OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 52,1(1990)0305-9049 $3.00

An Asymptotically Distribution-Free Test for Sen'sWelfare Index

John A. Bishop, S. Chakraborti and Paul D. Thistle

I. INTRODUCTION

Inequality and welfare comparisons of income distributions are closelyrelated. Welfare comparisons incorporate 'efficiency preference' orpreference for higher mean income, as well as inequality aversion. In welfarecomparisons, greater inequality may be compensated for by sufficientlyhigher mean income. To completely order a set of income distributionsrequires specifying a welfare index.' Sen (1974, 1976) provides an axiomaticbasis for a welfare index based on the Gini coefficient.

The purpose of this paper is to provide statistical procedures that allowóne to make probabiistically valid statements regarding Sen's index. Much ofthe data used in empirical analyses of income inequality and welfare arebased on samples of the population. Mean incomes, income shares, andinequality and welfare indices calculated from such data are sample statistics,and are therefore subject to sampling variability. Comparisons of thecalculated values of such statistics (i.e., point estimates), commonplace in theliterature, are hypothesis tests where the probability of type I error is one.Application of sound statistical inference procedures requires knowledge ofthe sampling distribution of the statistic. In this paper we examine the largesample distribution of Sen's welfare index.

Section II briefly discusses the interpretations of Sen's index. Section IIIdiscusses that statistical properties of Sen's index. The analysis draws on theasymptotic distribution theory for Lorenz curves developed by Gail andGastwirth (1978a), Beach and Davidson (1983), Beach and Richmond(1985) and Gastwirth and Gail (1985).2 We provide asymptoticallydistribution-free confidence intervals and tests for Sen's index. In Section IV,the procedures are applied to data on US state income distributions. SectionV concludes the paper.

'As Blackorby and Donaldson (1978, 1984) and Ebert (1987) show, the choice of aninequality index implicitly specifies a welfare index.

2 see Gastwirth (1974), and Gail and Gastwirth (1 978b).

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SEN'S WELFARE INDEX

Sen's welfare index is based on the Gim coefficient. This leads to threecomplementary interpretations of the index.

Let income, Y, have the distribution function F. We assume the support ofF is a bounded subset of R. The mean of F is u, where u >0, and the Ginicoefficient is G. Then Sen's index is

S=1u(1G) (1)

Sheshinski (1972) derives the welfare function for the Gini coefficient. Sen(1974) provides an axiomatic basis for the Gim coefficient as an incomeinequality measure. The index in (1) can be interpreted as the equally-distributed-equivalent income measure for the Gini coefficient.

Sen (1976) considers welfare basis of indices of real national income. Inthis case the social preference relation is defined on the distribution ofcommodities, and only indirectly on the distribution of income. Sen showsthat the index (1), calculated from the income distribution, is a subrelation ofthe social preference relation. This provides a second interpretation of theindex.

The third interpretation of Sen's index is geometric, and follows from thegeometric interpretation of the Gini. As is well known, the Gini coefficient istwice the area between the Lorenz curve and the 45-degree 'equality line'.Then 1 - G is twice the area below the Lorenz curve. Rescaling the Lorenzcurve by mean income yields the generalized Lorenz curve (Shorrocks,1983), so Sen's index is twice the area below the generalized Lorenz curve.

THE PROPOSED TEST

The geometric interpretation of Sen's welfare index allows us to applyavailable results on the distribution of Lorenz and generalized Lorenzordinates, and leads to a straightforward asymptotic distribution theory forSen's index.

Income, Y, is a positive random variable with distribution F. Let Pi'be a set of Kfractions, where °=Po <j<... <PK<PK+1 = 1. Correspondingto these K fractions we have a set of K income quantiles, ¿k' whereF(k)=pk. Let yk=E(YI Yk) be the conditional mean, or mean ofincomes less than ¿k. Let )=E((Yyk)2I Yk) be the conditionalvariance. Observe that the (unconditional) mean and variance can be written

i'Iv V<1- \ ,i 2 L'(Iv_ \2 V<.as YK+1 I _,K+l/an 'K-f-1 II YK1/ K+1The vector of K population Lorenz ordinates at the abcissae Pk is (01,

02,..., 0K), where 0k =PkYkI/ The generalized Lorenz ordinates are notnormalized by the mean. Then the vector of K + 1 freely variable populationgeneralized Lorenz ordinates is Ø=(Ø, ø2,...øK1) where øk=/0k=PkYkThe Gini coefficient can be calculated from the Lorenz ordinates as

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K+1

G 12 ak(pkOk), (2)k=1

where ak = 2/(K + 1), k= 1,..., K, and aK + = 1 ¡(K + 1 ).3 Similarly, Sen'sindex can be calculated from the generalized Lorenz ordinates as

K+1

S= 12 aç/),, (3)k1

Now suppose that we have a random sample of N observations from F.The quantities , , çb,, and ) are all consistently estimated by thecorresponding sample quantities; hats denote estimators. The asymptoticdistribution of follows directly from the asymptotic distribution of thesample generalized Lorenz ordinates.

Lemma: If F is continuously differentiable and strictly monotonic, then- Ø) has a limiting K + 1 - variate normal distribution with mean zero

and variance-covariance matrix Q = [w,], where, for ij,

+ (1-pi) (, - y) (- y1) + ( - y) (y.- y)] (4)

This is proved in Beach and Davidson (1983, p. 726). As Bishop,Charkraborti and Thistle (1989) point out, the Lemma provides a basis fortests for generalized Lorenz dominance.

Under the same assumptions, we have the following result.

Theorem: JN( - S) has a limiting normal distribution with mean zero andvariance

K+1 K+1

o2= 12 aa1w,i=1 j=1

I K+1 K+1

=(4 12 12 w_3wK+lK+l)/(K+1)2. (5)i=1 j=1

This follows directly from the fact that is a linear function of the samplegeneralized Lorenz ordinates.

If two independent random samples of N1 and N2 observations areavailable, an asymptotically distribution-free test of the null hypothesisH0:S1 =2 can be based on

Z=(1 -52)/(ô/N1 + ô/N2)'I2. (6)

formulae in (2) and (3) assume the Pk are equally spaced. This simplifying assumptioncan easily be relaxed. The Lorenz and generalized Lorenz ordinates at successive Pk areconnected by linear interpolation. Then (2) and (3) give the lower bound Gini and the upperbound Sen index, respectively.

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Then, for example, if the alternative hypothesis is one sided, HA : S1 > S2, anapproximately size a test is to reject H0 in favor of HA if Z> Za, where Za isthe upper a quantile of the standard normal distribution. Finally, from theabove results one can easily construct confidence intervals around the Senindex.

III. EMPIRICAL EXAMPLE

While many studies have used the Gim coefficient, the only previousapplications of Sen's welfare index are Sen (1976) and Berrebi and Silber(1987). We point out that neither Sen nor Berrebi and Silber providestandard errors or confidence intervals.

We use data on US state income distributions.4 The sample is a one-quarter percent (one in 400) sample of households drawn from the 1980 USCensus. Sample sizes range from 597 households in Alaska to 36,842 house-holds in California. The US sample, formed by pooling the state samples,contains 356,393 households. The income measure is total household moneyincome. We use the procedure in Tremblay (1986) to adjust for cost-of-livingdifferences.5

Table 1 reports the estimates of mean income, the Gini coefficient, andSen's index for each of the 50 states, the District of Columbia, and the US as awhole. The Gini and Sen's index are estimated at vigintiles (K + 1 20).Column 1 reports the Gim coefficients. The Gus range from 0.3 584(Wyoming) to 0.4464 (District of Columbia). The Gini for the US as a wholeis 0.4056. Colunm 2 gives the mean incomes, which range from Maine's$14,758 to Maryland's $25,068; the US average is $19,645.

The estimates of Sen's index are reported in Column 4, and range from$9122 (Maine) to $15,579 (Maryland). The value of Sen's index for the US asa whole is $11,677. Given the definition of Sen's index in (1), one wouldexpect the index to be correlated with mean income. However, empiricallythe state mean incomes and Sen indices are uncorrelated, the samplecorrelation is - 0.0175. Importantly, this suggests that mean income may bemisleading as an indicator of standards of living.

Columns 5 and 6 give the lower (L1) and upper (U1) end points of the 95percent confidence intervals. The confidence intervals are based on theestimated asymptotic standard errors, calculated from (5) and the samplesizes, N1, reported in Column 7 The estimated asymptotic standard errorscan easily be recovered as â1/,JN1 = ( U1 - L. )/ 1.96, and the test statistics in (6)calculated.

No two states have the same value of the Sen index. To give an idea of theimportance of statistical tests, we tested for equality of the US Sen index with

4See, e.g., Kuznets (1963), Al-Sammarie and Miller (1967), Sale (1974), Nord (1984), andBishop, Formby, and Thistle (1989) for previous studies of US state income distributions.

Bishop, Formby, and Thistle for a more detailed description of the sample.

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TABLE 1Confidence Bands for Sen Index

109

State Gini MeanSENindex

Confidenceinterval

Samplesize

Alabama (46) 0.4277 18155 10390 10123-10658 6061Alaska(3) 0.3841 21927 13504 12498-14511 597Arizona(27) 0.3980 19237 11580 11262_11899* 4313Arkansas (51) 0.4302 17001 9687 9378-9997 3674California(12) 0.4069 21190 12568 12446-12691 36842Colorado(6) 0.3945 22110 13387 13045-13729 4835Connecticut(11) 0.3886 20871 12760 12447-13074 4981Delaware(5) 0.4102 22718 13399 12588-14211 968Dist.Columbia(44) 0.4424 18780 10472 9853-11091 1150Florida(37) 0.4164 18547 10823 10669-10978 16738Georgia(24) 0.4108 19792 11662 11417_11907* 8506Hawaii(34) 0.3956 18306 11064 10526-11602 1351!daho(30) 0.3803 18408 11406 10893_11921* 1482illinois (7) 0.3955 21932 13257 13075-13439 18240Indiana(1O) 0.3742 20407 12770 12533-13008 8666Iowa(14) 0.3850 20365 12525 12207-12844 4766Kansas(21) 0.4004 19769 11853 11511_12196* 3938Kentucky(40) 0.4207 18355 10631 10358-10906 5816Louisiana(36) 0.4405 19571 10950 10664-11237 6361Maine(52) 0.3819 14758 9122 8754-9491 1789Maryland(1) 0.3785 25068 15579 15249-15911 6715Massachusetts(29) 0.3955 18975 11470 11255_11686* 9326Michigan(8) 0.3866 21073 12926 12731-13123 14327Minnesota(16) 0.3889 20399 12466 12193-12740 6561Mississippi (49) 0.4259 17155 9848 9530-10168 3784Missouri(39) 0.4089 17998 10639 10416-10863 8027Montana(31) 0.3757 18264 11401 10850_11953* 1292Nebraska(25) 0.3919 19158 11650 11243_12058* 2550Nevada(4) 0.3792 21746 13499 12873-14125 1387NewHampshire(32) 0.3734 18186 11395 10903_11888* 1504NewJersey(13) 0.3928 20648 12537 12325-12750 11504NewMexico(41) 0.4071 17885 10604 10164-11045 2011New York (50) 0.4165 16765 9781 9664-9900 25875NorthCarolina(35) 0.4048 18451 10982 10770-11196 9169NorthDakota(33) 0.3940 18447 11179 10554_11805* 1040Ohio(15) 0.3858 20383 12519 12348-12691 17268Oklahoma(28) 0.4248 19999 11503 11196_11811* 5068Oregon(22) 0.3957 19358 11697 11384_12012* 44421nnsy1vania(45) 0.3906 17158 10456 10320-10593 18863Rhode Island (47) 0.3906 16860 10273 9801-10746 1523SouthCarolina(38) 0.4070 18249 10822 10522-11123 4650South Dakota (48) 0.3981 16747 10080 9526-10634 1107Tennessee (42) 0.4195 18227 10580 10337-10823 7087

All tests performed at 5 percent signficance level. indicates that the Sen index is notsignificantly different from the US average.

each state Sen index. Twenty states have Sen indices significantly greater thanthe US Sen index, while 19 states have significantly lower Sen indices. Twelveof the states have values of the Sen index that are not significantly differentfrom the US as a whole.6 This is nearly one-fourth of the states.

Figure 1 presents a different view of the information in Table 1. The statesand US are ranked by their estimated Sen indices. The ranks are given inparentheses next to the state name in Table 1. Figure 1 plots the upper andlower 95 percent confidence bounds as a function of the state ranks. Then,for example, Maryland (rank 1) has a Sen index that is significantly greaterthan that of any other state; the confidence interval for Maryland does notoverlap that of any other state. On the other hand, the confidence intervalsfor 'Wyoming (rank 2) and Alaska (rank 3) do overlap; the Sen indices forthese states are not significantly different. Thus, Figure 1 allows one todetermine, by inspection, if the Sen indices for any two states are significantlydifferent.

It is important to observe that many of the confidence intervals overlap,implying the Sen indices for those states are not significantly different. Forexample1 the Sen indices for Alaska and Wisconsin are not significantlydifferent, despite the eleven hundred dollar difference in their values.Similarly, there is a difference of over one thousand dollars in the Sen indicesfor Wyoming and Connecticut, but this difference is not significant. In fact,Maryland (rank 1) is the only state for which the Sen index is significantly

6 states with larger values are AK, CA, CO, CN, DE, IL, IN, IA, MD, MI, MN, NJ, NV,OH, TX, UT, VA, WA, WI, and WY. The states with smaller values are AL, AR, DC, FL, HI,KY, LA, ME, MS, MO, NM, NY, NC, PA, RI, SC, SD, TN, and VT. The states with values notsignificantly different from the US are AZ, GA, ID, KS, MA, MT, NE, NH, ND, OK, OR, andWV.

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TABLE 1 - contd

SEN Confidence SampleState Gini Mean index interval size

Texas(20) 0.4190 20599 11967 11809-12126 20985Utah (9) 0.3646 20130 12791 12326-13257 2026Vermont(43) 0.3894 17248 10531 9909-11153 789Virginia(17) 0.3985 20643 12415 12168-12664 8368Washington(19) 0.3872 19891 12188 11925-12451 6933West Virginia(26) 0.4052 19566 11637 11244_12032* 3079Wisconsin(18) 0.3838 20108 12389 12138-12641 7321Wyoming(2) 0.3584 21563 13833 12990-14677 738

UnitedStates(23) 0.4056 19645 11677 11641-11714 356393

SEN

17,000

16,000

15,000

14,000

13,000-

12,000-

11,000 -

10,000 -

9000 -

8000

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ItulIltitILIItIJIttu!' 11JJq1

"iiiI'I IIij

1 3 5 7 9 1113 15 17 19 21 2325 272931 3335373941 43454749 51RANK

Fig. 1.

different from all other states. Clearly, the use of statistical tests, such as thosedeveloped here, can have an important effect on the conclusions drawn fromthe analysis.

V. CONCLUSION

This paper provides simple asymptotically distribution-free statisticalinference procedures for Sen's (1974, 1976) welfare index. Sen's index is theequally-distributed-equivalent income for the Gim coefficient, and is equal tothe area under the generalized Lorenz curve. This allows us to obtain thelarge sample distribution of Sen's index in a straightforward manner, andpropose procedures for constructing tests of hypotheses and confidenceintervals.

We apply the inference procedures to data on US state incomedistributions, providing confidence bands for our estimates of Sen's index.Empirically, the estimates of Sen's index are uncorrelated with mean income.This implies mean income may be a misleading indicator of standards ofliving. We test for equality of the state and overall US Sen indices, finding that20 states have significantly greater values and 19 states have significantly

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lower values of the Sen index. However, twelve of the states have índices thatare not significantly different from the US value. Only one state has a Senindex that is significantly different from those of all other states. The conclu-sion of 'not different' cannot be obtained when indices are comparednumerically.

Department of Economics, East Carolina University,Department of Mgt Science &Statistics, University ofAlabama,Department of Economics, Finance &Legal Studies, University of Alabama

Date of Receipt of Final Manuscript: October 1989

REFERENCES

Al-Sammarie and Miller (1967). 'State Differentials in Income Concentration',American EconomicReview, Vol. 57, pp. 175-84.

Beach, C. M. and Davidson, R. (1983). 'Distribution-Free Statistical Inference withLorenz Curves and Income Shares', Review of Economic Studies, Vol. 50, pp.723-35.

Beach, C. M. and Richmond, J. (1985). 'Joint Confidence Intervals for Lorenz Curvesand Income Shares', International Economic Review, Vol. 26, pp. 439-50.

Berrebi, Z. M. and Silber, J. (1987). 'Regional Differences and the Components ofGrowth and Inequality Change', Economics Letters, Vol. 25, pp. 295-98.

Bishop, J. A., Chakraborti, S. and Thistle, P. D. (1989). 'Asymptotically Distribution-Free Statistical Inference for Generalized Lorenz Curves', Review of Economicsand Statistics, forthcoming.

Bishop, J. A., Formby, J. P. and Thistle, P. D. (1989). 'Statistical Inference, IncomeDistributions, and Social Welfare', in Slottje, D. (ed.), Research on EconomicInequality, Vol. I, JAl Press: Greenwich.

Blackorby, C. and Donaldson, D. (1978). 'Measures of Relative Inequality of TheirMeaning in Terms of Social Welfare', Journal of Economic Theory, Vol. 18, pp.59-80.

Blackorby, C. and Donaldson, D. (1984). 'Ethically Significant Ordinal Indices ofRelative Inequality', in Basmann, R L and Rhodes, G. F., Jr (eds.), Advances inEconometrics, Vol. 3, JAl Press, Greenwich.

Ebert, U. (1987). 'Size and Distribution of Incomes as Determinants of SocialWelfare', Journal of Economic Theory, Vol.41, pp. 23-33.

Gail, M. H. and Gastwirth, J. L. (1 978a). 'A Scale-Free Goodness-of-Fit Test for theExponential Distribution Based on the Lorenz Curve', Journal of the AmericanStatisticalAssociation, Vol. 73, pp. 787-93.

Gail, M. H. and Gastwirth, J. L. (1 978b). 'A Scale-Free Goodness-of-Fit Test for theExponential Distribution Based on the Gini Statistic', Journal of the RoyalStatistical Society, Series B, Vol. 40, pp. 350-57.

Gastwirth, J. L. (1974). 'Large Sample Theory of Some Measures of IncomeInequality', Econometrica, Vol.42, pp. 19 1-96.

Gaswirth, J. L. and Gail, M. H. (1985). 'Simple Asymptotically Distribution-FreeMethods of Comparing Lorenz Curves and Gini Indices Obtained from Complete

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Data', in Basmann, R. L. and Rhodes, G. F., Jr (eds.) Advances in Econometrics,Vol. 4, JAl Press, Greenwich.

Kuznets, S. (1963). 'Quantitative Aspects of the Economic Growth of Nations:Distribution of Income by Size', Economic Development and Cultural Change, Vol.ll,pp. 1-80.

Nord, S. (1984). 'An Economic Analysis of Changes in the Relative Shape of theInterstate Size Distribution of Income', American Economist, Vol. 28, pp. 18-25.

Sale, T. (1974). 'Interstate Analysis of the Size Distribution of Income, 1950-70',Southern Economic Journal, Vol. 40, pp.434-41.

Sen, A. K. (1974). 'Informational Bases of Alternative Welfare Approaches:Aggregation and Income Distribution', Journal of Public Economics, Vol. 3, pp.387-403.

Sen, A. K. (1976), 'Real National Income', Review of Economic Studies, Vol. 43, pp.19-39.

Sheshinski, E. (1972). 'Relation Between a Social Welfare Function and the GiniIndex of Income Inequality', Journal of Economic Theory, Vol.4, pp. 98-100.

Tremblay, C. H. (1986). 'Regional Wage Differentials: Has the South Risen Again:Comment', Review of Economics and Statistics, Vol. 68, pp. 175-78.

US Department of Commence, Bureau of the Census (1983). Census of Populationand Housing, 1980: Public Use Microdata Sample A, Bureau of Labor Statistics,Washington, DC.