Applied Mathematics, 05, 6, 57-70 Published Olie Jauary 05 i SciRes. http://www.scirp.or/joural/am http://dx.doi.or/0.436/am.05.6007 The Distributio of the Cocetratio Ratio for Samples from a Uiform Populatio Giovai Giroe, Atoella Naavecchia Faculty of Ecoomics, Uiversity of Bari, Bari, Italy Email: 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). http://creativecommos.or/liceses/by/4.0/ 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, 57-70. http://dx.doi.or/0.436/am.05.6007
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
( ( ) ( ) ( ) ) 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) 7 78 9683 5596 3375 = +, for 0 < R <, 3 4 ( + R) 5 50 + R 5 + R 5 + R 5 5 5 5 5 5 + 7853 44953 + 67747 5596, for < R <, 44R 3 4 5 5 75 + R 400 + R 5 5 5 5 5 5 3 3 + 33773 44953 + 9683, for < R < 3, 40R 3 5 5 75 + R 75 + R 400 + R 5 + R 5 5 5 5 7 + 437 3 + 7853 78, for 3 < R < 4, 40R 5 5 400 + R 75 + R 75 5 5 5 5 5 59
Fiure. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 6 from a uiform populatio. 7 + 3 5 + 7, r 4 < R <. 3 44R 5 450 + R 400 + R 75 50 + R 5 5 5 5 fo Characteristic values of the distributio are: 696lo 43lo 3 50lo 5 mea E( R ) = 3 + + = 0.35, 5 3 69056 lo 49 lo 3 85lo 5 secod momet E( R ) = 6 + = 0.376, 5 50 3 368 678688lo 56006 lo 3 65lo 5 third momet E( R ) = + + = 0.05785, 5 5 5 4 4 8937 5434496 lo 769436 lo 3 590 lo 5 fourth momet E( R ) = + = 0.0606, 65 35 35 3 stadard deviatio σ 0.444, idex of skewess γ ( R) = 0.0793, idex of kurtosis γ ( R) = 0.6767. 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: 7649 7649 05884 3764768 ( R) = + 3 4 58400 + R 60 + R 640 + R 405 + R 6 6 6 6 3676535 05884 +, for 0 < R <, 5 00( + R) 6 0736 6 60
Fiure. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 7 from a uiform populatio. 9 435653 450309 09905 0R 3 03680 + R 80 + R 8 + R 6 6 6 4930947 3676535 +, for < R <, 4 5 6 6 60 0736 6 6 35 7853 46303907 7450309 90R 0368 + R 590 + R 80 + R 6 6 6 09905 3764768 3 +, for < R <, 3 4 6 6 8 + R 405 + R 6 6 ( ) = + R 3 0640 33773 46303907 90R 405 + R 5840 + R 590 + R 6 6 6 450309 05884 3 4 +, for < R <, 3 6 6 80 640 6 6 ( ) = R 9 463 0640 7853 3 90R 80 34 + R 5840 + R 6 6 6 435653 7649 4 5 +, for < R <, 6 6 03680 + R 60 + R 6 6 + 6
9 3 35 ( ) = R 4 3 800 + R 80 + R 405 + R 0368 + R 6 6 6 6 9 7649 5 +, for < R <. 0R 6 58400 6 Characteristic values of the distributio are: 7 3507 lo 797 lo 3 359375lo 5 83543lo 7 mea E( R ) = + + = 0.3495, 5 60 88 440 763 38068lo 455lo 3 359375lo 5 564737 lo 7 secod momet E( R ) = + + = 0.39, 36 405 80 7 6480 third momet 3 879 7744 lo 46635lo 3 8546875lo 5 576480lo 7 E( R ) = + + = 0.0547, 6 35 30 78 960 fourth momet 4 6593 4060588lo 7638867 lo 3 6334375lo 5 89594lo 7 E( R ) = + + + = 0.034, 648 5 30 555 77760 stadard deviatio σ 0.0367, idex of skewess γ ( R) = 0.8545, idex of kurtosis γ ( R) = 0.4535. 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 = 6. 5. 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: 89 0975 66355 5368709 ( R) = 3 4 995 + R 33075 + R 45 + R 9845 + R 7 7 7 7 8000000 698693 3768944 + +, for 0 < R <, 5 6 05( + R) 7 33 + R 5 + R 7 7 6807 963463 548367503 38686368 86400R 3 6650 + R 43360 + R 665 + R 7 7 7 73035887 900769787 698693, for < R <, 4 5 6 7 7 84670 6650 + R 5 + R 7 7 7 ( ) = + R 8 435653 309405 86007487 0960R 490 + R 665 + R 43360 + R 7 7 7 9697496 73035887 8000000 3 + +, for < R <, 3 4 5 7 7 330 + R 84670 + R 33 7 7 7 6
Fiure 3. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 8 from a uiform populatio. 565 7453 46303907 68853 + 3040R 5406 + R 665 + R 9845 + R 7 7 7 86007487 38686368 5368709 3 4, for < R <, 3 4 7 7 43360 665 + R 9845 + R 7 7 7 64 4779 54307 46303907 3 3040R 665 + R 43360 + R 665 + R 7 7 7 309405 548367503 66355 4 5 + +, for < R <, 3 7 7 665 + R 43360 + R 45 + R 7 7 7 8 5949 4779 7453 ( ) = + R 4 3 56800 + R 6650 + R 43360 + R 665 + R 7 7 7 7 435653 963463 0975 5 6, for, 0960 + 7 7 6650 3305 7 < R < R 7 7 8 64 565 5 4 3 98450 + R 56800 + R 665 + R 5406 + R 7 7 7 7 8 6807 89 6 + +, for < R <. 86400R 7 490 995 7 7 Characteristic values of the distributio are: 63
3475456 lo 775303lo 3 4875lo 5 348lo 7 mea E( R ) = 4 + + = 0.34747, 35 560 008 70 90 384576 lo 877643lo 3 44065lo 5 788483lo 7 secod momet E( R ) = + + = 0.985, 7 63 39 39 360 third momet 3 606 658405376 lo 497559 lo 3 99875lo 5 3596863lo 7 E( R ) = + + = 0.0560, 49 545 7440 5488 40 fourth momet 4 45475 379880384 lo 3074009lo 3 533984375lo 5 985303lo 7 E( R ) = + + = 0.06, 343 5435 400 4406 90 stadard deviatio σ 0.09544, idex of skewess γ ( R) = 0.6867, idex of kurtosis γ ( R) = 0.84. 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 7. 6. 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: 5344 5344 046035303 5764 ( ) = + R 3 4 0070400 + R 00 + R 86700 + R 35 + R 8 8 8 8 458885 046035303 43766455463 54495584 +, for 0 < R <, 5 6 7 5( + R) 8 4688 + R 00 + R 409600 8 8 8 04 9897949 5483 44664793 85739568 475R 3 4 7744400 + R 600 + R 98978 + R 945 + R 8 8 8 8 6604483874 35849660787 43766455463, for < R <, 5 6 7 8 8 86700 + R 90700 + R 409600 8 8 8 ( ) = + R 83543 963463 75845860 694489 75600R 05900 7744400 8640 8 8 8 5875077 56844988749 6604483874 046035303 3 +, for < R <, 3 4 5 6 8 8 3096576 75600 86700 00 8 8 8 8 + 8 83689449 309405 930968877 7560R 40 + R 3096576 + R 3096576 + R 8 8 8 9753639 5875077 85739568 458885 +, for 3 < R < 4, 3 4 5 8 8 8640 + R 3096576 + R 945 + R 4688 + R 8 8 8 8 64
Fiure 4. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 9 from a uiform populatio. 785 783437 57570699 68853 + 3 680R 98978 + R 60480 + R 3096576 + R 8 8 8 930968877 694489 44664793 5764 +, for 4 < R < 5, 3 4 8 8 3096576 + R 8640 + R 98978 + R 35 8 8 8 8 4 90957 959360 57570699 4 3 475 + R 7744400 + R 30400 + R 3096576 + R 8 8 8 8 309405 75845860 5483 046035303 +, for 5 < R < 6, 7560R 3 8 8 7744400 + R 600 + R 86700 8 8 8 8 5 90957 783437 + 5 4 3 86700 + R 75600 + R 7744400 + R 60480 + R 8 8 8 8 83689449 963463 9897949 5344 6 7 +, for < R <, 75600R 8 8 3096576 + R 7744400 + R 00 + R 8 8 8 8 4 785 6 5 4 3 6350400 + R 86700 + R 475 + R 98978 + R 8 8 8 8 8 83543 04 5344 7 +, for < R <. 475R 8 40 + R 05900 + R 0070400 8 8 8 65
Characteristic values of the distributio are: 9 8457 lo 94867 lo 3 9535lo 5 09539 lo 7 mea E( R ) = + + = 0.34589, 35 0 576 5760 secod momet 083 376874 lo 09048769 lo 3 5039065lo 5 007699lo 7 E( R ) = + + = 0.754, 3 05 35840 96 50 third momet 3 43983 696579 lo 394860663lo 3 9476565lo 5 365459lo 7 E( R ) = + + + = 0.04969, 56 84 57344 4576 880 fourth momet 4 7385 5863649 lo 366609453lo 3 3608359375lo 5 73943063lo 7 E( R ) = + = 0.003, 4096 35 45875 58984 474560 stadard deviatio σ 0.08889, idex of skewess γ ( R) = 0.5559, idex of kurtosis γ ( R) = 0.467. 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 8. 7. 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: 785 0000000 34375 50000000 769394535 ( ) = + R 3 4 5 087648 + R 3489 + R 56 + R 4597 + R 73483 + R 9 9 9 9 9 30000000 4503750785 6440000000 + 463035, for 0 < R <, 6 7 8 568( + R) 9 7 5488 3489 9 9 9 59049 60360983 9486677807 98739 889079385 508800R 3 4 6497800 + R 3546400 + R 80 94058496 9 9 9 9 + 48309387 786068377306 + + 46574908078869 6440000000, for < R <, 5 6 7 8 9 9 985400 793800 + R 6497800 + R 3489 + R 9 9 9 9 3768 9897949 335583049 870900R 4875 + R 985400 + R 9 9 + 3495644367439 74686777 44884558986787 3 4 66 + R 68040 3359300 + R 9 9 9 39437806583 868450499 4503750785 3 + +, for < R <, 5 6 7 9 9 985400 + R 358400 + R 5488 + R 9 9 9 66
Fiure 5. Probability desity fuctio of the cocetratio ratio R for radom samples of size = 0 from a uiform populatio. 576480 60556649 75845860 870900R 3359300 + R 985400 + R 9 9 373859538 67366766434 3770730 + + 3 837080 + R 94058496 + R 34000 9 9 9 44884558986787 48309387 30000000 3 4 +, for < R <, 4 5 6 9 9 3359300 + R 985400 + R 7 + R 9 9 9 3 8593507 43055453 ( ) = + R 3 560 + R 94058496 + R 837080 + R 9 9 9 930968877 4904053 67366766434 + 3483648R 98540 R 94058496 + + R 9 9 74686777 889079385 769394535 4 5 + for 3 68040 94058496 4 +,, 9 9 73483 5 < R < + R + R 9 9 9 785 95677 95383538 ( ) = R 4 3 94058496 + R 68040 + R 94058496 + R 9 9 9 7055453 930968877 373859538 + + 3483648R 837080 + R 837080 + R 9 9 3495644367439 98739 50000000 5 6 +, for < R <, 3 4 9 9 94058496 80 + R 4597 + R 9 9 9 67
64 404606309 77963 R + 5 4 3 4875 + R 3546400 + R 48600 + R 9 9 9 ( ) = 95383538 43055453 75845860 870900R 94058496 + R 837080 + R 9 9 335583049 94 + 86677807 + 34375, for 6 < R < 7, 3 9 9 985400 3546400 + R 56 + R 9 9 9 3 8073 404606309 ( ) = R 6 5 4 508800 + R 6497800 + R 3546400 9 9 9 95677 8593507 60556649 + + 3 68040 + R 94058496 + R 985400 + R 9 9 9 9897949 60360983 + 0000000, for 7 < R < 8, 870900R 9 9 6497800 + R 3489 + R 9 9 3 64 ( ) = + R 7 6 5 57900 + R 508800 + R 4875 + R 9 9 9 785 3 576480 4 3 94058496 + R 560 + R 3359300 + R 9 9 9 3768 59049 785 + + 508800R 4875 087648 9 9 8, for < R <. 9 Characteristic values of the distributio are: 6864944 lo 88683579 lo 3 60546875lo 5 40353607 lo 7 mea E( R ) = 5 + = 0.3446, 835 0 6048 80 secod momet 4977475650505 550600944989793lo 3870789470673lo 3 E( R ) = 666598976 5786800 5736040 350944535lo 5 5533880564063lo 7 + = 0.574, 54384 866860 third momet 3 56500 306650669 lo 6093753lo 3 E( R ) = 43 76545 0 99484875lo 5 33859449 lo 7 + = 0.0483, 489888 74960 fourth momet 68
4 658554 039897685 lo 8990777 lo 3 038748885lo 5 E( R ) = 656 4009 560 9840464 89583980753lo 7 + = 0.0933, 78730 stadard deviatio σ ( = 0.0835, idex of skewess γ ( R) = 0.4505, idex of kurtosis γ ( R) = 0.0366. 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 9. 8. 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
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, 93-35. [3] Glasser, G.J. (96) Variace Formulas for the Mea Differece ad the Coefficiet of Cocetratio. Joural of the America Statistical Associatio, 57, 648-654. http://dx.doi.or/0.080/06459.96.0500553 [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, 379-44. [6] Deltas, G. (003) The Small-Sample Bias of the Gii Coefficiet: Results ad Implicatios for Empirical Research. Review of Ecoomics ad Statistics, 85, 6-34. http://dx.doi.or/0.6/rest.003.85..6 [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, -7. http://dx.doi.or/0.98/jbes.009.000 [8] Davidso, R. (009) Reliable Iferece for the Gii Idex. Joural of Ecoometrics, 50, 30-40. http://dx.doi.or/0.06/j.jecoom.008..004 [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, 45-435. http://dx.doi.or/0.06/j.spl.0.04.03 [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, 3-5. 70