The Distribution of an Index of Dissimilarity for Two Samples from a Uniform Population ()
1. Introduction
In 1915 Corrado Gini [1] introduced, as an index of dissimilarity between two groups of observations, the mean of the differences in absolute value between co-ranked observations. Fifty years later, in 1965, Gini [2] published a comprehensive and systematic overview of his own and of other authors results about dissimilarity.
We assume that two groups have equal number of observations n. Let 
and
, 
be the set of the observations in the first and in the second group respectively. Let
, 
and 
be the observations arranged in non-descending order of magnitude, respectively in the first and in the second group. Gini’s simple index of dissimilarity is given by
If the two groups of observations are random samples, the distribution of the Gini index with samples drawn from the same population is of great interest for inferential purposes (e.g. to verify the homogeneity of two populations).
As far as we know, the distribution of the index of dissimilarity between two samples has not been studied in depth yet, maybe for the complex structure of the index which considers observations arranged in order of magnitude, co-ranked pairs and differences in absolute value. Instead the distribution of the index of dissimilarity between a sample and its population was obtained by Forcina and Galmacci [3] for the particular case of a discrete, equispaced and equidistributed population. The mean and variance of the distribution of the index of dissimilarity, both between two samples and between a sample and its population, have been obtained by Herzel [4] and Bertino [5] for some particular cases.
The purpose of this note is to study the sample behavior of the index of dissimilarity in the case of two samples from the same uniform population. In this paper we present the analytical results obtained for the exact distribution of the index of dissimilarity for sample sizes n ≤ 8 as well as for the exact general expressions of the main characteristic values (mean, variance, moments) of such a distribution for any sample size and their limiting values.
2. The Distribution of a Linear Combination of Spacings
Let
, be a random sample from a uniform population expressed by the probability density function (p.d.f.)
(1)
The variables of the sample in nondescending order 
determine 
spacings
denoted by
. These spacings are exchangeable random variables.
Let
be their linear combination. Huffer and Lin [6] showed that the p.d.f. of Z is
(2)
in which 
are the distinct values of the weights ai with frequencies 
while the sign +, appearing at the numerator of the ratio in the brackets, indicates that the function is nonzero only if it is positive. This result will be used for determining the p.d.f. of the index of dissimilarity.
3. The Index of Dissimilarity as a Linear Combination of Spacings
The case of two independent random samples from uniform population with p.d.f. given by expression (1), both of size n, can be brought to the case of one random sample of size 2n from the same uniform population. Let 
be the 2n variables of the pooled sample arranged in nondescending order of magnitude. They determine 
spacings
which we denote by
. Given all the 
subdivisions of the pooled sample into two samples of equal size n, let 
and 
be the ranks in the pooled sample of the variables in the first and in the second sample respectively. Moreover 
and 
, for
. It is easy to verify that the index of dissimilarity is given by
or, when the differences are expressed in terms of spacings
by
clearly D is equal to the linear combination of spacings given by
in which ah is the relative frequency of the n intervals 
that contain the hth spacing. The above procedure for each of the 
subdivisions of the pooled sample should be used to determine the p.d.f. of the index of dissimilarity. To reduce the amount of computation it is appropriate to aggregate the subdivisions which bring to the same coefficients ah. Due to the exchangeability of thespacings it is also appropriate to aggregate all subdivisions which bring to the same set offrequencies. In practice the 
subdivisions are aggregated into groups characterized by the same set of frequencies. Denoting by 
the distinct values of coefficients ah and by 
their frequencies (see Paragraph 2) we can apply formula (2) with the only variant of replacing n with 2n. Taken together the subdivisions which have the same values 
it is possible to proceed in an aggregate way by applying formula (2) only to homogeneous groups of subdivisions. These are in number of 
which allows to reduce considerably the amount of computation. Obviously, regard should be given to frequencies of the homogeneous subdivisions.
4. The Distributions of the Index of Dissimilarity for Sample Size Up to 8
To generate the sampling distribution of the index of dissimilarity two programs have been developed for Mathematica software. The first one generates, for each value of n, the subdivisions of the first 2n natural numbers into two subsets of size n and proceeds to identify homogeneous typologies and to define their frequencies. The second one starts, for each value of n, with such typologies and frequencies and gives, by applying formula (2), the p.d.f. of the index of dissimilarity. It should be said that heaviness of both calculation and resulting expressions led us to stop at
. The sampling densities of the index of dissimilarity (see Paragraph 2) are splines of order 2n − 1 with knots at points i/n for 
They are also unimodal between 0 and 1, extreme values in which they assume value 0. The only exception is the degenerate case 
For 
For 
For 
For 
For 
For 
For 
For 
The sampling distribution of the index of dissimilarity for 
is shown in Figure 1.
5. The Moments of the Sampling Distribution of the Index of Dissimilarity
On thee basis of the sampling distributions of the index of dissimilarity (see Paragraph 4), we obtained the values ofthe mean and of the second, third and fourth moment about the origin reported in Table 1.
Using the moments of the Dirichelet distribution, which is the model of the joint distribution of uniform spacings, as well as the expressions of the moments of a linear combination of equally distributed random variables, we obtained the mean and the second, third and fourth moment of the sampling distribution of the index of dissimilarity:
Figure 1. Sampling distribution of the index of dissimilarity for n = 7.
Based on these moments, we could study the main features of the sampling distribution of the index of dissimilarity starting from the mean. The trend of the mean with increasing sample size is shown in Figure 2.
It can be seen, the mean decreases for increasing n and it goes to zero for
. The trend of the mean multiplied by 
is increasing and converges to 
(Figure 3).
The variance of the sampling distribution of the index of dissimilarity can be expressed as follows
The trend of the standard deviation with increasing sample size is shown in Figure 4.
It can be seen that the standard deviation decreases with increasing n and it goes to zero for
. The trend of the standard deviation multiplied by 
is also
Table 1. Mean and moments of the sampling distribution of the index of dissimilarity for sample size up to 8.
Figure 2. Trend of the mean with increasing sample size.
Figure 3. Trend of the mean multiplied by 
with increasing sample size.
Figure 4. Trend of the standard deviation with increasing sample size.
decreasing and converges to 
(Figure 5).
By converting moments about the origin into central moments it is possible to obtain indices of skewness and excess of kurtosis of the sampling distribution of the index of dissimilarity for a uniform population. Their trends with increasing sample size are shown in Figures 6 and 7.
It can be seen that distributions of the dissimilarity index are positively skewed and the skewness increases for increasing sample sizes up to the limit given by
Figure 5. Trend of the standard deviation multiplied by 
with increasing sample size.
Figure 6. Trend of the skewness index with increasing sample size.
Figure 7. Trend of the excess of kurtosis index with increasing sample size.
It is also clear that such distributions are leptokurtic and this leptokurtosis increases for increasing sample sizes up to the limit given by
Table 2. Thresholds of the sampling distribution of the index of dissimilarity for sample size up to 8.
6. Thresholds of the Sampling Distributions of the Index of Dissimilarity
Thresholds at usual levels 
are required to test statistical hypothesis that two samples come from the same uniform population. Thresholds have been calculated on the exact sampling distributions of the index of dissimilarity (see Paragraph 4) and their values are reported in Table 2. For larger sample sizes can be used the approximate conservative thresholds
7. Conclusions
In this note we obtained the distributions of the index of dissimilarity in the case of two random samples of small size from a uniform population. This result was obtained by expressing the index of dissimilarity as a linear combination of spacings of the pooled sample. The exact expressions, in the form of splines of order 2n − 1, were obtained for samples of size ≤ 8. Although it would be possible to go further we stopped at this size because of the heaviness in processing and in the resulting expressions. This limit could be overcome by using some already set calculus programs. The obtained results allow to achieve the exact expressions of the moments and, therefore, to highlight the main features of the sampling distributions of the index of dissimilarity.
Beyond the problem of dealing with larger sample sizes, open problems are also those considering the case of two samples with different sizes as well as the case of other distribution models for population.
We believe that the present work has re-opened a research strand able to enhance also the inferential statistical aspects about one of the fundamental contributions of Gini.