Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol101.Issue2.2001:

Journal of Econometrics 101 (2001) 337}356

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 inference 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: buhong.zheng@cudenver.edu (B. Zheng).

0304-4076/01/$ - see front matter ( 2001 Published by Elsevier Science S.A.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 8 8 - 9

338

B. Zheng / Journal of Econometrics 101 (2001) 337}356

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 quantile (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 international poverty comparisons and intranational 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
poverty threshold under the proposed concept must be expressed as a percentage of median expenditures on these categoriesa (p. 148). This suggests the use of
the relative approach in forming the basic components of the poverty line.
A relative approach, argued the panel, `recognizes the social nature of economic
derivation and provides a way to keep the poverty line up to date with overall
economic changes in a societya (p. 125). Compared with absolute poverty lines,

1 For a survey on this literature, see Foster (1984), Chakravarty (1990), or Zheng (1997a).

B. Zheng / Journal of Econometrics 101 (2001) 337}356

339

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 determined 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.3 Speci"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).4 We 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 structure and show that the structure can be consistently estimated. Consequently,
asymptotically nonparametric distribution-free statistical inference can be established 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
Consider a continuous income distribution with c.d.f. F(x) where x is de"ned
over (0,R).5 We further assume that F(x) is di!erentiable and the "rst two
moments of x exist and are "nite. For a given poverty line z, a person is poor if

2 Expert 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.
3 The inference procedures for poverty measures with absolute poverty lines have been well
established in the literature (e.g., JaK ntti (1992), Kakwani (1994), Bishop et al. (1995), for decomposable 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.
4 Using 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.
5 Although critical to our derivations at several places, the assumption of continuity of F(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 of F(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.

340

B. Zheng / Journal of Econometrics 101 (2001) 337}356

her income is below z. A decomposable (additively separable) poverty measure,
in its continuous form, is

P

P(F; z)"

=
p(x, z) dF(x),
0

(1)

where p(x, z) is the individual poverty deprivation function and is continuous in
both x and z with p(x, z)"0 for x'z, Lp(x, z)/Lx)0 and L2p(x, z)/Lx2*0 over
(0, z).6 We also assume that p (x, z),Lp(x, z)/Lx is bounded and that
x
p (x, z),Lp(x, z)/Lz } the increase in p(x, z) when the poverty line z is increased

z
by an in"nitesimal amount } exists and is uniformly continuous over (0,R).
Consequently, we may reasonably assume that a,:z p (x, z) dF(x)"LP(F, z)/Lz
0 z
} the increase in P(F, z) when the poverty line z is increased } exists and is "nite.
Assume a random sample of size n, x , x ,2, x , is drawn from a population
1 2
n
with c.d.f. F(x), then the decomposable poverty measure de"ned in (1) can be
estimated as
1 n
PK " + p(x , z( )I(x !z( ),
(2)
i
i
n
i/1
where z( is the sample estimate of z and the indicator variable I(y) is de"ned as

G

I(y)"

1 if y(0,

0 if y*0.

(3)

Hence, I(x !z( ) is one if x (z( and zero otherwise.
i
i
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 line z is a mean poverty line if z"ak where k is mean
income and a'0; a poverty line z is a quantile poverty line if z"am where m is
q
q
a quantile of order q, i.e., m "supMx D F(x))qN. The sample estimate of z"ak

q
is z( "ax6 with x6 "(1/n)+n x ; the sample estimate of z"am is z( "ax where
i/1 i
q
(r)
x is the rth order statistic of (x , x ,2, x ) with r"[nq]. If q"1, z is
2
(r)
1 2
n
a median poverty line.

6 These 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).

B. Zheng / Journal of Econometrics 101 (2001) 337}356

341

It is well known that x6 converges almost surely to k and x converges almost
(r)
surely to m (see, e.g., Theorems 2.2.1A and 2.3.1 of Ser#ing, 1980). Hence,
q
z( converges almost surely to z for 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 that PK can be expressed as
1 n
1 n
PK " + p(x , z)I(x !z)# + [p(x , z( )!p(x , z)]I(x !z)
i
i
i
i
i

n
n
i/1
i/1
1 n
(4)
# + p(x , z( )[I(x !z( )!I(x !z)]"(i)#(ii)#(iii).
i
i
i
n
i/1
Applying the mean-value theorem to p(x , z( )!p(x , z), we may write (ii) of (4) as
i
i
1 n
1 n
+ [p(x , z( )!p(x , z)]I(x !z)" + p (x , z8 )(z( !z)I(x !z)
i
i
i
z i
i
n
n
i/1
i/1
1 n
+ p (x , z8 )I(x !z) , (5)
"(z( !z)
z i
i
n
i/1
where z8 is a value between z( and z.
For each x between z and z( , the application of the one-term Young's form of
i
Taylor's expansion (Ser#ing, 1980, p. 45) to p(x , z( ) entails
i

C

D

p(x , z( )"p(z, z( )#p (z, z( )(x !z)#o(Dx !zD).
i
x
i
i

(6)

Hence, we can express (iii) of (4) as

H

G

1 n
+ [I(x !z( )!I(x !z)]
i
i
n
i/1
1 n
# p (z, z( )
+ (x !z)[I(x !z( )!I(x !z)]
x
i
i
i
n
i/1
1 n
#
+ o(Dx !zD)[I(x !z( )!I(x !z)] .
(7)
i
i
i
n
i/1
Since +n [I(x !z( )!I(x !z)] in (7) is the (signed) number of observations
i/1
i
i
between z( and z and (1/n)+n I(x !z) converges almost surely to F(z) (Ser#ing,
i/1 i
1980, Section 2.1.1), it follows that +n [I(x !z( )!I(x !z)] can be approxii/1
i
i
mated by n(F(z( )!F(z)) at a rate of convergence o(n1@2), i.e.,
(iii)"p(z, z( )

G

G

H

H

n
+ [I(x !z( )!I(x !z)]"n(F(z( )!F(z))#o(n1@2).
i
i
i/1

(8)

342

B. Zheng / Journal of Econometrics 101 (2001) 337}356

Further applying the one-term Taylor expansion to (F(z( )!F(z)), we have
1 n
+ [I(x !z( )!I(x !z)]"f (z)(z( !z)#o(n~1@2),
i
i
n
i/1
where f (z) is the density function of F(x).
For the second term of the right-hand side of (7), we have

K

(9)

K

1 n
+ (x !z)[I(x !z( )!I(x !z)]
i
i
i
n
i/1
1 n
)Dz!z( D
+ [I(x !z( )!I(x !z)]
i
i
n
i/1
" f (z)Dz!z( D2#o(n~1@2),

K

K
(10)

here we have used Dx !zD)Dz!z( D since x is between z( and z. Similarly, for the
i
i
third term of the right-hand side of (7),

K

K

1 n
+ o(Dx !zD)[I(x !z( )!I(x !z)] )f (z)o(Dz!z( D2)#o(n~1@2). (11)
i
i
i
n
i/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 p (z, z( ) is
x
bounded) since z( converges almost surely to z. Therefore, part (iii) of (4) can be
approximated as
1 n
+ p(x , z( )[I(x !z( )!I(x !z)]&p(z, z( ) f (z)(z( !z)#o(n~1@2),
(12)
i
i
i
n
i/1
where u (x)&v (x) denotes that u (x)!v (x) converges in probability to zero. It
n
n
n
n
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

C

1 n
1 n
+ p (x , z8 )I(x !z)
PK & + p(x , z)I(x !z)#(z( !z)
i
i
z i
i
n
n
i/1
i/1
# p(z, z( ) f (z)(z( !z)#o(n~1@2).

D
(13)

By assumption, p (x, z) is uniformly continuous in x and z, thus
z
(1/n)+n p (x , z8 )I(x !z) converges almost surely to a,:z p (x, z) dF(x). This
0 z
i/1 z i
i
result, together with the fact that p(z, z( ) also converges almost surely to p(z, z),
entails the following approximation of PK :
1 n
PK & + p(x , z)I(x !z)#(z( !z)[a#p(z, z) f (z)]#o(n~1@2).
i
i
n
i/1

(14)

B. Zheng / Journal of Econometrics 101 (2001) 337}356

343

Clearly, PK is a consistent estimator of P. It is also easy to see that
lim
E(PK )"E[p(x, z)I(x!z)]"P(F; z), which establishes the asymptotic
n?=
mean of PK . The asymptotic normality of PK can be directly veri"ed by applying
the Kolmogorov (strong) law of large numbers and the Lindeberg}LeH vy central
limit theorem.
If z"ak, (14) becomes
1 n
PK & + p(x , ak)I(x !ak)#a(x6 !k)[a#p(ak, ak) f (ak)]#o(n~1@2).
i
i
n
i/1
(14a)
If z"am , (14) becomes
q
1 n
PK & + p(x , am )I(x !am )#a(x !m )[a#p(am , am ) f (am )]
i q
i
q
(r)
q
q q
q
n
i/1
# o(n~1@2).
(14b)
Using the Bahadur representation (see, e.g., Bahadur, 1966; Ghosh, 1971) which
states the relationship between population quantiles and sample quantiles,
q!(1/n)+n I(x !m )
i/1 i
q #o (n~1@2),
x !m "
1
(r)
q
f (m )
q
we have

(15)

1 n
aq[a#p(am , am ) f (am )]
q q
q
PK & + p(x , am )I(x !am )#
i q
i
q
n
f (m )
q
i/1
a[a#p(am , am ) f (am )] 1 n
q q
q ] + I(x !m )#o (n~1@2).
(16)
!
i
q
1
f (m )
n
q
i/1
Now suppose k di!erent decomposable poverty measures are considered, and
one wishes to use all of them in a poverty comparison. Denote these measures
as Pm if z"ak and Pd if z"am , j"1, 2,2, k. Also denote C as the vector of
j
q
j
these poverty indices with two alternative poverty lines, i.e., C"
(Pm,2, Pm, Pd ,2, Pd ). It is easy to see that the vector of estimates, CK , also tends
k
k 1
1
to a normal distribution. Through some direct calculations, one can verify the
main result of this paper.7

7 In 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.

344

B. Zheng / Journal of Econometrics 101 (2001) 337}356

Theorem 1. Under the conditions that F(x) is diwerentiable and has xnite xrst two
moments, CK is a consistent estimator of C and is asymptotically normally distributed
in that n1@2(CK !C) tends to a normal distribution with mean zero and the covariance
matrix
n
u
jl ,
R
" jl
2kC2k
u
u
jl
jl
where for j, l"1, 22, k,

C

P

n "
jl

ak
0

D

p (x, ak)p (x, ak) dF(x)!PmPm
j l
j
l

CP
CP

# bm
l

# bm
j

P

u "
jl

(17)

ak

0
ak
0

D
D

xp (x, ak) dF(x)!kPm
j
j

xp (x, ak) dF(x)!kPm #bmbmp2,
j l x
l
l

(18)

amq

p (x, am )p (x, am ) dF(x)!PdPd
j l
j
q l
q
0
! bdPd(1!q)!bdPd(1!q)#bdbdq(1!q)
j l
j j
l l

(19)

and

P

u "
jl

.*/(ak,amq )

0

.*/(ak,mq )

CP
CP

! bd
l

# bm
j

p (x, ak)p (x, am ) dF(x)!PmPd
j l
j
l
q

0
amq
0

D

p (x, ak) dF(x)!qPm
j
j

D

CP

xp (x, am ) dF(x)!kPd !amad
l
q
l
j l

mq

0

D

x dF(x)!qk ,
(20)

where bm"a[am#p (ak, ak) f (ak)], bd"a[ad#p (am , am ) f (am )]/f(m ) and
j
j q q
q
q
j
j
j
j
am and ad are, respectively, the values of a,:z p (x, z) dF(x) with z"ak and
0 z
j
j
z"am . In particular, the variances of n1@2PK m and n1@2PK d are
j
j
q
ak
n " p2(x, ak) dF(x)!(Pm)2
jj
j
j
0
ak
# 2bm
(21)
xp (x, ak) dF(x)!kPm #(bm)2p2
j
j x
j
j
0

P

CP

D

B. Zheng / Journal of Econometrics 101 (2001) 337}356

345

and

P

u "
jj

amq

0

p2(x, am ) dF(x)!(Pd)2!2bdPd(1!q)#(bd)2q(1!q),
j
j j
j
j
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 variance terms can also be somewhat simpli"ed. Table 1 documents several commonly used poverty measures with the corresponding coe$cients
a,:z p (x, z) dF(x).8 Also p(z, z)"0 for all poverty measures except the head0 z
count ratio.9 For the headcount ratio, p(z, z)"1.
Remark 2. It is worth noting the di!erence in the asymptotic variance of
a poverty estimate with z being absolute and being relative. If z is 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 and u are not negligible, it is
jj
jj
important to take them into account when poverty lines are relative.10
Remark 3. In developing the asymptotic distribution of the poverty estimates,
we have assumed samples to be simple random. Since many income and
earnings data are not simple random, it would be useful to extend the results

8 The 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 parameters b, c and j. See Zheng (1997a, 2000b) for
interpretations of these parameters in poverty measurement.
9 It is easy to construct poverty measures with p(z, z)O0 and aO0. An example is
p(x, z)"(c!x/z)2 with c'1. See Zheng (1999) for more discussions.
10 Preston (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.

346

B. Zheng / Journal of Econometrics 101 (2001) 337}356

Table 1
Decomposable poverty measures
Poverty measure

p(x, z)

The value of a

The
The
The
The
The
The

1
1!x/z
(1!(x/z))b, b*2
1!(x/z)c, 0(c(1
ln z!ln x
ej(z~x)!1, j'0

0
(1/z2):z x dF(x)
0
(b/zb`1):z x(z!x)b~1 dF(x)
0
(c/zc`1):z xc dF(x)
0
F(z)/z
j:z ej(z~x) dF(x)
0

headcount ratio
poverty gap ratio
FGT measure
CHU measure
Watts measure
CDS measure

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 estimated, 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$cients bm in (18) contain density f (ak) and bd in (19)
j
j
contain f (am ) and f (m ). Thus, we need to estimate these densities. In the
q
q
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 density f (z) is generally given by

A

B

z!x
1 n
i ,
+ K
(23)
fK (z)"
h
nh
i/1
where K is a kernel function and h is a `window widtha that depends on the
sample size n. In computing fK (z), one needs to choose a speci"c kernel function
K and a window width h. Silverman (1986) documents several kernel functions
and window width functions. Under certain conditions on K, Parzen (1962)
showed that fK (z) converges in probability to f (z). Stronger consistency of fK (z) has

B. Zheng / Journal of Econometrics 101 (2001) 337}356

347

also been established in the literature (see, for example, Silverman, 1978). Thus
fK (ak), fK (am ) and fK (m ) using (23) are consistent estimators of f (ak), f (am ) and
q
q
q
f (m ), respectively, provided that f is continuous at ak, am and m .
q
q
q
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, A and B,
using k decomposable poverty measures with two poverty lines (mean and
quantile). Denote the vector of poverty indices for these two distributions as
C and C . The generalized Wald method, as developed by Kodde and Palm
A
B
(1986) and Wolak (1989), can be used to test the following two types of
hypotheses:
H : C "C vs. H : C )C
0 A
B
1 A
B
and
H : C )C vs. H : C lC ,
0 A
B
1 A
B
where C )C means that distribution A has less poverty than distribution B by
A
B
all poverty measures with both poverty lines.
Assume two samples of sizes n and n are drawn independently from the two
A
B
populations. The sample estimates of C and C are CK and CK and the
A
B
A
B
estimated covariance matrices are RK and RK , respectively. Further denote
A
B
*C"C !C and R "R /n #R /n . The critical step in using the Wald
B
A
AB
A A
B B
test is to solve the following minimization problem:11
min (*CK !l)RK ~1(*CK !l)@.
(24)
AB
lw0
Denoting the solution to this minimization problem as l8, we can compute the
following two Wald test statistics:
c "*CK RK ~1*CK @!(*CK !l8)RK ~1(*CK !l8)@
AB
AB
1
11 If only one poverty line is used, then either R"[n ] or R"[u ] should be used.
jj
jj

(25)

348

B. Zheng / Journal of Econometrics 101 (2001) 337}356

and
(26)
c "(*CK !l8)RK ~1(*CK !l8)@.
AB
2
Next compare c or c with the lower and upper bounds of the critical value for
1
2
a pre-selected signi"cance level (Kodde and Palm, 1986, provide a table of these
values). If c (c ) is below the lower bound then H is accepted; if c (c ) is above
1 2
0
1 2
the upper bound then H is rejected. If c (c ) falls between the lower bound and
0
1 2
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.12 To make the procedures developed in Sections 2 and 3
more applicable, it is important to determine the minimum sample size necessary 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 distribution, the uniform distribution, 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 described 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)

In our simulations, we let g"1.697 and q"8.368 as estimated by McDonald
for the 1980 US income distribution. We also let f"1 since it is a scale
parameter. The support of the uniform distribution is [0, 1] and the lognormal

12 Such 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.

B. Zheng / Journal of Econometrics 101 (2001) 337}356

349

Table 2
Size simulations
Poverty line

Distribution

Sample size
50

One-half mean

One-half median

100

300

500

700

1000

Unit exp.
Uniform
Singh}Maddala
Lognormal

11.2
11.7
11.4
9.7

9.8
9.8
9.7
9.5

9.9
9.3
9.2
10.1

9.5
9.4
9.5
9.3

9.8
10.2
10.0
9.3

9.5
10.2
10.1
9.5

Unit exp.
Uniform
Singh}Maddala
Lognormal

14.7
15.2
14.5
6.6

12.5
12.9
12.3
5.8

11.6
11.5
11.3
13.1

11.1
11.1
10.8
12.4

11.2
11.2
11.5
9.3

10.5
10.9
10.3
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 with b"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 Microsoft 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 normality 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
The data of a strati"ed sample are collected via a two-step procedure: "rst, the
population is grouped into many mutually exclusive subpopulations or strata
(such as racial or ethnic groups); second, a simple random sample is drawn from
each stratum. The sample size of each stratum does not need to be proportional
to its population size; if it does, the strati"ed sample is said to be proportional.

350

B. Zheng / Journal of Econometrics 101 (2001) 337}356

Suppose the population with c.d.f. F(x) is divided into M strata and the c.d.f.
of the jth stratum is F (x), j"1, 2,2, M. Assume that the population sizes of
j
these strata are N , N ,2, N . A simple random sample of size n ,
1 2
M
j
xj , xj ,2, xj j , is extracted from stratum j. Thus, n"+M n is the size of the
n
j/1 j
1 2
strati"ed sample. We also assume that nPR implies each n PR and that
j
n /N is very small so that no "nite population adjustment will be needed.
j j
Denote the poverty index of the jth stratum as P (F ; z), then the poverty
j j
index of the overall population can be expressed as
M N
M
P(F; z)" + j P (F ; z)" + h P (F ; z),
j
j
j j j
N
j/1
j/1
where h "N /N. Thus, an unbiased estimator of P is
j
j
M
PK " + h PK
j j
j/1
with
1 nj
PK "
+ p(xj , z( )I(xj !z( ).
j n
i
i
j i/1
Using the same arguments as in the derivation of (14), we have

(28)

(29)

(30)

1 nj
(31)
+ p(xj , z)I(xj !z)#(z( !z)[a #p(z, z) f (z)]#o(n~1@2),
PK &
j
i
j
j
i
j n
j i/1
where f (x) is the density function of F (x) and a ":z p (x, z) dF (x). Substitu0 z
j
j
j
j
ting (31) into (29), we further have

C

D

M
1 nj
(32)
+ p(xj, z)I(xj !z) #s(z( !z)#o(n~1@2)
PK & + h
i
i
j n
j i/1
j/1
with s"+M h [a #p(z, z) f (z)].
j/1 j j
j
For a strati"ed sample, the population mean k can be consistently estimated
as (Raj, 1968)
M
x6 " + h x6 ,
(33)
j j
j/1
where x6 is the sample mean income of the jth stratum. Thus for z"ak, (32)
j
becomes

C

D

M
M
1 nj
+ p(xj, ak)I(xj !ak) #asm + h (x6 !k )#o(n~1@2),
PK & + h
j j
j
i
i
j n
j i/1
j/1
j/1
(32a)

B. Zheng / Journal of Econometrics 101 (2001) 337}356

351

where sm"+M h [a #p(ak, ak) f (ak)] and k is the population mean income
j/1 j j
j
j
of the jth stratum.
For z"am , we need to estimate m and extend the Bahadur representation to
q
q
samples that are not simple random. Following Woodru! (1952), we can
estimate m as a weighted sample quantile and the weight assigned to each
q
observation is proportional to the inverse of its selection probability. For
a strati"ed sample, if it is proportional, mK is simply the rth-order statistic of the
q
pooled sample with r"[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 representation is
q!+M h [(1/n )+nj I(xj!m )]
i
q #o (n~1@2).
j/1 j
j i/1
(34)
x !m "
1
(r)
q
f (m )
q
Thus, (32) with z"am becomes
q
M
1 nj
asd M
1 n
PK & + h
+ h
+ I(xj!m )
+ p(xj, am )I(xj !am ) !
j n
i
q
j
i
q
i q
f (m )
n
j i/1
q j/1
j i/1
j/1
aqsd
#o (n~1@2),
(32b)
#
1
f (m )
q
where sd"+M h [a #p(am , am ) f (am )].
j/1 j j
q q j q
From (32a) and (32b), it is clear that PK is a consistent and asymptotically
unbiased estimator of P. The asymptotic normality of PK also follows directly as
each n PR. Based upon (32a) and (32b), and because samples from di!erent
j
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 of PK with z"ak is

C

D

C

D

CP

ak
p2(x, ak) dF (x)!P2
j
j
0
ak
M h2
M h2
xp(x, ak) dF (x)!k P #(asm)2 + j p2,
# 2asm + j
j
j j
n j
n
0
j/1 j
j/1 j
where p2 is the variance of x of the jth stratum.
j
M h2
+ j
n
j/1 j

D

CP

D

(35)

4.3. Large sample properties of poverty estimates with cluster samples
To extract a cluster sample, the population is "rst collapsed into many
clusters (e.g., communities or counties) and, then, a set of clusters are selected in

352

B. Zheng / Journal of Econometrics 101 (2001) 337}356

a simple random manner. Suppose the population is composed of M clusters
and from them m clusters are randomly selected. Denote N , N ,2, N the
1 2
m
population sizes of these m clusters and n"+m N the sample size.13 To
j/1 j
ensure the asymptotic normality of the poverty estimates, we need to require
that both M and m be large so that the law of large numbers may apply.
Denote the poverty index of the jth cluster as P (F ; z), then the poverty index
j j
of the overall population can be expressed as
m
m N
j P (F ; z)" + 0 P (F ; z),
P(F; z)" +
j j j
j
j
n
j/1
j/1
where 0 "N /n. Thus, an unbiased estimator of P is
j
j
M
PK " + 0 PK ,
j j
j/1
where
1 Nj
PK "
+ p(xj, z( )I(xj !z( ).
j N
i
i
j i/1
Paralleling to (31), we have

(36)

(37)

(38)

1 Nj
PK &
+ p(xj, z)I(xj !z)#(z( !z)[a #p(z, z) f (z)].
j N
i
j
j
i
j i/1
Substituting (39) into (37), we further have

C

D

C

D

(39)

1 Nj
M
+ p(xj , z)I(xj!z) #w8 (z( !z)
(40)
PK & + 0
i
i
j N
j i/1
j/1
with w8 "+m 0 [a #p(z, z) f (z)].
j/1 j j
j
Because as mPR, w8 Pw"+M h [a #p(z, z) f (z)] and z( !zP0, thus
j/1 j j
j
the di!erence between using w8 and w is negligible in the approximation of P.
Consequently, we can replace w8 with w in (40). That is,
M
1 Nj
PK & + 0
(41)
+ p(xj , z)I(xj !z) #w(z( !z).
j N
i
i
j i/1
j/1
Since the population mean k can be consistently estimated as (Raj, 1968)
m
x6 " + 0 k ,
j j
j/1

(42)

13 For 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.

B. Zheng / Journal of Econometrics 101 (2001) 337}356

353

where k is the mean income of the jth cluster, then for z"ak, (41) becomes
j

C

D

m
m
1 Nj
PK & + 0
+ p(xj, ak)I(xj!ak) #awm + 0 (k !k),
j N
j j
i
i
j i/1
j/1
j/1

(41a)

where wm"+m 0 [a #p(ak, ak) f (ak)].
j/1 j j
j
For z"am , mK is simply the rth order statistic of the cluster sample with
q q
r"[nq]. The Bahadur representation for a cluster sample is
q!+m 0 [(1/N )+Nj I(xj !m )]
j/1 j
q #o (n~1@2).
j i/1 i
x !m "
(r)
q
1
f (m )
q

(43)

Thus, (41) with z"am is
q

D
D

C

m
1 Nj
PK & + 0
+ p(xj , am )I(xj !am )
i q
i
q
j N
j i/1
j/1
!

C

aqsd
awd m
1 Nj
+ I(xj !m ) #
#o (n~1@2),
+ 0
1
i
q
j
f (m )
f (m )
N
q
q j/1
j i/1

(41b)

where wd"+m 0 [a #p(am , am ) f (am )].
j/1 j j
q q j q
From (41a) and (41b), it is clear that PK is a consistent and asymptotically
unbiased estimator of P. The asymptotic normality of PK 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 of PK with z"ak is
gm
M

G

M
M
+ h2(P !P)2#2asm + h2(P !P)(k !k)
j j
j
j j
j/1
j/1

H

M
#(asm)2 + h2(k !k)2 ,
j j
j/1

(44)

where g"1!(m/M) and sm is de"ned as in the case of strati"ed samples
(because wm"sm when m"M).

5. Summary and conclusion
In poverty studies, relative poverty lines such as one-half mean income and
one-half median income have been routinely used. The US government is also
experimenting with a new measure of the poverty line which uses a relative
approach in forming the basic components of the poverty threshold (Short et al.,

354

B. Zheng / Journal of Econometrics 101 (2001) 337}356

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 distributionfree 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 dominance (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
I am grateful to two referees, an Associate Editor and Professor Cheng Hsiao
for very constructive suggestions. I also thank Stephen Jenkins, Fanhui Kong
and Ian Preston for useful comments and conversations on an earlier version of
the paper.

B. Zheng / Journal of Econometrics 101 (2001) 337}356

355

References
Atkinson, A., 1987. On the measurement of poverty. Econometrica 55, 244}263.
Bahadur, R., 1966. A note on quantiles in large samples. Annals of Mathematical Statistics 37,
577}580.
Bishop, I.A., Formby, J.P., Thistle, P., 1992. The convergence of South and non-South income
distribution, 1969}1979. American Economic Review 82, 262}272.
Bishop, J., Chow, K.V., Zheng, B., 1995. Statistical inference and decomposable poverty measures.
Bulletin of Economic Research 47, 329}340.
Bishop, J., Formby, J., Zheng, B., 1997. Statistical inference and the Sen index of poverty. International Economic Review 38, 381}387.
Blackburn, M., 1994. International comparisons of poverty. American Economic Review 84 (Papers
and Proceedings), 371}374.
Blackburn, M., 1990. Trends in poverty in the United States, 1967}84. Review of Income and Wealth
36, 53}66.
Chakravarty, S.R., 1990. Ethical Social Index Numbers. Springer, New York.
Chakravarty, S.R., 1983. A new index of poverty. Mathematical Social Science 6, 307}313.
Clark, S., Hemming, R., Ulph, D., 1981. On indices for the measurement of poverty. Economic
Journal 91, 515}526.
Expert Committee on Family Budget Revisions, 1980. New American family budget standards.
Institute for Research on Poverty, University of Wisconsin, Madison.
Fisher, G., Willson, D., Xu, K., 1998. An empirical analysis of term premiums using signi"cance tests
for stochastic dominance. Economics Letters 60, 195}203.
Foster, J., 1984. On economic poverty: a survey of aggregate measures. In: Basmann, R.L., Rhodes,
G.F. (Eds.), Advances in Econometrics, Vol. 3. JAI Press, Connecticut.
Foster, J., Greer, J., Thorbecke, E., 1984. A class of decomposable poverty measures. Econometrica
52, 761}766.
Foster, J., Shorrocks, A., 1988a. Poverty orderings and welfare dominance. Social Choice and
Welfare 5, 179}198.
Foster, J., Shorrocks, A., 1988b. Poverty orderings. Econometrica 56, 173}177.
Francisco, C., Fuller, W., 1991. Quantile estimation with a complex survey design. The Annals of
Statistics 19, 454}469.
Fuchs, V.R., 1967. Rede"ning poverty and redistributing income. Public Interest 5, 88}95.
Ghosh, J.K., 1971. A new proof of the Bahadur representation of quantiles and an application. The
Annals of Mathematical Statistics 42, 1957}1961.
Heckman, J., Ichimura, H., Smith, J., Todd, P., 1998. Characterizing selection bias using experimental data. Econometrica 66, 1017}1098.
Howes, S., 1994. Testing for dominance: inferring population rankings from sample data. World
Bank discussion paper.
JaK ntti, M., 1992. Poverty dominance and statistical inference. Research Report, Stockholm
University.
Jenkins, S., Lambert, P., 1997. Three `Ias of poverty curves, with an analysis of UK poverty trends.
Oxford Economic Papers 49, 317}327.
Johnson, P., Webb, S., 1992. O$cial statistics on poverty in the United Kingdom. Poverty
Measurement for Economies in Transition in Eastern European Countries. Polish Statistical
Association and Polish Central Statistical O$ce, Warsaw.
Kakwani, N., 1994. Poverty measurement and hypothesis testing. In: Creedy, J. (Ed.), Taxation,
Poverty and Income Distribution. Edward Elgar, Aldershot.
Kodde, D.A., Palm, F.C., 1986. Wald criteria for jointly testing equality and inequality restrictions.
Econometrica 50, 1243}1248.
Kolm, S.C., 1976. Unequal inequalities: I. Journal of Economic Theory 12, 416}442.

356

B. Zheng / Journal of Econometrics 101 (2001) 337}356

McDonald, J., 1984. Some generalized functions for the size distribution of income. Econometrica
52, 647}663.
OECD, 1982. The OECD list of social indicators, Paris.
O'Higgins, M., Jenkins, S., 1990. Poverty in the EC: estimates for 1975, 1980, and 1985. In: Teekins,
R., van Praag, B.M.S. (Eds.), Analysing Poverty in the European Community: Policy Issues,
Research Options, and Data Sources. O$ce of O$cial Publications of the European Communities, Luxembourg.
Panel on Poverty and Family Assistance, 1995. Measuring poverty: a new approach. National
Academy Press, Washington, DC.
Parzen, E., 1962. On estimation of a probability density function and mode. Annals of Mathematical
Statistics 33, 1065}1076.
Preston, I., 1995. Sampling distributions of relative poverty statistics. Applied Statistics 44, 91}99.
Raj, D., 1968. Sampling Theory. McGraw-Hill, New York.
Sen, A.K., 1976. Poverty: an ordinal approach to measurement. Econometrica 44, 219}231.
Sen, P.K., 1968. Asymptotic normality of sample quantiles for m-dependent processes. Annals of
Mathematical Statistics 39, 1724}1730.
Sen, P.K., 1972. On the Bahadur representation of sample quantiles for sequences of /-mixing
random variables. Journal of Multivariate Analysis 2, 77}95.
Ser#ing, R.J., 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York.
Shorrocks, A., 1998. Deprivation pro"les and deprivation indices. In: Jenkins, S., Kapteyn, A., van
Praag, B. (Eds.), The Distribution of Welfare and Household Production: International Perspectives. Cambridge University Press, London.
Short, K., Garner, T., Johnson, D., Doyle, P., 1999. Experimental poverty measures: 1990 to 1997.
US Census Bureau, Current Population Reports, Consumer Income, P60-205 (US Government
Printing O$ce, Washington, DC).
Silverman, B.W., 1978. Weak and strong uniform consistency of the kernel estimate of a density
function and its derivative. Annals of Statistics 6, 177}184.
Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis. Chapman & Hall,
London.
Smeeding, T.M., 1991. Cross-national comparisons of inequality and poverty position. In: Osberg,
L. (Ed.), Economic Inequality and Poverty: International Perspectives, M.E. Sharpe, Inc.,
Armonk.
Thon, D., 1979. On measuring poverty. Review of Income and Wealth 25, 429}440.
Watts, H., 1968. An economic de"nition of poverty. In: Moynihan, D.P. (Ed.), On Understanding
Poverty. Basic Books, New York.
Wolak, F.A., 1989. Testing inequality constraints in linear econometric models. Journal of Econometrics 41, 205}235.
Wolfson, M., Evans, J.M., 1989. Statistics Canada's low income cut-o!s: methodological concerns
and possibilities } a discussion paper. Research Paper Series, Statistical Canada, Ottawa.
Woodru!, R., 1952. Con"dence intervals for medians and other position measures. Journal of the
American Statistical Association 47, 635}646.
Zheng, B., 1997a. Aggregate poverty measures. Journal of Economic Surveys 11, 123}162.
Zheng, B., 1997b. Statistical inference for poverty measures with relative poverty lines. Mimeo,
University of Colorado at Denver.
Zheng, B., 1999. On the power of poverty orderings. Social Choice and Welfare 16, 349}371.
Zheng, B., 2000a. Minimum distribution-sensitivity, poverty aversion and poverty orderings. Journal of Economic Theory 95, 116}137.
Zheng, B., 2000b. Poverty orderings. Journal of Economic Surveys 14, 427}466.