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In the present paper we derived, with direct method, the exact expressions for the sampling probability density function of the Gini concentration ratio for samples from a uniform population of size
*n* = 6, 7, 8, 9 and 10. Moreover, we found some regularities of such distributions valid for any sample size.

In 1914 Corrado Gini [

Various aspects of the Gini index have been taken into account. One of the most interesting topics regards the estimation of the concentration ratio (Hoeffding, 1948 [

Girone (1968) [

In the present note (Section 2), we calculate the joint probability density function (p.d.f.) of the random sample of size n and, then, the joint p.d.f. of the n order statistics. Hence, we transform one of the order statistics in their average and the remaining n ‒ 1 order statistics are divided by the same average. We calculate the joint p.d.f. of the new n variables and integrating with respect to the average we obtain the joint p.d.f. of the other n ‒ 1 variables. One of these variables is transformed in the concentration ratio. We calculate the joint p.d.f. of the concentration ratio and of the other n ‒ 2 variables and at last we integrate this p.d.f. with respect to the n ? 2 variables obtaining the marginal p.d.f. of the concentration ratio. The main difficulty of this procedure consists in the identification of the region of integration of the n ‒ 2 variables, for two reasons: firstly the need to decompose this region into subregions which allow identifying directly the limits of integration and secondly the growing number of such subregions that makes the derivation heavy.

In Sections 3-7, using the software Mathematica, we derive the exact distributions of the concentration ratio for samples from a uniform distribution of size n = 6, 7, 8, 9 and 10. Moreover (Section 8), we find some regularities of such distributions valid for any sample size.

Let random variables

The joint p.d.f. of the variables is

The joint p.d.f. of the order statistics

By transforming the variables

whose Jacobian is

we obtain the joint p.d.f. of the variables S and

We integrate expression [

By transforming the variable

from which we get

the Jacobian of the transformation is

and the joint p.d.f. of the variable R and

for

By integrating expression [

The procedure indicated in Section 2 is used to obtain the following p.d.f. (

Characteristic values of the distribution are:

mean

second moment

third moment

fourth moment

standard deviation

index of skewness

index of kurtosis

The distribution of the concentration ratio R for samples of size n = 6 from a uniform population shows a slight positive skewness and platykurtosis.

The procedure indicated in Section 2 is used to obtain the following p.d.f. (

Characteristic values of the distribution are:

mean

second moment

third moment

fourth moment

standard deviation

index of skewness

index of kurtosis

The distribution of the concentration ratio R for samples of size n = 7 from a uniform population shows slight positive skewness and platykurtosis, both lower than those obtained for samples of size n = 6.

The procedure indicated in Section 2 is used to obtain the following p.d.f. (

Characteristic values of the distribution are:

mean

second moment

third moment

fourth moment

standard deviation

index of skewness

index of kurtosis

The distribution of the concentration ratio R for samples of size

The procedure indicated in Section 2 is used to obtain the following p.d.f. (

Characteristic values of the distribution are:

mean

second moment

third moment

fourth moment

standard deviation

index of skewness

index of kurtosis

The distribution of the concentration ratio R for samples of size n = 9 from a uniform population shows slight positive skewness and platykurtosis, both lower than those obtained for samples of size n = 6, 7 and 8.

The procedure indicated in Section 2 is used to obtain the following p.d.f. (

Characteristic values of the distribution are:

mean

second moment

third moment

fourth moment

standard deviation

index of skewness

index of kurtosis

The distribution of the concentration ratio R for samples of size n = 10 from a uniform population shows slight positive skewness and platykurtosis, both lower than those obtained for samples of size

The analysis of the p.d.f. for

● The p.d.f. of the concentration ratio R, for

● Furthermore, the p.d.f. of the concentration ratio R, for

● The density of the concentration ratio R, for

● The density of the concentration ratio R, for

● The jth term of the density of the concentration ratio R, denoted as

The coefficients of the

These results are valid for every sample size and may allow reducing the heavy calculation to determine the p.d.f. of the concentration ratio R.

In the present paper we obtain the distributions of the Gini concentration ratio R for samples of size

The obtained results show that the p.d.f. of the concentration ratio R is given by hyperbolic splines with degree 2 and with nodes in

Beyond the possibility to obtain similar results for samples of larger size, open problems are the derivation of the exact expression for the mean and the other features of the distribution of the concentration ratio R for random samples of size n drawn from a uniform population.