The Distribution of the Concentration Ratio for Samples from a Uniform Population

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i SciRes. http://www.scirp.or/joural/am http://dx.doi.or/0.436/am.05.

it, aavecchia@lum.it Received 4 October 04; revised 0 November 04; accepted 6 December 04 Copyriht 05 by authors ad Scietific Research Publishi Ic.

1 Applied Mathematics, 05, 6, Published Olie Jauary 05 i SciRes. The Distributio of the Cocetratio Ratio for Samples from a Uiform Populatio Giovai Giroe, Atoella Naavecchia Faculty of Ecoomics, Uiversity of Bari, Bari, Italy iovai.iroe@uiba.it, aavecchia@lum.it Received 4 October 04; revised 0 November 04; accepted 6 December 04 Copyriht 05 by authors ad Scietific Research Publishi Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract I the preset paper we derived, with direct method, the exact expressios for the sampli probability desity fuctio of the Gii cocetratio ratio for samples from a uiform populatio of size = 6, 7, 8, 9 ad 0. Moreover, we foud some reularities of such distributios valid for ay sample size. Keywords Gii Cocetratio Ratio, Uiform Distributio, Order Statistics, Probability Desity Fuctio. Itroductio I 94 Corrado Gii [] itroduced the cocetratio ratio R for the measure of iequality amo values of a frequecy distributio. The Gii idex is widely used i fields as diverse as socioloy, health sciece, eieeri, ad i particular, ecoomics to measure the iequality of icome distributio. Various aspects of the Gii idex have bee take ito accout. Oe of the most iteresti topics reards the estimatio of the cocetratio ratio (Hoeffdi, 948 []; Glasser, 96 [3]; Cuccoi, 965 [4]; Dall Alio, 965 [5]). More recetly, Deltas (003) [6] discussed the sources of bias of the Gii coefficiet for small samples. This has implicatios for the compariso of iequality amo subsamples, some of which may be small, ad the use of the Gii idex i measuri firm size iequality i markets with a small umber of firms. Barret ad Doald (009) [7] cosidered statistical iferece for cosistet estimators of eeralized Gii idices. The empirical idices are show to be asymptotically ormally distributed usi fuctioal limit theory. Moreover, asymptotic variace expressios are obtaied usi ifluece fuctios. Davidso (009) [8] derived a approximatio for the estimator of the Gii idex by which it is expressed as a sum of IID radom variables. This approximatio allows developi a reliable stadard error that is simple to compute. Fakoor, Ghalibaf ad Azaroosh (0) [9] cosidered oparametric estimators of the Gii idex based o a sample from leth-bi- How to cite this paper: Giroe, G. ad Naavecchia, A. (05) The Distributio of the Cocetratio Ratio for Samples from a Uiform Populatio. Applied Mathematics, 6,

2 ased distributios. They showed that these estimators are stroly cosistet for the Gii idex. Also, they obtaied a asymptotic ormality for the correspodi Gii idex. Giroe (968) [0] focused o the study of the sampli distributio of the Gii idex ad i 97 [] derived the exact expressio for samples draw from a expoetial populatio. I 97 Giroe [] obtaied, with direct method, the sampli distributio fuctio of the Gii ratio for samples of size 5 draw from a uiform populatio. I the preset ote (Sectio ), we calculate the joit probability desity fuctio (p.d.f.) of the radom sample of size ad, the, the joit p.d.f. of the order statistics. Hece, we trasform oe of the order statistics i their averae ad the remaii order statistics are divided by the same averae. We calculate the joit p.d.f. of the ew variables ad iterati with respect to the averae we obtai the joit p.d.f. of the other variables. Oe of these variables is trasformed i the cocetratio ratio. We calculate the joit p.d.f. of the cocetratio ratio ad of the other variables ad at last we iterate this p.d.f. with respect to the variables obtaii the marial p.d.f. of the cocetratio ratio. The mai difficulty of this procedure cosists i the idetificatio of the reio of iteratio of the variables, for two reasos: firstly the eed to decompose this reio ito subreios which allow idetifyi directly the limits of iteratio ad secodly the rowi umber of such subreios that makes the derivatio heavy. I Sectios 3-7, usi the software Mathematica, we derive the exact distributios of the cocetratio ratio for samples from a uiform distributio of size = 6, 7, 8, 9 ad 0. Moreover (Sectio 8), we fid some reularities of such distributios valid for ay sample size.. The Procedure to Derive the Distributio of the Cocetratio Ratio Let radom variables X, X,, X from a uiform populatio have p.d.f. The joit p.d.f. of the variables is (,,, x ) h x x f ( x) The joit p.d.f. of the order statistics X( ), X ( ),, X( ) is By trasformi the variables, 0 < x <, = () 0, elsewhere., 0 < xi <, for i =,,,, = () 0, elsewhere.!, 0 < x( ) < x( ) < < x ( ) <, h( x( ), x( ),, x( ) ) = (3) 0, elsewhere. S = X + X + + X ( ) ( ) ( ), whose Jacobia is X ( i) D( ) =, for i = i,,,, S J = S, we obtai the joit p.d.f. of the variables S ad D( ), D ( ),, D( ) that ca be writte as ( ( ) ( ) ( ) ) = ( ) sd,, d,, d! s, (4) ( ) for 0 < sd < sd < < sd < s d d d <. ( ) ( ) ( ) ( ) ( ) ( ) We iterate expressio [4] with respect to the variable S ad obtai the joit p.d.f. of the variables D, D,, D that ca be writte as ( ) ( ) ( ) 58

3 ( ( ) ( ) ( ) ) for 0 < d < d < < d < d d d. ( )! f d, d,, d =, ( ) ( ) ( ) ( ) ( ) ( ) ( d( ) d ( ) d( ) ) By trasformi the variable D( ) i the variable R i.e. the cocetratio ratio R = ( i) D( ), i i= (5) from which we et ( )( R) ( ) ( i) D( ) = i D i=, the Jacobia of the trasformatio is J = ad the joit p.d.f. of the variable R ad D( ), D ( ),, D( ) is for ( ( ) ( ) ( ) ) ( )! h d, d,, d, R =, (6) ( )( R) + ( i ) d ( i ) i= ( )( ) R R 0 < d < d < < d < i d < + i d. (7) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i= i= By iterati expressio [6] with respect to the variables D( ), D ( ),, D( ) over the reios determied by iequalities [7], we et the marial p.d.f. of the cocetratio ratio R. 3. The Distributio of the Cocetratio Ratio for = 6 The procedure idicated i Sectio is used to obtai the followi p.d.f. (Fiure ) of the cocetratio ratio R for radom samples of size = 6: ( R) = +, for 0 < R <, 3 4 ( + R) R 5 + R 5 + R , for < R <, 44R R R , for < R < 3, 40R R 75 + R R 5 + R , for 3 < R < 4, 40R R 75 + R

4 Fiure. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 6 from a uiform populatio , r 4 < R <. 3 44R R R R fo Characteristic values of the distributio are: 696lo 43lo 3 50lo 5 mea E( R ) = = 0.35, lo 49 lo 3 85lo 5 secod momet E( R ) = 6 + = 0.376, lo lo 3 65lo 5 third momet E( R ) = + + = , lo lo lo 5 fourth momet E( R ) = + = , stadard deviatio σ 0.444, idex of skewess γ ( R) = , idex of kurtosis γ ( R) = The distributio of the cocetratio ratio R for samples of size = 6 from a uiform populatio shows a sliht positive skewess ad platykurtosis. 4. The Distributio of the Cocetratio Ratio for = 7 The procedure idicated i Sectio is used to obtai the followi p.d.f. (Fiure ) of the cocetratio ratio R for radom samples of size = 7: ( R) = R 60 + R R R , for 0 < R <, 5 00( + R)

5 Fiure. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 7 from a uiform populatio R R 80 + R 8 + R , for < R <, R R R 80 + R , for < R <, R R 6 6 ( ) = + R R R R R , for < R <, ( ) = R R R R , for < R <, R 60 + R

6 ( ) = R R 80 + R R R , for < R <. 0R Characteristic values of the distributio are: lo 797 lo lo lo 7 mea E( R ) = + + = , lo 455lo lo lo 7 secod momet E( R ) = + + = 0.39, third momet lo 46635lo lo lo 7 E( R ) = + + = , fourth momet lo lo lo lo 7 E( R ) = = 0.034, stadard deviatio σ , idex of skewess γ ( R) = , idex of kurtosis γ ( R) = The distributio of the cocetratio ratio R for samples of size = 7 from a uiform populatio shows sliht positive skewess ad platykurtosis, both lower tha those obtaied for samples of size = The Distributio of the Cocetratio Ratio for = 8 The procedure idicated i Sectio is used to obtai the followi p.d.f. (Fiure 3) of the cocetratio ratio R for radom samples of size = 8: ( R) = R R 45 + R R , for 0 < R <, ( + R) R 5 + R R R R R , for < R <, R 5 + R ( ) = + R R R R R , for < R <, R R

7 Fiure 3. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 8 from a uiform populatio R R R R , for < R <, R R R R R R , for < R <, R R 45 + R ( ) = + R R R R R , for, < R < R R R R R , for < R < R Characteristic values of the distributio are: 63

8 lo lo lo 5 348lo 7 mea E( R ) = = , lo lo lo lo 7 secod momet E( R ) = + + = 0.985, third momet lo lo lo lo 7 E( R ) = + + = , fourth momet lo lo lo lo 7 E( R ) = + + = 0.06, stadard deviatio σ , idex of skewess γ ( R) = , idex of kurtosis γ ( R) = The distributio of the cocetratio ratio R for samples of size = 8 from a uiform populatio shows sliht positive skewess ad platykurtosis, both lower tha those obtaied for samples of size = 6 ad The Distributio of the Cocetratio Ratio for = 9 The procedure idicated i Sectio is used to obtai the followi p.d.f. (Fiure 4) of the cocetratio ratio R for radom samples of size = 9: ( ) = + R R 00 + R R 35 + R , for 0 < R <, ( + R) R 00 + R R R R R R , for < R <, R R ( ) = + R R , for < R <, R 40 + R R R , for 3 < R < 4, R R R R

9 Fiure 4. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 9 from a uiform populatio R R R R , for 4 < R < 5, R R R R R R R , for 5 < R < 6, 7560R R R R R R R , for < R <, 75600R R R 00 + R R R R R , for < R <. 475R R R

10 Characteristic values of the distributio are: lo lo lo lo 7 mea E( R ) = + + = , secod momet lo lo lo lo 7 E( R ) = + + = 0.754, third momet lo lo lo lo 7 E( R ) = = , fourth momet lo lo lo lo 7 E( R ) = + = 0.003, stadard deviatio σ , idex of skewess γ ( R) = , idex of kurtosis γ ( R) = The distributio of the cocetratio ratio R for samples of size = 9 from a uiform populatio shows sliht positive skewess ad platykurtosis, both lower tha those obtaied for samples of size = 6, 7 ad The Distributio of the Cocetratio Ratio for = 0 The procedure idicated i Sectio is used to obtai the followi p.d.f. (Fiure 5) of the cocetratio ratio R for radom samples of size = 0: ( ) = + R R R 56 + R R R , for 0 < R <, ( + R) R R R , for < R <, R R R R R R R R , for < R <, R R R

11 Fiure 5. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 0 from a uiform populatio R R R R R , for < R <, R R 7 + R ( ) = + R R R R R R R for ,, < R < + R + R ( ) = R R R R R R R , for < R <, R R

12 R R R R ( ) = R R R , for 6 < R < 7, R 56 + R ( ) = R R R R R R , for 7 < R < 8, R R R ( ) = + R R R R R R R R , for < R <. 9 Characteristic values of the distributio are: lo lo lo lo 7 mea E( R ) = 5 + = , secod momet lo lo 3 E( R ) = lo lo 7 + = 0.574, third momet lo lo 3 E( R ) = lo lo 7 + = , fourth momet 68

13 lo lo lo 5 E( R ) = lo 7 + = , stadard deviatio σ ( = , idex of skewess γ ( R) = , idex of kurtosis γ ( R) = The distributio of the cocetratio ratio R for samples of size = 0 from a uiform populatio shows sliht positive skewess ad platykurtosis, both lower tha those obtaied for samples of size = 6,7,8 ad Some Reularities of the Distributios The aalysis of the p.d.f. for =,3,,0 shows some reularities: The p.d.f. of the cocetratio ratio R, for 0< R < ad for samples of size, ca be expressed by ( ) R = i ; i + i ( ) ( ) i= (! ) Furthermore, the p.d.f. of the cocetratio ratio R, for ( ) expressed by ( ) R = ( ) ( i) i= (! ) < R < ad for samples of size, ca be + i i ; i The desity of the cocetratio ratio R, for 0< R < ad for samples of size, is ive by The desity of the cocetratio ratio R, for ( ) 0 ( R) d R = ; (! ) < R < ad for samples of size, is ive by ( ) d ; (! ) R R = The jth term of the desity of the cocetratio ratio R, deoted as a i, j, verifies the followi symmetry a. i, j = aji, The coefficiets of the a terms of the p.d.f. of the cocetratio ratio R for samples of size ii, multiplied by ( ) become the coefficiets of the a i +, i + terms of the same p.d.f. for sample of size. These results are valid for every sample size ad may allow reduci the heavy calculatio to determie the p.d.f. of the cocetratio ratio R. 9. Cocludi Remarks I the preset paper we obtai the distributios of the Gii cocetratio ratio R for samples of size = 6,7,8,9 ad 0 draw from a uiform populatio. We use the same method used by Giroe [] to derive the same distributios for samples of size 5. We obtai the p.d.f. of the cocetratio ratio R calculati a multiple iteral i dimesios for each reio from ( k ) ( ) to k ( ) for k =,,,. The limits of iteratio are defied by solvi the iequalities of the order statistics divided by the sample 69

14 k for k =,,,. Such distributios are uimodal with mea tedi to 3, which is the value of the cocetratio ratio R for the populatio, ad have decreasi stadard deviatio. Moreover, the distributios show a sliht positive skewess ad platykurtosis that ted to decrease as icreases. Beyod the possibility to obtai similar results for samples of larer size, ope problems are the derivatio of the exact expressio for the mea ad the other features of the distributio of the cocetratio ratio R for radom samples of size draw from a uiform populatio. mea ad expressed i terms of the cocetratio ratio R for the values assumed i each of such reios. The calculatio of the limits of iteratio is particularly heavy ad requires a very lo processi time. The obtaied results show that the p.d.f. of the cocetratio ratio R is ive by hyperbolic splies with deree ad with odes i ( ) Refereces [] Gii, C. (94) L ammotare e la composizioedellaricchezzadelleazioi. Bocca, Torio. [] Hoeffdi, W. (948) A Class of Statistics with Asymptotically Normal Distributio. Aals of Mathematical Statistics, 9, [3] Glasser, G.J. (96) Variace Formulas for the Mea Differece ad the Coefficiet of Cocetratio. Joural of the America Statistical Associatio, 57, [4] Cuccoi, O. (965) Sulla distribuzioecampioaria del rapporto R di cocetrazioe. Statistica, 5, 9. [5] Dall Alio, G. (965) Comportametoasitoticodellestimedelladiffereza media e del rapporto di cocetrazioe. Metro, 4, [6] Deltas, G. (003) The Small-Sample Bias of the Gii Coefficiet: Results ad Implicatios for Empirical Research. Review of Ecoomics ad Statistics, 85, [7] Barrett, G.F. ad Doald, S.G. (009) Statistical Iferece with Geeralized Gii Idices of Iequality, Poverty, ad Welfare. Joural of Busiess & Ecoomic Statistics, 7, [8] Davidso, R. (009) Reliable Iferece for the Gii Idex. Joural of Ecoometrics, 50, [9] Fakoor, V., Ghalibaf, M.B. ad Azaroosh, H.A. (0) Asymptotic Behaviors of the Lorez Curve ad Gii Idex i Sampli from a Leth-Biased Distributio. Statistics ad Probability Letters, 8, [0] Giroe, G. (968) Sulcomportametocampioariosimulato del rapporto di cocetrazioe. Aalidella Facoltà di Ecoomia e Commerciodell Uiversitàdeli Studi di Bari, 3, 5-. [] Giroe, G. (97) La distribuzioe del rapporto di cocetrazioe per campioicasuali di variabiliespoeziali. Studi di Probabilità, Statistica e Ricercaoperativa i oore di Giuseppe Pompilj, Oderisi, Gubbio. [] Giroe, G. (97) La distribuzioe del rapporto di cocetrazioe per piccolicampioiestratti da uapopolazioeuiforme. Aalidell Istituto di Statisticadell Uiversitàdeli Studi di Bari, 36,

Topic 9: Sampling Distributions of Estimators

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