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Improving Finite Sample Confidence Intervals for

Welfare Measures using Empirical Saddlepoint

Approximations

Christian Schluter

University of Southampton

January 2003

Abstract

It is shown that standard confidence intervals for inequality and povertymeasures have actual coverage errors in finite samples which substantially ex-ceed their nominal levels. We propose much improved confidence intervals basedon empirical saddlepoint approximations to the finite sample distribution of thestudentised welfare measure.

Keywords: inequality and poverty measures, saddlepoint approximations,higher order expansions, finite sample inference.

JEL classification: C10, C14, D31, D63, I32

Funding from the ESRC under grant R000223640 is gratefully acknowledged. The R-code usedin the simulation studies is available from the author on request.

Department of Economics, University of Southampton, Highfield, Southampton, SO17 1BJ,UK. Tel. +44 (0)2380 59 5909, Fax. +44 (0)2380 59 3858. Email: [email protected]://www.economics.soton.ac.uk/staff/schluter/

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1 Introduction

Most attention in the statistical literature on inequality and poverty measures (wel-fare measures for short) has focused on the asymptotic properties of their estimators

(see e.g. Davidson and Duclos, 1997 and Cowell, 1989). Their finite sample proper-ties have rarely been considered, and it is this gap which we address. It is commonpractice to use confidence intervals based on the well-known asymptotic properties,and invert the Gaussian limit distribution in order to obtain the confidence limitsirrespective of sample size. We therefore investigate first the performance of theseusual confidence intervals in finite samples. It turns out that these typically grosslyoverstate the precision of the estimate. In some of the cases considered, the actualcoverage error rate can exceed the nominal rate by a factor of 4.5. Large discrepanciespersist even in samples containing 500 observations. This poor performance is stemsfrom the substantial skewness of the actual finite sample distributions relative to theGaussian limit distribution. Having diagnosed the great need for improved methodsand the root cause of the problem, we propose improved methods based on empiricalsaddlepoint approximations to the finite sample distribution of the studentised welfaremeasure. These new confidence intervals are shown to perform significantly better,exhibiting often actual coverage error levels in good agreement with their nominallevels.

The issues raised in this paper are of practical relevance, for instance when eval-uating policies in terms of welfare or when seeking to identify geographical areas orsocio-economic groups as policy targets. These exercises require testing, such as pos-sibly comparing inequality or poverty before and after intervention, or cross-regioncomparisons. Using the usual confidence intervals, which are too short and thus less

precise than suggested by their nominal levels, can lead to the wrong inference, andtherefore to wrong targeting or the adoption of the wrong policy. Another exam-ple is the case of decomposition analyses. Applied work is frequently interested inthe welfare of socio-economic subgroups and their contributions to overall welfare.Although the overall sample may be large, the partition into subgroups can resultin subsamples which are comparatively small. The final example is the focus of thenew macroeconomics growth literature on the distribution and inequality of incomesamongst nations. Inequality indices are frequently computed using the Summers andHeston data set contains which about 122 countries.

The paper is organized as follows. Section 2 introduces the specific welfare mea-

sures to be considered, and defines the usual (first order) confidence intervals. Theirperformance is examined in section 3, which reveals a great need for improvement.New confidence intervals, based on empirical saddlepoint approximations are proposedin section 4, and section 5 demonstrates that they indeed yield substantial improve-ments. Section 6 concludes. The technical appendix contains a detailed derivation ofthe key coefficients of the cumulant expansions.

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2 Welfare Measures and First Order Confidence

Intervals

The objective is to construct confidence intervals for a given welfare measure I (in-

equality or poverty measure) based on a random sample of incomes Xi, i = 1, . . . , n,of size n from an income distribution FX. The measure I is a functional that mapsa distribution FX into a scalar, and the commonly used estimator bI = I( bFX), simplyuses the empirical distribution function (EDF) bFX ofFX, bFX (x) = n1 Pi 1 (Xi x).In many cases I(FX) only depends on the moments of the distribution and the EDFestimator is then obtained by replacing the population moments by the empiricalmoments. The asymptotic variance 2 = V ar(n1/2(bI I)), is obtained by the deltamethod, and estimated by an EDF-based estimator, denoted by b2.

The standard first order approach to constructing confidence intervals uses thestudentised welfare measure,

S = n1/2 bI Ib

!, (1)

which, under standard assumptions, has a distribution that converges asymptoticallyto the Gaussian distribution, denoted by . This leads to the basic nominal 100%percent confidence interval for the welfare measure typically used in applied research:

bIbn1

1 +

2

I Ib

n1

1

2

. (2)

The case of a nominal level of 95% is examined extensively below in Section 3.2.Confidence intervals based on the Gaussian quantiles 1.96 turn out to be very poorin practice because the actual coverage failure is be much larger than the nominal 5%.In short, the actual finite sample distribution of S, denoted below by G, is poorlyapproximated by the Gaussian distribution . In particular, the distribution of S isheavily skewed. In Section 4 we propose an improved approximation to G based onempirical saddlepoint approximation techniques.

2.1 Specific Welfare Indices

We consider two leading classes of inequality and poverty: the Generalised Entropyindices, the Lorenz curve, and the poverty indices proposed in Foster, Greer andThorbecke (FGT, 1984).

Generalised Entropy indices are defined by

GE(F) =1

2

"(F)

1(F) 1

#for 6= {0, 1}, (3)

where is a sensitivity parameter and (F) =R

xdF(x) is the moment functional.This inequality index is of particular interest because it is the only inequality measurethat simultaneously satisfies the property of scale independence, and the principles

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of transfer and decomposability, and the population principle. The smaller is thesensitive parameter , the larger is the sensitivity of the inequality index to the lowertail of the income distribution. The index, however, is not monotonic in . If = 2the index equals half the coefficient of variation squared. The limit cases of = 0

and = 1 are better known as Theil indices. We therefore denote them by T1and T0. However, rather treating these two special cases separately, we exploit thecontinuity of the index in , and approximate them by GE1.05 and GE0.05. Anotherpopular inequality measure is the Atkinson (1970) index, defined by A (F) = 1 [1 (F)]

1/1 1 (F)1, where 0 is a parameter defining (relative) inequality

aversion. However, since the Atkinson index can be mapped into the Generalised

Entropy index by setting 1 = , GE = [2 ]1h

1 A11 1

i, it will not

consider separately. For an extensive discussion of the properties of the GeneralisedEntropy index see Cowell (1980, 2000). First order methods for these inequalitymeasures have been considered in e.g. Cowell (1989) and Thistle (1990).

Another popular measure of inequality is the Lorenz curve, which depicts thecumulative income share of the least well-off fraction of the population. Let xp andp denote a quantile of the income variable and its population share, xp = F

1(p). Acoordinate of the Lorenz curve is a pair (p; L(p; F)) where

L(p; F) =1

1(F)

Zxp0

xdF(x). (4)

The Lorenz curve and the associated Lorenz dominance criterion are a centre pieceof inequality analysis: in order to compare inequality between two distributions onedraws their Lorenz curves and concludes that inequality is unanimously higher in one

distribution if its Lorenz curve is everywhere below the curve of the other distribu-tion. Any inequality measure which satisfies the principles of transfer, of anonymity,and of mean independence will rank the two distributions in the same way as theLorenz curves (Atkinson, 1970). First order methods for the Lorenz curve have beendeveloped in Beach and Davidson (1983).

The FGT poverty indices are of the form

P(F) =Z

1 xz

1 (x z) dF(x), for > 0, (5)

where z denotes the poverty line, which we assume to be distribution invariant, and1 (.) is the indicator function. is the sensitivity parameter, which determines the

weight of an income shortfall from the poverty line. This large class of indices includesthe Headcount index when = 0 and the Poverty Gap index when = 1. First ordermethods have been proposed by Kakwani (1993).

3 Simulation Evidence: The Need for Improved

Confidence Intervals

In order to investigate the finite sample performance of the usual first order confidenceintervals given by (2), we have carried out a simulation study varying over income

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distributions, sample sizes, and welfare measures. In particular, we have examinedthe performance of the GE2, and the Theil indices, the Lorenz curve at populationshares 0.25, 0.5, and 0.75, as well as of the Headcount, the Poverty Gap, and P2.

3.1 Simulation Design: Income Distributions

We consider three classes of income distributions that are often used in practice, andfit real-world data reasonably well.1

1. The Singh-Maddala (1976) distribution SM (a,b,c). Its density

f(x; a,b,c) =bcxb1

abh1 + (x/a)b

ic+1 ,is a special case of the Generalized Beta distribution (McDonald, 1984). We

also note that the Singh-Maddala distribution is heavy-tailed, its tails decayingslowly like power functions (to be precise, the index of the right tail equals bc).The population moments are given by = ca

(c /b) (1 + /b) / (1 + c)

where denotes the Gamma function, yielding immediately the population Gen-eralised Entropy index. The population Lorenz curve ordinate L(p) is given byIB1(1p)1/c (1/b + 1, c 1/b) where IB (, ) is the incomplete Beta function.

2. The lognormal LN(, 2Y), which implies a population inequality index equal toGE = (

2 )1 [exp (0.52Y( 1)) 1], independent of. The populationLorenz curve ordinate L(p) is given by (1 (p) Y).

3. The Gamma distribution G (x; r,) with shape parameter r,scale parameter ,and density exxr1r/(r). The population moments are = (r + )

/ (r),which imply a population inequality index independent of the scale parameter, GE = (

2 )1 (r( + r)/(r) 1). The population Lorenz curveordinate L(p) is [ (r + 1) / (r)] 11 G (G

1 (p; r,) ; r + 1,).

In case of the poverty measures, given the truncation of the income distributionat the poverty line, numerical methods have to be used. We have chosen a povertyline such that 20% of the population are in poverty.

The actual distributions used in the investigations are SM(100, 2.8, 1.7), LN(1, 0.52),and G (3, 0.15).

3.2 The Need for Improved Confidence Intervals

Table 1 records the incidence of coverage failures of symmetric nominal 95%-confidenceintervals based on first order methods, given by equation (2). The experiment involved100, 000 repetitions and the three sample sizes 100, 250, and 500. The Theil indicesT1 and T0 are approximated by GE1.05 and GE0.05.

1The key parameters used in our simulations are similar to the ones reported in Biewen (2001)and Brachmann et al.(1996) for German income data. Cowell and Feser (1996) have used the sameparametrisation of the Gamma distribution.

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sample sizeGE2

T1

T0

SM(100, 2.8, 1.7)100 250 500

22.6 18.2 15.713.1 10.1 8.7

9.4 7.2 6.4

LN(1;0.52)100 250 500

16 11.9 9.511.1 8.3 7.1

8.6 6.6 6.0

G (3, 0.15)100 250 5009.9 7.5 6.57.7 6.2 5.7

7.6 6.1 5.5

sample sizeLC(0.25)

LC(0.5)LC(0.75)

SM(100, 2.8, 1.7)100 250 5006.8 5.0 5.66.5 5.6 5.48.5 8.2 6.1

LN(1;0.52)100 250 5006.5 4.8 5.36.5 5.7 5.37.9 7.6 5.7

G (3, 0.15)100 250 5006.8 5.4 5.56.0 5.5 5.26.6 6.7 5.2

sample sizeP2P1P0

100 250 50010.4 6.7 6.4

7.4 6.1 5.76.7 5.1 5.2

100 250 5009.4 7.1 6.77.4 5.8 5.56.6 5.1 5.0

100 250 5008.9 6.9 5.47.4 5.7 5.67.0 5.0 5.4

Table 1: Actual coverage failure in per cent of usual nominal 95 per cent confidenceintervals for the welfare indices. Based on 100,000 replications.

Consider the inequality measures first. The table makes abundantly clear thatempirical coverage failures are substantially larger than the nominal value of 5 percent:in one case (GE2 and n=100) up to 4.5 times the nominal value. The coverages failuresfall as the sensitivity parameter of the Generalised Entropy Index falls (giving lessweight to the upper tail of the income distribution), the right tail of the incomedistributions decay more rapidly, and the sample size increases, but the extent of the

failure remains considerable.By contrast, the confidence interval for the Lorenz curve ordinates perform com-paratively well. The discrepancy between the actual and nominal error levels doesnot exceed the factor 2.

Coverage failures are also expected to be better for the poverty indices, given theirsimpler linear structure and the truncation of the income distribution at the povertyline (so that the speed of decay of the right tail of the income distribution becomesimmaterial). In particular P0 is expected to perform well since only a proportion isestimated. While Table 1 shows that this is indeed the case, the coverage failure canstill be twice the nominal rate (e.g for P2 and n=100).

The worst performer, GE2 estimated with n=100 and incomes drawn from theheavy-tailed Singh-Maddala distribution, is further examined in Figure 1 Depicted isthe simulated finite sample density of the studentised inequality measure2, and theGaussian limit distribution. The difference between these two densities is substantial.In particular, the finite sample distribution is heavily skewed.

Figure 1 about here.

In summary, the results show a great need for improved confidence intervals. All

2The density has been estimated using kernels, whose bandwidth was chosen using a cross-validation method.

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first order confidence intervals are too short, and in some cases the actual coveragefailure rate can be as much as 4.5 times higher than the nominal rate. Inference basedon such confidence intervals can therefore be seriously flawed in practice, and it isprecisely the first order confidence intervals which applied researcher typically use.

4 Confidence Intervals based on Empirical Saddle-

point Approximations

4.1 Saddlepoint Approximations

Saddlepoint approximations often yield remarkably accurate approximations to den-sities and cumulative distribution functions of certain statistics of interest. Daniels(1954) first developed these methods in the case of the mean of a sample of iid randomvariables. Reid (1988) provides an extensive survey, and an annotated bibliography.We proceed to summarise the results relevant to our analysis.

Let S1,...,SN be an iid sample from a continuous distribution G. Denote byMS() = E{exp(S)} the moment generating function, and the cumulant generatingfunction (CGF) by KS() = log MS(). Consider the sample mean S = N

1 Pi Si,

whose CGF is K() = N KS(N1). Denote its first and second derivatives by K0 ()

and K00 ().The saddlepoint approximation to the density of S at s is

gS(s) =n

2K00bo1/2

exp

Kb

bs

, (6)

where the saddlepoint b satisfies the equation K0 b = s. The approximation to thedistribution function of S is

GS (s) =

w +1

wlog

v

w

, (7)

with w = signb h2 bs Kbi1/2 and v = b hK00 bi1/2. These approximations

are typically remarkably accurate, especially in the tails of the distribution. Forinstance, denoting the actual density by g, we have g (u) /gS(u) = 1 + O (N

1). Theerror is relative rather than absolute.

4.2 Empirical Saddlepoint Approximations for Welfare Mea-

sures and Confidence Intervals

We proceed to apply the saddlepoint approximations to the case of the studentisedwelfare measure. S denotes again the studentised welfare measure computed froman iid sample of size n of incomes X1,...,Xn drawn from distribution FX. The exactcumulant generating function of S, KS(), for unknown FX is not known. However,we can construct an approximation to KS(). The first four cumulants of S admit

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expansions in powers of n:

KS,1 = n1/2k1,2 (FX) + O

n3/2

,

KS,2 = 1 + n1k2,2 (FX) + O(n

3/2),

KS,3 = n1/2k3,1 (FX) + O n3/2 ,

KS,4 = n1k4,1 (FX) + O

n3/2

.

The key quantities k1,2 - k4,1 depend on the population moments of FX and arederived explicitly in the Appendix. These cumulant expansions can be used to ap-proximate KS() by

K(; FX) =1

22 + n1/2

k1,2 (FX) +

1

63k3,1 (FX)

(8)

+n1 122k2,2 (FX) + 1244k4,1 (FX) .

A similar approximation to the CGF has been considered by Easton and Ronchetti(1986). Finally, we approximate K(; FX) by the empirical CGF K

; bFX , which

uses the empirical distribution function bFX, thus replacing in the coefficients ki,jpopulation moments by sample moments.

The empirical saddlepoint approximation to the density and distribution functionof S is now obtained by using (6) and (7) with K

; bFX and N = 1. Easton and

Ronchetti (1986) have shown that when S is itself a function of a sample of sizen, as in our case, the relative error of the saddlepoint approximation then becomes

O (n

1). The simulation studies reported below investigate how accurate the resultingapproximations are compared to the exact simulated distributions of S for a rangeof sample sizes and income distributions FX.

The confidence limits for a nominal 100 percent confidence interval are obtainedby inverting (7). The resulting confidence interval for the population welfare measureI is therefore

bIbn

G1S

1 +

2

I bIb

nG1S

1

2

. (9)

The actual coverage probability of the usual first order confidence intervals (2), based

on the Gaussian quantiles, is approximated by

GS

1

1 +

2

GS

1

1

2

. (10)

Remark 1 The empirical saddlepoint approximation becomes exact when n .In this case KS,1 = KS,3 = KS,4 = 0, KS,2 = 1, and K

; bFX = 0.52. It follows

immediately that gS(u) = (2)1/2 exp(0.5u2) and GS(u) = (u).

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5 Empirical Saddlepoint Approximations: Simu-

lation Evidence

The simulation design is the one of Section 3. The algorithms have been implemented

in R3, and are available from the author on request.First, we consider the approximations to the actual coverage failures of the usual

first order confidence intervals, computed using equation (10). We take the nominallevel to be 95 per cent, hence we use the Gaussian quantiles 1.96. The results arereported in Table 2, which have to be compared to the actual coverage failures re-ported in Table 1. The approximations are remarkable accurate for the Theil indices,irrespective of the income distribution and sample size. For GE2, the approximationsare still good for the lognormal and the gamma distributions. Only in the case ofthe heavy-tailed Singh-Maddala distribution do we observe some noticeable discrep-ancy. As regards the Lorenz curve ordinates and the poverty indices, the saddlepoint

approximations yield very good predictions of the actual coverage behaviour of thestandard confidence intervals. Table 2 and Table 1 are in close agreement for allincome distributions and sample sizes considered.

sample sizeGE2

T1T0

SM(100, 2.8, 1.7)100 250 500

11.8 11.3 10.710.1 9.1 8.3

9.2 7.5 6.6

LN(1;0.52)100 250 500

11.2 10.0 9.19.8 8.3 7.38.7 7.0 6.2

G (3, 0.15)100 250 5009.0 7.6 6.87.8 6.5 5.87.9 6.3 5.7

sample sizeLC(0.25)

LC(0.5)LC(0.75)

SM(100, 2.8, 1.7)

100 250 5005.6 5.3 5.26.6 5.9 5.68.7 7.0 6.4

LN(1;0.52)

100 250 5005.2 5.1 5.16.1 5.5 5.37.9 6.3 5.8

G (3, 0.15)

100 250 5004.7 4.8 4.95.9 5.4 5.28.5 6.4 5.8

sample sizeP2P1P0

100 250 50011.0 7.8 6.4

8.1 6.2 5.66.2 5.4 5.2

100 250 50010.9 7.8 6.4

7.9 6.2 5.66.2 5.4 5.2

100 250 50010.3 7.4 6.2

7.8 6.1 5.56.2 5.4 5.2

Table 2: Predicted actual coverage failure in per cent confidence intervals based onthe empirical saddlepoint approximation and the Gaussian quantiles 1.96. Basedon 10,000 replications.

Since the confidence intervals based on the Gaussian quantiles are too short, im-proving the performance of confidence intervals requires the use of quantiles furtherinto the tails of the actual distribution. Using an approximation to the actual distri-bution might therefore be expected to yield benefits which diminish the further one

3This GNU S statistical programme language is freely available and open source, seehttp://cran.r-project.org/ for details.

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has to move out into the tails.4 In the case of the empirical saddlepoint approxima-tion this shortcoming is not observed. Table 3 reports the actual coverage failures ofimproved confidence intervals with 95% nominal level, defined in equation (9). Theconfidence intervals for the Theil indices exhibit a coverage behaviour in close agree-

ment with the nominal level. For GE2, lognormal and gamma income distributionsalso lead to good actual coverage behaviour. Only for the heavy-tailed Singh-Maddalado we observe again a sizable discrepancy. However, the actual coverage failure halvesthe failure rate of the first order methods. The problem is further illustrated in Fig-ure 1, which not only depicts the simulated distribution of S, but also a saddlepointapproximation based on one random sample of size 100. The approximation capturesthe skewness of the actual distribution, but has still insufficient mass in the left tail.This leads to confidence intervals which are still too short.

The new confidence intervals for the Lorenz curve ordinates and the poverty indicesperform very well. The coverage errors are close to their nominal levels, irrespective

of the income distribution and sample size considered.

sample sizeGE2

T1T0

SM(100, 2.8, 1.7)100 250 500

14.2 12.0 9.97.7 6.9 5.95.0 5.1 5.3

LN(1;0.52)100 250 500

10.2 7.1 6.16.6 5.3 4.75.1 5.1 5.3

G (3, 0.15)100 250 5005.4 5.5 5.15.0 5.1 5.24.7 4.7 5.0

sample sizeLC(0.25)

LC(0.5)LC(0.75)

SM(100, 2.8, 1.7)100 250 5006.6 5.4 5.15.2 4.7 4.74.6 5.8 5.3

LN(1;0.52)100 250 5005.9 5.1 5.05.1 5.0 5.14.7 5.7 4.7

G (3, 0.15)100 250 5007.1 5.9 5.54.6 4.7 5.02.8 4.5 4.3

sample sizeP2P1P0

100 250 5005.1 4.1 4.54.0 4.6 5.14.7 4.9 5.6

100 250 5004.4 4.2 4.63.9 4.7 5.14.4 5.1 4.7

100 250 5004.9 4.3 4.83.9 4.2 4.84.4 5.1 4.8

Table 3: Actual coverage failure in per cent of nominal 95 per cent confidence intervalsbased on the empirical saddlepoint approximation. Based on 10,000 replications.

In summary, confidence intervals based on empirical saddlepoint approximations

do indeed lead to substantial improvements over first order methods. Actual coveragelevels typically closely agree with their nominal levels.

4For instance, this affects confidence intervals based on (empirical) Edgeworth expansions, whosebehaviour we have also investigated. Unreported results show that their performance is substantiallyinferior to confidence intervals based on saddlepoint approximations, stemming from the formerspoor tail behaviour. The theoretical explanation of this observation is that the formers error isabsolute rather than relative.

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6 Conclusions

Standard confidence intervals for inequality and poverty measures, based on the Gaus-sian limit distribution, are too short in finite samples. Actual coverage error levels lie

substantially above the nominal levels, in some cases up to a factor of 4.5. We haveproposed new confidence intervals based on empirical saddlepoint approximations.The simulation studies show that actual and nominal error levels are now typicallyin good agreement.

The approach proposed in this paper is based on a good approximation to theactual finite sample distribution of the studentised welfare measure. An alternativeapproach is to find and apply a normalising transform, such that its distribution iscloser to the Gaussian limit distribution. This alternative is pursued in van Garderenand Schluter (2002).

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References

Atkinson, A. B. (1970) On the measurement of inequality, Journal of Economic

Theory, 2, 244263.Beach, C.M. and Davidson, R. (1983), Distribution-free statistical inferencewith Lorenz curves and income shares, Review of Economic Studies, 50, 723-735.C.M. Beach and R. Davidson and G.A. Slotsve (1995), Distribution freestatistical inference for Lorenz Dominance with crossing Lorenz curves, GreqamDiscussion Paper 95A03.Biewen, M. (2001) Bootstrap inference for inequality, poverty and mobility mea-surement, Journal of Econometrics, vol. 108, 2, 317-342.Brachmann, K. and Stich, A. and M. Trede (1996) Evaluating parametricincome distribution models, Allgemeines Statistisches Archiv, 80, 285298.

Cowell, F. A. (1980) On the structure of additive inequality measures, Reviewof Economic Studies, 47, 521-531.Cowell, F. A. (1989) Sampling variance and decomposable inequality measures,Journal of Econometrics 42, 27-41.Cowell, F. A. (2000), Measurement of inequality, In A.B. Atkinson and F. Bour-guignon (Eds.), Handbook of Income Distribution, Chapter 2. Amsterdam: NorthHolland.Daniels, H. E. (1954), Saddlepoint approximations in statistics, Annals of Math-ematical Statistics, Vol. 25, No. 4., 631-650.Davidson, R. and J.-Y. Duclos (1997) Statistical inference for the measurementof the incidence of taxes and transfers, Econometrica, 65, 1453-1465.Easton G. S. and E. Ronchetti (1986), General saddlepoint approximationswith applications to L-statistics, Journal of the American Statistical Association,Vol. 81, No. 394, 420-430.Foster, J. E., J. Greer, and E. Thorbecke (1984) A class of decomposablepoverty measures, Econometrica, 52, 761-776.van Garderen, K.J. and C. Schluter (2002), Improving finite sample confi-dence intervals for inequality and poverty measures, University of Southampton.Kakwani, N. (1993) Statistical inference in the measurement of poverty, TheReview of Economics and Statistics, 75, 632-639.McDonald, J.B. (1984) Some generalized functions for the size distribution of

income, Econometrica, 52, 647663.Reid, N (1988), Saddlepoint methods and statistical inference, Statistical Science,Vol. 3, No. 2., 213-227.Singh, S.K. and G.S. Maddala (1976) A Function for size distribution of in-comes, Econometrica, 44, 963970.Thistle, P. D. (1990) Large sample properties of two inequality indices, Econo-metrica, 58, 725-728.

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A Technical Details: The Inequality Measure GE

This appendix contains the derivation of the key second order quantities k1,2, k3,1,and the key third order quantities k2,2 and k4,1. These are given in equations (19),

(21), (22), and (23) below. The method of derivation is based on obtaining first astochastic expansion of the studentised welfare measure S. We then consider theexpansions of the first four moments, which are finally used to give the expansions ofthe first four cumulants ofS. All derivations have been carried out using the symbolicmanipulation capabilities of Maple.

A.1 A Stochastic Expansion for the Studentised Inequality

Index

We first need to obtain an asymptotic expansions of the moments of S. As a compact

notation, we use Sq to denote a term of an expansion of S which is of order nq

.Hence the desired stochastic expansion of S is given by

S = S0 + S1/2 + S1 + Op(n3/2). (11)

Our approach is based on the decomposition ofS in two parts, S = AB1/2, whichin turn are expanded as A = A0 + A1/2 + A1 + Op

n3/2

, and B = B0 + B1/2 + B1

+ Op(n3/2). These constituent parts are sums of some random variables

A0 = n1/2 P

i Y1,iA1/2 = n

3/2

PiPj Y2,iY3,j B1/2 = n1

Pi Z1,iA1 = n2.5 PiPj Pk Y4,iY5,jY6,k B1 = n2 PiPj Z2,iZ3,j + Z4,iZ5,j (12)The precise definitions of the variables B0, Z1,i - Z5,i and Y1,i - Y6,i will be statedbelow. In terms of these consituent parts the stochastic expansion (11) is given by

S0 = B1/20 A0

S1/2 = 1

2B3/20 A0B1/2 + B

1/20 A1/2 (13)

S1 = 1

2B3/20 A0B1 +

3

8B5/20 A0

hB1/2

i2 12

B3/20 B1/2A1/2 + B

1/20 A1.

We first derive explicitly the stochastic expansion for S, given by (11) and theprecise variable definitions B0, Z1,i - Z5,i and Y1,i - Y6,i, given by equations (15),(16), (17), and (18) below. We then derive the relevant expansions for the first fourmoments and cumulants.

A.2 The Stochastic Expansion: Details

Recall our notation for population and sample moments, (FX) =R

ydF(y) andm = ( bFX). We derive the stochastic expansion in four steps:

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1. Centre the inequality index to obtain

n1/2([GE GE) = n1/2h2

i1

1 m

1 [

1 m m1 ] .

2. Derive the asymptotic variance by applying the delta-method

2 = V ar(n1/2([GE GE)) =1

(2 )21

2+21B0, (14)

with

B0 =h22

2 21+1 + 212 (1 )2 212

i, (15)

or, equivalently, using the covariance function a,b = Cov

Xa, Xb

= a+b ab, B0 =

21, + ()

21,1 21,1. This is estimated by B using

the corresponding moments. The expansion of B is denoted by B0 + B1/2

+ B1 + Op n3/2. As regards B1/2 it can be shown that, after centering andcollecting terms of the same order, B1/2 = n

1 Pi Z1,i with

Z1,i = 2

12 +1 (1 )2 12

(Xi 1)+22

X2i 2

(16)

+222 1+1 (1 )2 21

(Xi )

21

X+1i +1

+ 21

X2i 2

.

Similar but tedious manipulations reveal that B1 = n2

PiPj Z2,iZ3,j + Z4,iZ5,jwithZ2,i = (X1,i 1) , (17)Z3,i =

2 (1 )2 2

(X1,i 1)

+2+1 2 (1 )2 1

X1,i

2

X+11,i +1

+21

X21,i 2

,

Z4,i = X1,i

Z5,i = 22 (1 )2 21 X1,i

+22

X21,i 2

21

X+11,i +1

3. Combine the results from steps 1 and 2, and cancel common terms to get

S = n1/2hmm1 1 m+11

iB1/2.

Hence, in terms of the basic decomposition S = AB1/2, A = n1/2

hmm1 1 m+11

i.

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4. Expand m and m

1 to order Op (n1) to get the stochastic expansion S =

S0 + S1/2 + S1 + Op

n3/2

given by (11), the precise variables definitions forY1,i - Y6,i being

Y1,i = 1 (X

i ) (Xi 1) , (18)Y2,i = (Xi 1) ,

Y3.i = (X

i ) ( + 1)

2

11 (Xi 1) ,

Y4,i = (X1,i 1)Y5,i = (X1,i 1)

Y6,i = 1

6

21

3

(X1,i 1)

and Z1,i - Z5,i given by (16) and (17).

A.3 The Cumulant Expansions: Second Order Terms k1,2 andk3,1

A.3.1 The Asymptotic Bias Term k1,2

Taking expectations of the individual terms of (11) yields immediately E(S0) =

n1/2B1/20 E{n

1 PY1,i} = 0, and E(S1/2) = n1/2(B1/20 E(Y3Y4)0.5B3/20 E(Y1Z1)).It follows from the definition of k1,2 that E{S} = n

1/2k1,2 + O (n1) with

k1,2 = B

1/2

0 E(Y3Y4) 1

2B

3/2

0 E(Y1Z1) . (19)

A.3.2 The Asymptotic Skewness Term k3,1

In order to derive the asymptotic skewness term, we first need to obtain an expansionof the third moment of S. We take expectations of

S3 =

S0 + S1/23

= S30 + 3S20 S1/2 + Op

n1

by considering the constituent parts separately.

1. EnS20 S1/2o = n3/2B3/20 E{n4 PiPj Pk Pl Y1,iY1,jY3,kY4,l} 0.5n3/2B5/20E{n4

Pi

Pj

Pk

Pl Y1,iY1,jY3,kZ1,l}. Since we are only interested in the O

n1/2

term, we conclude that

En

S20 S1/2o

= n1/2B3/20 [E(Y1Y1) E(Y3Y4) + 2E(Y1Y3) E(Y1Y4)]

n1/2 32

E(Y1Z1) .

2. Consider S30 = n3/2B

3/20 n

3 (P

Y1,i)3. Hence E(S30 ) = n

1/2B3/20 E(Y

31 ) +

O (n1) .

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In summary

E(S3) = n1/2B3/20

E(Y31 ) (20)

+ 3 E(Y1Y1) E(Y3Y4) + 2E(Y1Y3) E(Y1Y4) 3

2 E(Y1Z1)+O

n1

.

Finally, since K3,n = E(S3)3E(S2) E(S)+2(E(S))3, and E(S2) = 1+O (n1),

we conclude that

k3,1 = B3/20

hE(Y31 ) + 6E(Y1Y3) E(Y1Y4) 3E(Y1Z1)

i. (21)

A.4 The Cumulant Expansions: The Third Order Terms k2,2and

k4,

1A.4.1 The Coefficient k2,2

We take expectations of

S2 = S20 + 2S0S1/2 +

2S0S1 + S21/2

+ Op

n3/2

by considering the constituent parts separately. For compactness, we make use of(13).

1. E{S20} = B10 E{A

20} = B

10 Ehn

1/2

Pi Y1,ii2

= B10 En[Y1,i]

2

o . Since En[Y1,i]2

o =B0 we haveEn

S20o

= 1.

2. En

S0S1/2o

= 0.5B20 En

A20B1/2o

+ B10 Enh

A0A1/2io

. As regards the last

term, Enh

A0A1/2io

= n2EnP

k

Pi

Pj Y1,kY2,iY3,j

o= n1E{Y1Y2Y3}, and for

the first term we have En

A20B1/2o

= n1E{Y1Y1Z1}. In summary,

En

S0S1/2o

= n11

2B20 E{Y1Y1Z1} + B

10 E{Y1Y2Y3}

.

3. E{S0S1} = E{12 B20 A20B1 + 38 B30 A20hB1/2

i2 12

B20 B1/2A0A1/2 + B10 A1A0}.

Similarly En

S21/2o

= B10 E{[12 B10 A0B1/2+A1/2]2} = B10 E{B10 A0A1/2B1/2+A21/2 +

14

B20 A20B

21/2}. Hence E{2S0S1 + S

21/2} = E{B20 A20B1 + B30 A20B21/2

2B20 B1/2A0A1/2 + 2B10 A0A1 + B

10 A

21/2}. For the sake of brevity we consider

only E{A1A0} explicitly. The other terms are dealt with in the same fashion.

E{A1A0} = n3E

nhPi

Pj

Pk

Pl Y4,iY5,jY6,kY1,l

io. Since we are only interested

in the O (n1) term, and since the random variables are centred, it followsthat the only contributions of interest come from the case of only two distinct

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indices and expectations taken over two random variables. Considering thepermutations over the indices, there is a total of 3 terms: E{Y4Y5}E{Y6Y1}+ E{Y4Y6}E{Y5Y1} + E{Y4Y1}E{Y5Y6}. Rather than using tensor notation,we use the compact shorthand

En[Y1,kY4,lY5,iY6,j][pairs of 2, 3 terms]o .Thus E(S2) = KS,2 = 1 + n

1k2,2 + O

n3/2

with

k2,2 =1

2B20 E{Y1Y1Z1} + B

10 E{Y1Y2Y3}

B20 E

n[Y1,kY1,lZ2,iZ3,j + Y1,kY1,lZ4,iZ5,j][pairs of 2, 3 terms]

o(22)

+B30 En

[Y1,kY1,lZ1,iZ1,j][pairs of 2, 3 terms]

o

2B20 En[Z1,kY1,lY2,iY3,j][pairs of 2, 3 terms]o+2B10 En[Y1,kY4,lY5,iY6,j][pairs of 2, 3 terms]o

+B10 En

[Y2,kY2,lY3,iY3,j ][pairs of 2, 3 terms]

o.

Some simplifications obtain by noting that, for instance, En

[Y1,kY1,lZ2,iZ3,j][pairs of 2, 3 terms]

o= B0E{Z2Z3} + 2E{Y1Z2}E{Y1Z3}.

A.4.2 The Coefficient k4,1

We begin by considering the expansion of the fourth moment of S, given by

S4

= S40 + 4S

30 S1/2 + 4S30 S1 + 6S20 S21/2 + Op n3/2 ,

and take expectations of the constituent parts.

1. E{S40} = Eh

B1/20 A0

i4= B20 E

n[Y1,kY1,lY1,iY1,j][2 pairs]

o+n1B20 [E{Y1Y1Y1Y1}

En

[Y1,kY1,lY1,iY1,j][2 pairs]

o]. But B20 E

n[Y1,kY1,lY1,iY1,j][2 pairs]

o= 3B20 [E{Y1Y1}]

2 =

3 since E{Y1Y1} = B0.

En

S40o

= 3 + n1hB20 E{Y1Y1Y1Y1} 3

i.

2. EnS30 S1/2o = B

20 En

12

B10 A40B1/2 + A

30A1/2o. Consider the constituent parts.EnA40B1/2o = n3E{PkPlPmPiPj Z1,kY1,lY1,mY1,iY1,j}. Since we are only

interested in the O (n1) term, and since the random variables are centred, itfollows that the only contributions of interest come from the case of only twodistinct indices and expectations taken over two and three random variables,such as E{Z1Y1}E{Y1Y1Y1} etc. Considering the permutations over the in-dices, there is a total of 10 of such terms. We use the compact shorthand

E{Z1,kY1,lY1,mY1,iY1,j}[pairs of 2+3, 10 terms] .

Similarly E

nA30A1/2

o= n1E{Y1,kY1,lY1,mY2,iY3,j}[pairs of 2+3, 10 terms].

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3. E{S30 S1} = E{12 B30 A40B1 + 38 B40 A40hB1/2

i2 12

B30 B1/2A30A1/2 + B

20 A

30A1}.

Similarly En

S20 S21/2

o= E{B20 A

20A

21/2+

14

B40 A40B

21/2B30 A30A1/2B1/2}. Again,

we are only interested in the O (n1) term, and since the random variables arecentred, it follows that the only contributions of interest come from the case ofonly three distinct indices and expectations taken over two random variables.Considering the permutations over the indices, there is a total of 15 of suchterms. We use the generic compact shorthand

E{Y1,kY2,lY3,mY4,nY5,iY6,j}[pairs of 2, 15 terms] ,

which includes for instance the contribution E{Y1Y2}E{Y3Y4}E{Y5Y6}.

In summary, E{S4} = 3 + n1e4,1 + O

n3/2

with

e4,1 = hB20 E{Y1Y1Y1Y1} 3i2B30 E{Z1,kY1,lY1,mY1,iY1,j}[pairs of 2+3, 10 terms]+4B20 E{Y1,kY1,lY1,mY2,iY3,j}[pairs of 2+3, 10 terms]

2B30 E{Y1,kY1,lY1,mY1,nZ2,iZ3,j}[pairs of 2, 15 terms]2B30 E{Y1,kY1,lY1,mY1,nZ4,iZ5,j}[pairs of 2, 15 terms]+

3

2B40 E{Y1,kY1,lY1,mY1,nZ1,iZ1,j}[pairs of 2, 15 terms]

2B30 E{Z1,kY1,lY1,mY1,nY2,iY3,j}[pairs of 2, 15 terms]+4B

20 E{Y1,kY1,lY1,mY4,nY5,iY6,j}[pairs of 2, 15 terms]

+6B20 E{Y1,kY1,lY2,mY3,nY2,iY3,j}[pairs of 2, 15 terms]

+3

2B40 E{Y1,kY1,lY1,mY1,nZ1,iZ1,j}[pairs of 2, 15 terms]

6B30 E{Y1,kY1,lY1,mY2,nY3,iZ1,j}[pairs of 2, 15 terms] .

Some simplifications obtain by noting, for instance, that E{Y1,kY1,lY1,mY1,nZ2,iZ3,j}

[pairs of 2, 15 terms] = 3B20 E{Z2Z3}+12B0E{Y1Z2}E{Y1Z3} , and E{Z1,kY1,lY1,mY1,iY1,j}

[pairs of 2+3, 10 terms] = 4E{Z1Y1}E{Y3

1 } + 6B0E{Z1Y2

1 }.

Finally, since K4

= E(S4)

4E(S3)E(S)

3 (E(S2))2

+ 12E(S2) (E(S))2

6 (E(S))4 it follows that with K4 = n1k4,1 + O(n3/2), we have

k4,1 = e4,1 4k1,2 [k3,1 + 3k1,2] 6k2,2 + 12 (k1,2)2 . (23)

B The Lorenz Curve and Poverty Indices

Similar derivations apply to the Lorenz curve and the poverty indices. As the detailsare fairly tedious, we merely state the main results, i.e. the definitions of the variables

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given in equation (12). As regards the Lorenz curve ordinate L(p) it can be shown,using the insights of Beach and Davidson (1983) and Beach et al. (1995), that

Y1 = (X1(X xp) pp) xp (1(X xp) p) L (X 1)

Y2 = Y3 = Y4 = Y5 = Y6 = 0,

and

B0 =h(1 p)p (xp)2 +pp 2xppp + 2p2pxp p22p

i+

2 21

L (p)2 2L (p) [pp +pxp1 xppp pp1]

Z1 = (1 2L(p))

hX21(X xp) pp

x2p (1(X xp) p)

i+

L(p)2 X2 2 +2

xp + L (p))

21

+ 2xpL(p) 1

1pp

![(X1(X xp) pp) xp (1(X xp) p)] +

2L(p)

1

1pp xpL(p) L(p)

21

!(X 1)

Z2 = [(X1(X xp) pp) xp (1(X xp) p)]

Z3 =

2

xp1

+221

![(X1(X xp) pp) xp (1(X xp) p)] +

221pp 4xpL(p)

1 42L(p)

21 ! (X 1) +2L(p)

1

1

X2 2

2 1

1

hX21(X xp) pp

x2p (1(X xp) p)

iZ4 = (X 1)

Z5 = L(p)

32

L(p)

21 2

21pp + 2xp

L(p)

1

!(X 1) +

2L(p)

1

hX21(X xp) pp

x2p (1(X xp) p)

i 2L(p)

2

X2 2

,

with pp = Rx1(X xp)dF(x), and pp = Rx21(X xp)dF(x).As regards the poverty indices, define the function g(x) = (1 x/z) 1 (x z).The variables are then defined as follows

Y1 = g (X) PZ1 = (g (X)

2 En

g (x)2o

) 2P [g (X) P]Z2 = Y1Z3 = Y1,

and all other variables equal zero.

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density estimates with n=100, SM(100,2.8,1.7)

centered income

density

-10 -8 -6 -4 -2 0 2

0.

0

0.1

0.

2

0.

3

0.

4

Gaussianstudentized GE2one sample saddlepoint approx.

Figure 1:

20

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