An (almost) unbiased estimator for the S-Gini index

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1 An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for nteger parameter values. Smulatons ndcate that these estmators perform consderably better then the usual estmators, especally for small sample szes. 1 The absolute and relatve S-gn ndces Assume that ncome s dstrbuted accordng to a contnuous and dfferentable cumulatve dstrbuton functon (cdf F : [0, ] [0, 1] wth fnte mean, µ, and contnuous populaton densty functon (pdf f. The absolute sngle-seres Gn (absolute S-Gn, A, and the Relatve sngle seres Gn (relatve S-Gn, R, wth parameter R ++ are gven by: A µ H and R 1 H µ, wth H 0 x (1 F (x 1 df (x. These ndces exst for all values of 1, but for values of < 1 t s possble that H reaches nfnty. From now on, we assume that H s well defned for all values of under consderaton. The parameter determnes the weght attached to the ncome of ndvduals at dfferent ponts n the ncome dstrbuton. As ncreases, more weght s gven to the bottom of the ncome dstrbuton. For equal to one, H 1 s equal to the mean µ and R 1 and A 1 are both equal to zero. For equal to 2, the ndces A 2 and R 2 reduce to the well-known absolute and relatve Gns. We refer to Donaldson and Weymark (1980, Ytzhak (1983 and Bossert (1990 for an n depth dscusson of the propertes related to the S-Gn ndex. I am pleased to acknowledge the nsghtful comments of Drk Van de gaer. Unversty of Ghent, Sherppa, Tweekerkenstraat 2, B-9000 Gent, Belgum. E-mal: thomas.demuynck@ugent.be 1

2 The most common fnte sample estmators for the S-Gns are gven by: A n µ n H n and R n 1 H n wth H n n µ n ( (n + 1 (n x Here x represents the th smallest value n the sample (the th order statstc and µ n s the sample mean, n x /n. The estmators A n and Rn are strongly consstent estmators for A and R and they are asymptotcally normally dstrbuted (Barrett and Pendakur, 1995; Ztks and Gastwrth, Unfortunately, they are not unbased and ther bas depends on the sample sze, n, the value of the parameter,, and the dstrbuton, F. The sample mean µ n s an unbased estmator for the populaton mean µ, hence, for the absolute S-Gn, A, we only need to construct an unbased estmator for the term H. Such estmator would also provde us wth an almost unbased estmator for R. Ths last estmator s not unbased because t s dvded by the sample mean whch s tself an estmator of the populaton mean. The next secton provdes an unbased estmator of H and the last secton provdes smulaton results to compare these estmators wth the estmators A n and Rn. n 2 A unbased estmator for H { n We denote by the strlng number of the second knd wth upper ndex n and lower k { n ndex k. The number represents the number of ways that a set of sze n can be k parttoned nto k subsets. We denote by the bnomal coeffcent wth upper ndex k n and lower ndex k,.e. the number of k element subsets of an n element set. Fnally, we denote by k the fallng factoral n(n 1... (n k + 1. The followng denttes 1 wll be used n ths secton: ( ( n n, R-1 k n k { { n n 1 k k 1 x r r + k 1 See Graham et al. (1989 for a proof of these denttes. { n 1, R-2 k { r x, R-3 2

3 ( ( n n k k, R-4 k (x + y n x y n. R-5 We focus on the case where the parameter takes only nteger values. Assume that we have a set of observatons {x 1,..., x n that s drawn..d. from the cdf F. The th order statstc x wll have pdf f ( equal to: f ( (x F (x 1 (1 F (x n f(x. The expected value of H n equals: E(H n 1 n ( n ((n + 1 (n xf (x 1 (1 F (x n df (x 0 In order to smplfy ths expresson we splt t up nto several parts: E(Hn 1 x (n + 1 F (x 1 (1 F (x n n 0 {{ A 1 {{ A (n F (x 1 (1 F (x n df (x. {{ (1 B 1 {{ B We have that: ( n A 1 (n + 1 ( n 1 n(n ( (n n n { ( + 1 n n + 1 ( n B 1 (n ( n 1 (R-4 n(n 1 { (R-1 n { (R-3, R-4 3 ( n 1 n n +1 ( n 1 1 (R-4 (R-1, R-3.(R-4, R-1

4 +1 { ( + 1 n. (R These results enable us to smplfy A and B: A +1 { ( + 1 n F (x 1 (1 F (x n { + 1 F (x 1 (1 F (x n (1 F (x 1. R-5 B 1 { ( n 1 +1 F (x 1 (1 F (x n F (x 1 (1 F (x n 1 +1 (1 F (x R (1 F (x 1. Substtutng A and B nto equaton (1 gves: E(Hn 1 +1 ({ { + 1 x n 1 (1 F 1 df (x x n 1 n 1 1 (1 F 1 df (x R-2 H. (2 Equaton (2 shows that the expected value of H n can be expressed as a weghted average of all ndces H m wth m. As such, the estmator H n wll not be unbased unless H m 4

5 s zero for all m. Equaton 2 allows us to construct an unbased estmator of H n a recursve way. For 1, we have that E(Hn 1 H 1 µ. Hence, Hn 1 s an unbased estmator of H. 1 Now, assume that we have an unbased estmator h m n of H m for all m n {1, 2,..., 1. Then we can construct followng estmator h n of H : ( h n 1 1 { n Hn h n. (3 Ths estmator s unbased: ( ( E(h 1 1 { n E n Hn h n ( 1 n E ( 1 Hn E ( h n H. The unbased estmator for A s then gven by a n µ n h n and the almost unbased estmator for R s gven by r n 1 h n/µ n. For the Gn ndex,.e. 2, t can be shown that r 2 n nr n/(n 1. Ths s n agreement to the frst order correcton for the Gn ndex found n the lterature (see Deaton, 1997; Deltas, 2003; Davdson, It can be shown that h n s equal to the followng expresson 2 : h n n 1 x. (4 The multplcators n 1 / sum to one 3 whch mples that, analogue to the estmators Hn, the estmators h n are a weghted average of the order statstcs x. Also, note that the weghts attached to the 1 hghest ncomes are equal to zero. Ths mples that the estmator h n does not use all avalable nformaton. For example, the value of h 10 n on a sample of sze 10 concdes wth the smallest value n the sample. Smple manpulaton of equaton (4 shows that we can wrte h n as a x, wth 2 See appendx A. 3 See appendx B /n ( f 1 a a n ( 1 for > 1. (5 5

6 For 1, as ncreases, the weghts attached to x decrease n an ncreasng rate untl they reach zero for x n +2. The recurson (5 shows that the estmator h n s very easy to calculate. It also makes t possble to defne h n for non-nteger values of. Unfortunately, ths extenson has the unwanted sde-effects that the weghts a do no longer sum to unty, although s wll approxmate unty f n s not to small, and that the estmator s no longer unbased. 3 Smulaton For our emprcal llustraton we used a lognormal dstrbuton wth parameters 9.85 and 0.6. Our populaton statstcs A and R were calculated on the bass of a random sample of 50 mllon observatons. We drew ndependent samples of sze m (m 10, 30, 50. For each of these samples, we calculated the estmators A m, a n, R m and r m. Table 1 presents the averages over these samples (standard errors are between brackets for the values 1.5; 2; 5; 7.5 and 10. Smulaton results for other parameter values and other dstrbutons gve smlar results. Table 1: smulaton results sample sze A n a n A Rn rn R ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

7 Table 1: smulaton results sample sze A n a n A Rn rn R (1942 (1947 ( ( NOTE: These smulatons were based on the lognormal dstrbuton: ln X N(9.85, 0.6. The statstcs R and A were based on a random sample of 10 mllon observatons. Each average was computed over a set of samples. Standard errors are between brackets. We observe followng regulartes: For nteger parameter values, the estmators r n and a n performs consderably better then the estmators R n and A n. For nonnteger parameter values one can clearly see that the estmator a n s no longer unbased although the bas decreases for larger sample szes and larger parameter values. Furthermore, the estmators r n and a n seem to perform consderably better n comparson to the estmators A n and R n. The standard errors for the estmators r n and a n are slghtly larger compared to the standard errors for the estmators R n and A n. References Barrett, G. F., Pendakur, K., The asymptotc dstrbuton of the generalzed gn ndces of nequalty. Canadan Journal of Economcs 28, Bossert, W., An axomatzaton of the sngle-seres gns. Journal of Economc Theory 50, Davdson, R., Relable nference for the gn ndex. GREQAM Document de Traval nr Deaton, A. S., The analyss of household surveys: a mcroeconometrc approach to development polcy. John Hopkns Unversty Press for the World Bank, Baltmore. Deltas, G., The small-sample bas of the gn coeffcent: results and mplcatons for emprcal research. The Revew of Economcs and Statstcs 85, Donaldson, D., Weymark, J. A., A sngle-parameter generalzaton of the gn ndces of nequalty. Journal of Economc Theory 22, Graham, R. L., Knuth, D. E., Patashnk, O., Concrete Mathematcs. Addson-Wesley. Ytzhak, S., Relatve deprvaton and the gn coeffcent. Internatonal Economc Revew 93,

8 Ztks, R., Gastwrth, J., The asymptotc dstrbuton of the s-gn ndex. Australan and New Zealand Journal of Statstcs 44, A Equvalence of equaton 3 and 4 The proof s by nducton on. For 1 we easly establsh that both equatons 3 and 4 reduce to µ n. Assume that the asserton holds for all m <. The proof follows f we can show that: n Hn h n. where h n s gven by equaton 4. n Hn (n + 1 (n x x x x { n + 1 x n (R-1 n 1 ((n + 1 (n + 1 n 1 h n. n 1 x 8

9 B h n s a weghted sum We show that the weghts n 1 sum to one. n 1 (n! n ( 1!(n + 1! ( /( n n 1 n 1 1 n 1 ( /( n n 1 n k k The last step uses the dentty: problem 1, p k0 m k0 ( / m k k m + 1 m + 1 n ( 1!(n! (n 1! (see Graham et al., 1989, 9