Statistical inference for poverty measures with
relative poverty lines
Buhong Zheng
*
Department of Economics, University of Colorado at Denver, Campus Box 181, P.O. Box 173364, Denver, CO 80217-3314, USA
Received 9 September 1998; received in revised form 7 August 2000; accepted 2 October 2000
Abstract
Relative poverty lines such as one-half median income have been increasingly used in poverty studies. This paper contributes to the literature by developing statistical infer-ence for testing decomposable poverty measures with relative poverty lines. The poverty lines we consider are percentages of mean income and percentages of quantiles. We show that the estimates of poverty indices with relative poverty lines are asymptotically normally distributed and that the covariance structure can be consistently estimated. As a consequence, asymptotically distribution-free statistical inference can be established in a straightforward manner. ( 2001 Published by Elsevier Science S.A.
JEL classixcation: C40; I32
Keywords: Relative poverty line; Decomposable poverty measure; Statistical inference
1. Introduction
Professor Sen's (1976) groundbreaking work on poverty measurement has fundamentally changed the way poverty is viewed and measured. It is now well recognized that a poverty measure needs to consider not only the incidence of
*Tel.:#1-303-556-4413; fax:#1-303-556-3547. E-mail address:[email protected] (B. Zheng).
1For a survey on this literature, see Foster (1984), Chakravarty (1990), or Zheng (1997a). poverty (the proportion of the people living below the poverty line) but also the income distribution within the poor. In the past two and half decades, a growing body of literature has been devoted to the way poverty should be measured. As a result, many new distribution-sensitive poverty measures, in addition to the one proposed by Sen (1976), have been introduced.1 The recent literature is replete with numerous empirical studies using these new poverty measures to address distributional issues.
The application of a poverty measure requires the speci"cation of a poverty line which separates population into poor and nonpoor. In the literature, there are three distinct ways to specify a poverty line: the absolute, relative, and subjective methods. The de"ned poverty lines are referred to as the absolute, relative, and subjective poverty lines, respectively. The absolute method sets the poverty line as a minimum amount of resources at a point in time and updates the line only for price changes over time. The poverty line used in the o$cial US poverty statistics is an example of the absolute poverty line. The relative method speci"es the poverty line as a point in the distribution of income or expenditure and, hence, the line can be updated automatically over time for changes in living standards. In practice, researchers often specify the relative poverty line as a percentage of mean income or expenditure (e.g., Expert Committee on Family Budget Revisions, 1980; O'Higgins and Jenkins, 1990; Johnson and Webb, 1992; Wolfson and Evans, 1989), as a percentage of median income or expenditure (e.g., Fuchs, 1967; Blackburn, 1990, 1994; Smeeding, 1991), or simply as a quan-tile (e.g., OECD, 1982). The subjective method derives the poverty line based on public opinion on minimum income or expenditure levels that can&get along'or
&make ends meet'. Compared with the "rst two approaches, the subjective method is relatively less popular and has been rarely used.
While absolute poverty lines have been used in most government poverty statistics, relative poverty lines have recently gained momentum in both interna-tional poverty comparisons and intranainterna-tional cross-time analyses of poverty. In an important report on measuring poverty threshold by the Panel on Poverty and Family Assistance (1995), a group of leading scholars strongly urged the US government to abandon the absolute approach that has been used since 1963. Among many recommendations on designing a new poverty line, the Panel stated that`2food, clothing, and shelter components of the reference family
2Expert budgets outline the standards of needs for a large number of goods and services by
&experts'such as home economists. In the US, examples of expert budgets include the Economy Food Plan developed by experts from the United States Department of Agriculture, the Family Budgets Program of the Bureau of Labor Statistics, and the Thrift Food Plan, which succeeded the Economy Food Plan.
3The inference procedures for poverty measures with absolute poverty lines have been well established in the literature (e.g., JaKntti (1992), Kakwani (1994), Bishop et al. (1995), for decompos-able poverty measures, and Bishop et al. (1997), for the Sen poverty measure). However, the task we are pursuing here is quite di!erent since poverty lines have to be estimated from samples, so one needs to take into account of the sampling variability of poverty lines.
4Using a well-known result on quantile statistics, Preston (1995) provided sampling distribution for the headcount ratio when the poverty line is a percentage of median income. I thank Professor Stephen Jenkins for this reference.
5Although critical to our derivations at several places, the assumption of continuity ofF(x) may not always be ful"lled in practice. This is especially the case at the lower end of the income scale in the region of poverty group. The discontinuity ofF(x) may largely be due to the discontinuous nature of various tax policies, unemployment bene"ts, and social assistance programs. I thank a referee for pointing these out.
relative poverty lines such as one-half median income are easy to understand, easy to calculate and easy to update; they avoid the di$culty of periodic reassessments needed for absolute poverty lines. Besides, poverty lines deter-mined by an absolute approach such as &expert budgets' also contain large elements of relativity, as pointed out by the Panel (p. 32).2
The purpose of this paper is to develop appropriate statistical inference for poverty measures with relative poverty lines.3Speci"cally, we consider the class of decomposable poverty measures and two types of relative poverty lines: percentages of mean income and percentages of quantiles (which include median income as a special case).4We show that poverty indices with relative poverty lines can be consistently estimated and that the estimates are asymptotically normally distributed. Furthermore, we derive the asymptotic covariance struc-ture and show that the strucstruc-ture can be consistently estimated. Consequently, asymptotically nonparametric distribution-free statistical inference can be es-tablished in a straightforward manner. To determine the minimum sample size required for the asymptotic theory to be applicable, we conduct Monte Carlo simulations with several parametric distributions and"nd that a sample of size about 1000 would be su$cient. Finally, we extend our main results to both strati"ed samples and cluster samples.
2. Large sample properties of decomposable poverty measures
6These three conditions re#ect three key axioms in poverty measurement: the focus axiom, the (weak version) monotonicity axiom, and the (weak version) transfer axiom. The focus axiom states that a change in a nonpoor person's income (x*z) does not change the poverty level. The monotonicity axiom requires poverty not to increase if a poor person's income increases. The transfer axiom says that a transfer of income from a richer person to a poor person should not increase the overall poverty. For a complete discussion of these and other poverty axioms, see Zheng (1997a).
her income is belowz. A decomposable (additively separable) poverty measure, in its continuous form, is
P(F; z)"
P
= 0p(x,z) dF(x), (1)
wherep(x,z) is the individual poverty deprivation function and is continuous in bothxandzwithp(x,z)"0 forx'z,Lp(x,z)/Lx)0 andL2p(x,z)/Lx2*0 over (0,z).6 We also assume that p
x(x,z),Lp(x,z)/Lx is bounded and that p
z(x,z),Lp(x,z)/Lz}the increase inp(x,z) when the poverty linezis increased by an in"nitesimal amount} exists and is uniformly continuous over (0,R). Consequently, we may reasonably assume thata,:z
0pz(x,z) dF(x)"LP(F,z)/Lz
}the increase inP(F,z) when the poverty linezis increased}exists and is"nite. Assume a random sample of sizen,x
1,x2,2,xn, is drawn from a population with c.d.f. F(x), then the decomposable poverty measure de"ned in (1) can be estimated as
PK"1 n
n
+
i/1 p(x
i,z()I(xi!z(), (2)
wherez( is the sample estimate ofzand the indicator variableI(y) is de"ned as I(y)"
G
1 if y(0,
0 if y*0. (3)
Hence,I(x
i!z() is one ifxi(z( and zero otherwise.
In this paper we consider two types of relative poverty lines: mean poverty lines and quantile poverty lines. The following de"nition formally speci"es these two poverty lines and their sample estimates.
Dexnition 1. A poverty linezis a mean poverty line ifz"akwherekis mean income anda'0; a poverty linezis a quantile poverty line ifz"am
qwheremqis a quantile of orderq, i.e.,m
q"supMxDF(x))qN. The sample estimate ofz"ak isz("ax6 withx6"(1/n)+ni/1x
i; the sample estimate ofz"amqisz("ax(r)where x
(r) is the rth order statistic of (x1,x2,2,xn) with r"[nq]. If q"12, z is
It is well known thatx6 converges almost surely tokandx
(r)converges almost
surely to mq (see, e.g., Theorems 2.2.1A and 2.3.1 of Ser#ing, 1980). Hence, z( converges almost surely tozfor both poverty lines;n1@2(z(!z) also tends to a normal distribution. In what follows we derive the large sample properties of PK for these two types of poverty lines.
First note thatPK can be expressed as
PK"1 Applying the mean-value theorem top(x
i,z()!p(xi,z), we may write (ii) of (4) as
i betweenzandz(, the application of the one-term Young's form of Taylor's expansion (Ser#ing, 1980, p. 45) top(x
i,z() entails p(x
i,z()"p(z,z()#px(z,z()(xi!z)#o(Dxi!zD). (6) Hence, we can express (iii) of (4) as
Further applying the one-term Taylor expansion to (F(z()!F(z)), we have
For the second term of the right-hand side of (7), we have
K
1 third term of the right-hand side of (7),K
1 Therefore, both the second and third terms of the right-hand side of (7) are negligible in the approximation of (iii) (under the assumption that px(z,z() is bounded) sincez( converges almost surely toz. Therefore, part (iii) of (4) can be approximated as follows from Slutsky's theorem (Theorem 1.5.4 of Ser#ing, 1980) that both sides of (12) have the same limiting distribution.
Substituting (5) and (12) into (4), we have
PK&1 entails the following approximation ofPK:
7In empirical studies, researchers may also need to consider multiple poverty lines. For example, in developing comparable poverty estimates for member countries of the European Community, O'Higgins and Jenkins (1990) specify poverty lines to be 40%, 50% and 60% of mean equivalent disposable income of households. Results similar to Theorem 1 can be derived from (14a) and (14b) for poverty estimates with multiple poverty lines.
Clearly, PK is a consistent estimator of P. It is also easy to see that lim
n?=E(PK)"E[p(x,z)I(x!z)]"P(F;z), which establishes the asymptotic
mean ofPK. The asymptotic normality ofPK can be directly veri"ed by applying the Kolmogorov (strong) law of large numbers and the Lindeberg}LeHvy central limit theorem.
Using the Bahadur representation (see, e.g., Bahadur, 1966; Ghosh, 1971) which states the relationship between population quantiles and sample quantiles,
x
Now supposekdi!erent decomposable poverty measures are considered, and one wishes to use all of them in a poverty comparison. Denote these measures asPmjifz"akandPdjifz"am
q, j"1, 2,2,k. Also denoteCas the vector of
these poverty indices with two alternative poverty lines, i.e., C"
(Pm1,2,Pmk,Pd1,2,Pdk). It is easy to see that the vector of estimates,CK, also tends
8The headcount ratio is simply the proportion of the poor people (falling below the poverty line) in a distribution. The poverty gap ratio is the normalized income gap between the average income of the poor people and the poverty line. The FGT measure was proposed by Foster et al. (1984), the CHU measure was proposed by Clark et al. (1981) and the Watts measure was proposed by Watts (1968). The version of the CHU measure given here was also proposed by Chakravarty (1983). The original CHU measure is a transformation of the Chakravarty measure. The CDS (constant distribution sensitivity) poverty measure is similar to the inequality measure that Kolm (1976) introduced. This measure was recently characterized by Zheng (2000a) as the poverty measure that possesses constant distribution sensitivity or poverty aversion. To use the FGT, CHU and CDS measures, one needs to specify values for parametersb,c andj. See Zheng (1997a, 2000b) for interpretations of these parameters in poverty measurement.
9It is easy to construct poverty measures with p(z,z)O0 and aO0. An example is p(x,z)"(c!x/z)2withc'1. See Zheng (1999) for more discussions.
10Preston (1995) correctly observed that estimating the poverty line may increase as well as decrease sampling error, depending upon whether the two sources of sampling error tend to reinforce or o!set each other. In our empirical investigation using the Luxemburg Income Study data (not reported in this paper), however, we"nd that estimating the poverty line always increases sampling error.
and
u
jj"
P
amq0 p2
j(x,amq) dF(x)!(Pdj)2!2bdjPdj(1!q)#(bdj)2q(1!q),
(22)
respectively.
Remark 1. The results derived above can be applied to many commonly used poverty measures. For each speci"c poverty measure, the covariance and vari-ance terms can also be somewhat simpli"ed. Table 1 documents several com-monly used poverty measures with the corresponding coe$cients a,:z
0pz(x,z) dF(x).8Alsop(z,z)"0 for all poverty measures except the head-count ratio.9For the headcount ratio,p(z,z)"1.
Remark 2. It is worth noting the di!erence in the asymptotic variance of a poverty estimate withzbeing absolute and being relative. Ifzis absolute, then the asymptotic variance is simply the "rst term of (21) and (22) (see, e.g., Kakwani, 1994). Hence, the remaining two terms in both (21) and (22) can be attributed to the nature that z is relative and has to be estimated from the sample. Since the additional terms in n
jj and ujj are not negligible, it is important to take them into account when poverty lines are relative.10
Table 1
Decomposable poverty measures
Poverty measure p(x,z) The value ofa
The headcount ratio 1 0
The poverty gap ratio 1!x/z (1/z2):z0xdF(x)
The FGT measure (1!(x/z))b,b*2 (b/zb`1):z0x(z!x)b~1dF(x) The CHU measure 1!(x/z)c, 0(c(1 (c/zc`1):z0xcdF(x)
The Watts measure lnz!lnx F(z)/z
The CDS measure ej(z~x)!1,j'0 j:z0ej(z~x)dF(x)
above to other types of random samples. In Section 4, we will derive the asymptotic covariance matrices for both strati"ed samples and cluster samples.
3. Asymptotically distribution-free statistical inference
The covariance structure derived in the previous section depends upon the underlying distribution and, hence, the estimation of poverty measures is not distribution-free. However, if the covariance matrix can be consistently esti-mated, then asymptotically nonparametric distribution-free inference can be established. In what follows, we show how the covariance matrix can be consistently estimated.
First note that the coe$cientsbmjin (18) contain densityf(ak) andbdjin (19) contain f(amq) and f(mq). Thus, we need to estimate these densities. In the literature, there exist several nonparametric approaches to density estimation. Silverman (1986) provides a comprehensive survey on various methods
}ranging from the oldest method of histogram to some quite sophisticated ones. Among these di!erent approaches, the kernel estimation is probably the best known to economists. The method is popular because it is relatively easy to use and, more importantly, because the consistency of kernel estimation has been well established in the literature.
The kernel estimator of population densityf(z) is generally given by
fK(z)"1 nh
n
+
i/1
K
A
z!xih
B
, (23)whereKis a kernel function and his a`window widthathat depends on the
sample sizen. In computingfK(z), one needs to choose a speci"c kernel function
11If only one poverty line is used, then eitherR"[njj] orR"[ujj] should be used. also been established in the literature (see, for example, Silverman, 1978). Thus fK(ak), fK(amq) and fK(mq) using (23) are consistent estimators off(ak), f(amq) and f(mq), respectively, provided thatfis continuous atak,amq andmq.
It is easy to see that all remaining elements in the covariance matrix can also be consistently estimated. Therefore, by Slutsky's theorem, the whole covariance matrix can be consistently estimated.
To test poverty comparison using multiple poverty measures and multiple poverty lines, we need to construct a joint testing procedure. There are several ways to conduct such a joint test. For example, one can follow Bishop et al. (1992) to use a union}intersection test or follow Howes (1994) to use an intersection}union test. Both methods are easy to apply but may either have incorrect size or lack power. To avoid some of these problems, one may alternatively use the general Wald test outlined in Kodde and Palm (1986) and Wolak (1989). The procedure of this test is sketched below.
Consider the poverty comparison between two income distributions,AandB, using k decomposable poverty measures with two poverty lines (mean and quantile). Denote the vector of poverty indices for these two distributions as C
A andCB. The generalized Wald method, as developed by Kodde and Palm (1986) and Wolak (1989), can be used to test the following two types of hypotheses:
H
0: CA"CB vs.H1:CA)CB and
H
0: CA)CB vs.H1: CAlCB, whereC
A)CBmeans that distributionAhas less poverty than distributionBby all poverty measures with both poverty lines.
Assume two samples of sizesn
AandnBare drawn independently from the two populations. The sample estimates of C
A and CB are CKA and CKB and the estimated covariance matrices are RKA and RKB, respectively. Further denote
*C"C
B!CA andRAB"RA/nA#RB/nB. The critical step in using the Wald test is to solve the following minimization problem:11
min lw0
(*CK!l)RK~1AB(*CK!l)@. (24)
Denoting the solution to this minimization problem asl8, we can compute the following two Wald test statistics:
c
12Such a concern is also echoed in a recent paper on matching by Heckman et al. (1998). For the size of the sample used in their study, they"nd that these higher-order terms cannot be ignored and have to be included in approximation.
and
c
2"(*CK!l8)RKAB~1(*CK!l8)@. (26) Next comparec
1orc2with the lower and upper bounds of the critical value for
a pre-selected signi"cance level (Kodde and Palm, 1986, provide a table of these values). Ifc
1(c2) is below the lower bound thenH0is accepted; ifc1(c2) is above
the upper bound thenH
0is rejected. Ifc1(c2) falls between the lower bound and
the upper bound, then Monte Carlo simulations are required to complete the inference (for details, see Wolak, 1989, p. 215). The procedure described here has been used in testing stochastic dominance. A recent paper by Fisher et al. (1998), provides an example on how the Wald test is carried out (including Monte Carlo simulations).
4. Size simulations and extensions to non-simple random samples
The results derived in Section 2 are valid only asymptotically. In other words, the second and third terms of (7) may not be negligible when the sample size is not su$ciently large.12To make the procedures developed in Sections 2 and 3 more applicable, it is important to determine the minimum sample size neces-sary for the asymptotic theory to hold. In practice, this issue is usually investigated through Monte Carlo simulations. In this section, we conduct size simulations with four parametric distributions: the unit exponential dis-tribution, the uniform disdis-tribution, the Singh}Maddala distribution, and the lognormal distribution. We will also extend in this section the results of Section 2 to both strati"ed samples and cluster samples.
4.1. Size simulations
The unit exponential, uniform, and lognormal distributions are all well known in the statistical literature. The Singh}Maddala distribution was de-scribed by McDonald (1984) as the best"t for the US income data. The c.d.f. of the Singh}Maddala distribution is
F(x)"1![1/1#(x/f)g]q with f*0,g'0 and q'1/g. (27)
Table 2 Size simulations
Poverty line Distribution Sample size
50 100 300 500 700 1000
Unit exp. 11.2 9.8 9.9 9.5 9.8 9.5
One-half mean Uniform 11.7 9.8 9.3 9.4 10.2 10.2
Singh}Maddala 11.4 9.7 9.2 9.5 10.0 10.1
Lognormal 9.7 9.5 10.1 9.3 9.3 9.5
Unit exp. 14.7 12.5 11.6 11.1 11.2 10.5 One-half median Uniform 15.2 12.9 11.5 11.1 11.2 10.9 Singh}Maddala 14.5 12.3 11.3 10.8 11.5 10.3
Lognormal 6.6 5.8 13.1 12.4 9.3 10.1
distribution has mean 100 and variance 60. The nominal size is set at 10%. Thus, if the asymptotic normality of some estimate holds at a given sample size, about 10% of all runs with that sample size should lead to the rejection of the null hypothesis of equality.
Table 2 reports our simulation results for the FGT measure withb"3. We consider two di!erent poverty lines: one-half mean income and one-half median income. We also draw samples of sizes ranging from 50 to 1000. For each sample size and each parametric income distribution, we conduct 5000 independent trials. All random numbers are generated using the routine provided in Micro-soft FORTRAN. The percentages of rejections among these 5000 trials are reported in the table. An inspection of the table reveals that the paths to the asymptotic normality are not the same for the poverty measure with the two poverty lines: for the mean poverty line, the asymptotic normality can be achieved fairly fast with samples as small as 100; for the median poverty line, the process is much slower and a much larger sample size is required. Generally speaking, however, a sample of size 1000 will be able to reach the asymptotic normality for both poverty lines. Thus, we conclude that the asymptotic normal-ity of poverty estimates can be achieved with a sample of size 1000. For commonly used poverty data sets such as the Current Population Surveys, this requirement is not demanding at all.
4.2. Large sample properties of poverty estimates with stratixed samples
Suppose the population with c.d.f.F(x) is divided intoMstrata and the c.d.f.
strati"ed sample. We also assume thatnPRimplies eachn
jPRand that n
j/Denote the poverty index of theNj is very small so that no"nite population adjustment will be needed.jth stratum as P j(Fj;z), then the poverty index of the overall population can be expressed as
P(F;z)"+M
Using the same arguments as in the derivation of (14), we have
PKj&1 ting (31) into (29), we further have
PK&+M
wheresm"+M
j/1hj[aj#p(ak,ak)fj(ak)] andkjis the population mean income of thejth stratum.
Forz"am
q, we need to estimatemqand extend the Bahadur representation to samples that are not simple random. Following Woodru! (1952), we can estimate m
q as a weighted sample quantile and the weight assigned to each observation is proportional to the inverse of its selection probability. For a strati"ed sample, if it is proportional,mKqis simply therth-order statistic of the pooled sample withr"[nq]; if it is not proportional, the subsample from each stratum needs to be properly replicated to make it proportional before applying the above procedure. Sen (1968, 1972) and, more recently, Francisco and Fuller (1991) proved that the Bahadur representation also holds for nonsimple random samples under certain conditions. These conditions can easily be satis"ed by income data. Within the context of strati"ed samples, the Bahadur representa-tion is
From (32a) and (32b), it is clear that PK is a consistent and asymptotically unbiased estimator ofP. The asymptotic normality ofPK also follows directly as eachn
jPR. Based upon (32a) and (32b), and because samples from di!erent strata are independent, the asymptotic covariance matrix of a set of poverty estimates with both poverty lines can be directly derived. For example, the variance ofPK withz"akis
wherep2j is the variance ofxof thejth stratum.
4.3. Large sample properties of poverty estimates with cluster samples
13For simplicity, we use the same notations as in the case of strati"ed samples. That is, we view each cluster as a stratum in the derivation process.
a simple random manner. Suppose the population is composed ofMclusters and from them mclusters are randomly selected. Denote N
1,N2,2,Nm the
population sizes of these m clusters and n"+mj
/1Nj the sample size.13 To ensure the asymptotic normality of the poverty estimates, we need to require that bothMandmbe large so that the law of large numbers may apply.
Denote the poverty index of thejth cluster asP
j(Fj;z), then the poverty index of the overall population can be expressed as
P(F;z)"+m
Paralleling to (31), we have
PKj&1
Substituting (39) into (37), we further have
PK&+M the di!erence between usingw8 andwis negligible in the approximation of P. Consequently, we can replacew8 withwin (40). That is,
PK&+M Since the population meankcan be consistently estimated as (Raj, 1968)
x6"+m
j/1
0
wherekj is the mean income of thejth cluster, then for z"ak, (41) becomes
q, mKq is simply the rth order statistic of the cluster sample with r"[nq]. The Bahadur representation for a cluster sample is
x
From (41a) and (41b), it is clear that PK is a consistent and asymptotically unbiased estimator ofP. The asymptotic normality ofPK also follows directly as mPR. Based upon (41a) and (41b), the asymptotic covariance matrix of a set of poverty estimates with both poverty lines can be directly derived. For example, the variance ofPK withz"akis
gm
1999). A relative poverty line, as argued by the Panel on Poverty and Family Assistance (1995), provides a way to keep the poverty threshold up to date with overall economic changes in a society. It is also easy to understand, easy to calculate, and easy to update. Besides, poverty lines determined by an absolute approach such as `expert budgetsa also contain large elements of relativity.
This paper contributes to the literature by developing statistical inference for the class of decomposable poverty measures with relative poverty lines. The poverty lines we considered are percentages of mean income and percentages of quantiles (which include median income as a special case). Under certain regularities and assumptions, we showed that the poverty indices can be consistently estimated and that the poverty estimates are asymptotically normally distributed. We also derived the asymptotic covariance structure for the poverty estimates and showed that the covariance matrix can also be consistently estimated. Therefore, asymptotically distribution-free statistical inference can be established in a straightforward manner. To determine the minimum sample size required for the asymptotic results to hold, we conducted a series of Monte Carlo simulations and concluded that a sample of size 1000 will be su$cient. Finally, we extended the asymptotic results for simple random samples to both strati"ed samples and cluster samples.
Although we derived the asymptotic distribution only for decomposable poverty measures, the same methodology can be applied to other classes of poverty measures as well as to partial poverty ordering conditions. For example, Zheng (1997b), using the results reported in this paper and in Bishop et al. (1997), established the inference procedure for the Sen poverty measure with relative poverty lines. The procedure for the Sen measure can be modi"ed to test other rank-based poverty measures such as the Thon (1979) measure. Partial poverty ordering criteria such as (censored) generalized Lorenz dominance (Atkinson, 1987; Foster and Shorrocks, 1988a, b), deprivation curve dominance (Shorrocks, 1998; Jenkins and Lambert, 1997) and stochastic domi-nance (Atkinson, 1987; Foster and Shorrocks, 1988a, b; Zheng, 1999) have often been employed in poverty comparisons. The procedure outlined in this paper can also be used to test these dominance criteria when relative poverty lines are used.
Acknowledgements
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