 ep( α V ( ( By sing (4 we can be concded the weighted cmative distribtion fnction is given by ( [ ( A ( ] E 4ep( α n where ( 4 ( 4V ( ( ( ( s ( d n( ( V ( 3 3 ( ( ( 3 ( n 45

9 rthermore then we can obtain ep( 4 n ep( 4 α α B Sppose n n T et Then we get P n 3 P n n Therefore we have ep( 4 α T B Ths the weight poverty gap fnction is given by ep 4 α T B B Eampe 4 (The Eponentia distribtion răgesc aoveno [4] demonstrated that the distribtion of indivia income in the SA is eponentia Let y n y y be a compete rom sampe of size n from λ Ep as foows: ep R λ λ iven (4 after some agebraic cacations we obtain!! ep! ep n 4 ep( α α E rthermore since!! ep! ep λ λ λ λ (46 Jorna of Statistica Theory Appications Vo 6 No (arch

10 Jorna of Statistica Theory Appications Vo 6 No (arch Then we have In addition A ( 4 ep( α n 4 ep( α ep B ( ( ( ( ( (! ep 4ep( α n d ( (!! 4 ep( α!! ep ep! irecty by sing Eq (46 Eq (47 we can concde that!!!! 4 ep ep!! λ j λ λ B λ α λ λ λ λ j λ λ λ λ λ j j λ Ths the weight poverty gap fnction is given by (! i ni where (! /(! ϕ n λ n i n i λ λ ( B ( ( B ( λ λ! λ! ep λ λ! λ! λ ep ( λ ϕ( λ λ ( λ ϕ( λ Eampe 43 (The inite range distribtion Since the finite range distribtion have been sed to characterize the income between indivias The cmative distribtion fnction of the finite range distribtion is ( p p iven µ after some agebraic cacations we obtain whereas therefore E d ( p d R ( ep( α n pd ep( α n ( p pd ep is the generaized hypergeometric fnction hen d ( α ( d p d ( - - p - ( - d p - A ( 4 ep( α n ( 4 ep( α ( ( ( ( d p d ( ( p (47 47

11 Jorna of Statistica Theory Appications Vo 6 No (arch with the same steps we can concde that 4ep( α n d B ( According to Chadhry et a [5] we get 4dpep( α d ( ( p ( ( ( d p ( s B ( s ds 4dp ep( α ( d ps 4dp ep 4dp ep where Q( Q ( Q ( Q n Q n ( s n n ( d n ( n p ( α n n n! ( α ( d ( p ( (! ds Therefore the weight poverty gap fnction is given by ( B ( ( B ( ( d ( p ( (! ( d p To compare the weighted P fnctions of two grop poor peope et be two rom variabes (incomes of two poor grops with density fnction m distribtion fnction ( ( respectivey The foowing is the definition of P the weighted P fnctions orderings Then where is a poverty ine It can be written as whereas is said to be to in the poverty gap ordering ( if B ( In net theorem we can show that if Theorem 4: Let ( ( for a R ( B ( ( B ( ( ( ( respectivey E[ ( ] [ ( ] we obtain ( s B ( s ( s B ( s B ( ds ds ( m( ( is ess than or eqa to in P be two weighted rom variabes with C fnctions ( E are eists Then if is decreasing in we get ( Proof: Let then we can get 48

12 Jorna of Statistica Theory Appications Vo 6 No (arch since where ( ( / ( s Again we rewritten (48 as ( isincreasing for a R (48 ( s [ ( s ( s ] E ( s s ( ( ( [ ( s ( s ] E ( s s s ( s ( ( s [ ( s ( s ] [ ( s ( s ] ds ds ( ( ( isincreasing for a R s In addition (48 is occr when where ( ( / reqired rest foows emma 4: Let ( B ( ( ( ( ( s ( s ( s ( s ( ( / ds ds be increasing (decreasing in Then B is increasing (decreasing in Remar 4: If B ( is repaced by A ( ( ( since is decreasing in s The then emma 4(ii sti hods bt emma 4(i hods with ineqaity reversed (cf Na Jain [6] The foowing theorem shows that for ( increasing in the weighted rom variabes the origina rom variabe are ordered with respect to R ordering Theorem 4: Let ( be increasing in Then Proof: rom (44 we have ( / ( B ( / E[ ( ] which is increasing by emma 4(ii It gives the reqired rest Theorem 43: The reationship between RP the weighted reversed proportiona faire rate can be epressed as ~ ~ ~ p p ( ep ~ p p y dy Proof: Note that rp 49

13 Jorna of Statistica Theory Appications Vo 6 No (arch ~ p ( E E [ ( ] f ( / ( [ ( ] f ( / ( ~ ~ p p ( ( / ( ( ( rther since ( ep ~ p (4 e have ( t ep ~ p t then by (49 (4 gives the reqired rest The net rest stdies the reationship between the ( ( by sing the variance of the poverty gap as foows: Theorem 44: Let be a rom variabe with distribtion fnction Therefore the weighted poverty gap fnction can be epressed as ( ( ( σ ( /( ( where σ ( Proof: Since is variance of the poverty gap Then we can get that σ Then the reqired rest foows ( ( ( ( / ( /( ( ( s ( s ds ( s ( s ds / ( ( ( / t t dt Theorem 45: Let be a rom variabe with absotey contines distribtion fnction then the weighted variance of the poverty gap fnction can be epressed as σ ( / ( ( ( / B t t dt B Proof: The proof foows on sing (45 by noting the fact that σ ( / ( ( / t dt The importance the roe of power distribtion in basic income stdies is introced by imai eaai [7] In the net eampe we characterization for power distribtion based on a reationship between Eampe 44: Let ( y y C as foows y n be a compete rom sampe of size n from a power distribtion with ( y ( y / κ ς < y < κ κ ς R (49 5

14 Jorna of Statistica Theory Appications Vo 6 No (arch get is weighted rom variabe of with weighted fnction ( y 4ep( α n y oreover In addition by sing Eq (45 we have Then we can concde that E ( ( / κ ς d / ( / κ ς / ( ς ς ς [ ep( α n ] 4ep( α ς ( / κ /( ς B ( 4ep( α ς /( ς < ( /( ς > Then we can Eampe 45: Let V be two continos rom variabes with the distribtion fnctions ( / κ ς < < κ κ ς > ( / κ ς ς v < < κ κ ς > P fnctions ( ( weighted distribtion fnctions ( ( weighted poverty gap fnctions ( ( respectivey e can say: in poverty gap ordering (denoted as V if ς ς is smaer than in poverty gap ordering (denoted as ( ( ς ς if Note that: The comparison between two rom variabes (weighted rom variabes with power distribtion depend on power parameter (ς Acnowedgements The athor wod ie to epress her thans gratitde to the editor reviewers for their efforts great comments sggestions that add to more qaity of the manscript References [] AB Atinson On the measrement of poverty Econometrica 55 ( [] J E oster A Shorrocs Poverty orderings Econometrica 56 (988a [3] J E oster A Shorrocs Poverty orderings wefare dominance Soc Choice efare 5 (988b [4] Beznce J Ce J Riz Ordering asymptotic properties of resia income distribtions Sanhya Ser B 6 ( ( [5] A Sen Poverty: An Ordina Approach to easrement Econometrica 44( ( [6] Boc T Savits Singh The reversed hazard rate fnction Probab Eng Inform Sc ( [7] N Chra Roy Some rests on reversed hazard rate Probab Eng Inform Sc 5 ( 95- [8] S inestein A Note on some aging properties of the acceerated ife mode Reiab Eng Syst Safe 7 ( 9- [9] S inestein On the reversed hazard rate Reiab Eng Syst Safe 78 ( 7-75 [] S inestein On one cass of bivariate distribtions Stat Probab Lett 65 (3-6 [] R C pta R pta P L pta onotonicity of the (reversed hazard rate of the (maimm minimm in bivariate distribtions etria 63 (6 3-4 [] O Lorenz ethods of measring the concentration of weath J Am Stat Assoc 9 ( [3] J L astwirth A genera definition of the Lorenz crve Econometrica 39 ( [4] A Thompson isherman's c Biometrics 3 ( [5] B C Arnod Pareto istribtions (Internationa Co-operative Pbishing ose 983 5

15 Jorna of Statistica Theory Appications Vo 6 No (arch [6] I A Ahmad ayid Peerey rther rests invoving the IT order the IIT cass Probab Eng Inf Sci 9 ( [7] ayid I A Ahmad On the mean inactivity time ordering with reiabiity appications Probab Eng Inform Sc 8 ( [8] T Borjas The Reationship between wages weey hors of wor: The roe of division Bias J m Resor 5(3 ( [9] A i Crescenzo Some rests on the proportiona reversed hazard mode Stat Probab Lett 5 ( 33-3 [] Soon Intergenerationa Income obiity in the nited States Am Econ Rev 8(3 ( [] R Barrett Saes On three casses of differentiabe ineqaity measres Int Econ Rev 39 ( [] A Ichino eai T Nannicini rom temporary hep jobs to permanent empoyment: hat can we earn from matching estimators their sensitivity J App Econ 3 ( [3] S Loma Bsiness faires another eampe of the anaysis of faire data J Am Stat Assoc 49 ( [4] A răgesc V aoveno Eponentia power-aw probabiity distribtions of weath income in the nited ingdom the nited States Physica A 99 ( 3-- [5] A Chadhry A Qadir Srivastava RB Paris Etended hypergeometric confent hypergeometric fnctions App ath Compt 59 ( [6] A Na Jain Some weighted distribtion rest on nivariate bivariate cases J Stat Pan Inference 77 ( [7] C imai E eaai Identifiabiity of Income istribtions in the Contet of amage enerating odes Commn Stat-Theor 9(8 (