On the decomposition by subpopulations of the point and synthetic Zenga (2007) inequality indexes

Abstract

The Radaelli (Stat Appl V I(2):117–136, 2008) decomposition by k subpopulations of the Zenga (Stat Appl V(1):3–27, 2007) point index is based on the decomposition of the point uniformity measure. In this work we decompose, in a more direct way, the point inequality in a weighted mean of $k \times k$ relative differences between the upper mean of the subpopulation g and the corresponding lower mean of the subpopulation l; the weights are the product of their relative frequencies. From this decomposition, we obtain the decomposition of the point index into the within and the between components. The within component is given by the sum of k terms and the between component is the sum of $k \times (k-1)$ components. The decompositions proposed in this paper are applied to the net disposable income of the 8151 Italian households partitioned in three macroregions, supplied by the 2012 Bank of Italy sample survey on household income and wealth. This application shows that the values of the “relative frequencies” help in the interpretation of the $3 \times 3$ contributions.

Appendices

Appendix 1: Principal properties of the synthetic measure $I\left( Y\right)$ based on $I_{h}\left( Y\right)$

Zenga [40] showed that $I\left( Y\right)$ fulfills the following five properties.

• Property 1. In the case of absence of inequality $I\left( Y\right) =0.$

• Property 2. In the case of maximum inequality $I\left( Y\right) =1-\frac{1}{N^{2}}.$

• Property 3. If $X=a\cdot Y$   $(a>0)$, then $I\left( X\right) =I\left( Y\right) .$

• Property 4. If $X= b+Y$ $(b>0)$, then $I(X)<I(Y)$.

• Property 5. Let 0$\le y_{1}<y_{2}< \cdots<y_{i}<y_{i+1}<\cdots <y_{N}$. If a transfer of $c>0$ takes places from $y_{i+1}$ to $y_{i}$ $(i=1,\ldots ,N-1)$, subject to the restriction $c<\frac{1}{2}\left( y_{i+1}-y_{i}\right) ,$ then the value of $I\left( Y\right)$ decreases.

Moreover, [29, 30] showed that $I\left( Y\right)$ satisfies the population replication principle too.

Property 6. (Invariance to the population replication). Let $\alpha \ge 2$ be an integer. Let us replicate $\alpha$ times each value of Y observed on the N units of the initial distribution; the replicated distribution is furnished by $\left\{ \left( y_{h},\alpha \cdot n_{h.}\right) :h=1,\ldots ,r\right\} .$Then the r point measures and the synthetic index of the replicated distribution are equal to the corresponding values of the initial distribution.

Maffenini and Polisicchio [21] have compared in detail the effects of translations and of egalitarian transfers on the Zenga $I_{h}\left( Y\right)$ point measure and on the Lorenz curve. For simplicity, Maffenini and Polisicchio assume that $0\le y_{1}<y_{2}<\cdots<y_{i}<y_{i+1}<\cdots <y_{N}$.

Thus, in the case of the translation $X=b+Y,\;(b>0)$, the ordinates of the Lorenz curve are given by

$$\begin{aligned} L_{p_{i}}\left( X\right) =\frac{Q_{i}\left( Y\right) +i\cdot b}{\left[ M\left( Y\right) +b\right] \cdot N}=L_{p_{i}}\left( Y\right) \cdot \frac{M\left( Y\right) }{M\left( Y\right) +b}+p_{i}\cdot \frac{b}{M\left( Y\right) +b}, \end{aligned}$$

where $p_{i}=\frac{i}{N}.$ Consequently,

$$\begin{aligned} \left\{ L_{p_{i}}\left( X\right) -L_{p_{i}}\left( Y\right) \right\} =\frac{b}{M\left( Y\right) +b}\cdot \left[ p_{i}-L_{p_{i}}\left( Y\right) \right] \end{aligned}$$

is first an increasing function and subsequently a decreasing function of $p_{i}$.

After the translation, the ordinates of the Zenga [40] curve are given by

$$\begin{aligned} I_{p_{i}}\left( X\right) =\frac{\overset{+}{M}_{p_{i}}\left( Y\right) -\bar{M}_{p_{i}}\left( Y\right) }{b+\overset{+}{M}_{p_{i}}\left( Y\right) }=I_{p_{i}}\left( Y\right) \cdot \frac{\overset{+}{M}_{p_{i}}\left( Y\right) }{b+\overset{+}{M}_{p_{i}}\left( Y\right) }. \end{aligned}$$

The ratio $\frac{\overset{+}{M}_{p_{i}}\left( Y\right) }{b+\overset{+}{M}_{p_{i}}\left( Y\right) }$ assumes values in $\left( 0;1\right)$ and is an increasing function of $p_{i}$. In conclusion, the influence of the translation on $I_{p_{i}}\left( \cdot \right)$ curve decreases as $p_{i}$ increases.

The N values taken on by Y, after an egalitarian transfer from y$_{i+1}$ to y$_{i}$, are denoted by $X^{\left( i\right) }$, $i=1,\ldots ,N-1$. The N cumulative incomes $Q_{t}\left( X^{\left( i\right) }\right)$,$t=1,\ldots ,N$, of $X^{\left( i\right) }$ are given by

$$\begin{aligned} Q_{t}\left( X^{\left( i\right) }\right) ={\left\{ \begin{array}{ll} Q_{i}\left( Y\right) +c,\quad \,\text{ for } \text{ t } = \text{ i }\\ Q_{t}\left( Y\right) ,\quad \,\,\text{ for } \text{ t }\ne i \end{array}\right. }. \end{aligned}$$

Consequently,

$$\begin{aligned} \left\{ L_{p_{t}}\left( X^{\left( i\right) }\right) -L_{p_{i}}\left( Y\right) \right\} ={\left\{ \begin{array}{ll} \frac{c}{T},\quad \,\text{ for } \text{ t }= \text{ i }\\ 0\quad \text{ for } \text{ t } \ne \text{ i } \end{array}\right. }. \end{aligned}$$

Moreover, the N ordinates $I_{p_{t}}\left( X^{\left( i\right) }\right)$, for $1\le \text{ i }\le N-2$, are given by

$$\begin{aligned} I_{p_{t}}\left( X^{\left( i\right) }\right) = {\left\{ \begin{array}{ll} I_{p_{t}}\left( Y\right) , &{} \quad \text{ for }\,\text{ t }\ne \mathrm{i}\\ 1-\frac{N-i}{i}\cdot &{} \frac{Q_{i}\left( Y\right) +c}{T-\left( Q_{i}\left( Y\right) +c\right) },\quad \,\text{ for }\,\text{ t }= \text{ i } \end{array}\right. }, \end{aligned}$$

and for $i=N-1,$ are given by

$$\begin{aligned} I_{p_{t}}\left( X^{\left( N-1\right) }\right) ={\left\{ \begin{array}{ll} I_{p_{t}}\left( Y\right) , &{} \text{ for }\,\, 1\le \text{ t }\le \text{ N }-2\\ 1-\frac{1}{N-1}\cdot \frac{Q_{N-1}\left( Y\right) +c}{y_{N}-c}, &{} \text{ for }\,\,\,\text{ t }=\text{ N }-1\\ 1-\frac{M\left( Y\right) }{y_{N}-c}, &{} \text{ for }\,\, \,t=N \end{array}\right. }. \end{aligned}$$

Thus, for $1\le i\le N-2$

$$\begin{aligned} \left\{ I_{p_{t}}\left( Y\right) -I_{p_{t}}\left( X^{\left( i\right) }\right) \right\} ={\left\{ \begin{array}{ll} 0, &{} \text{ for }\,\,\,\text{ t }\ne \mathrm{i}\\ \frac{c}{T}\cdot \frac{N-i}{i}\cdot \frac{1}{\left( 1-L_{p_{i}}\left( Y\right) \right) \cdot \left( 1-L_{p_{i}}\left( Y\right) -\frac{c}{T}\right) }, &{} \text{ for }\,\,\,\text{ t }=i \end{array}\right. }, \end{aligned}$$

and for $i=N-1\,$

$$\begin{aligned} \left\{ I_{p_{t}}\left( Y\right) -I_{p_{t}}\left( X^{\left( i\right) }\right) \right\}= & {} {\left\{ \begin{array}{ll} 0,\;\;\qquad \quad \quad \quad \qquad \quad \quad \quad \text{ for }\,\, 1 {\le \text{ t }\le \text{ N }-2}\\ \frac{1}{N-1}\cdot \frac{T\cdot c}{\left( y_{N}\right) \cdot \left( y_{N}-c\right) }, &{} \text{ for }\,t=N-1.\\ \frac{T}{N}\cdot c/\left[ \left( y_{N}-c\right) \cdot y_{N}\right] &{} \text{ for } \,\,\text{ t }=\text{ N } \end{array}\right. } \end{aligned}$$

The following table reports the differences $\{ L_{p_{t}}\left( X^{\left( i\right) }\right) -L_{p_{t}}\left( Y\right) \}$ and $\{ I_{p_{t}}\left( Y\right) -I_{p_{t}}\left( X^{\left( i\right) }\right) \}$ for: $c=1$, $N=10$ and M(Y) $=$ 30.

i	$y_{i}$	$Q_{i}\left( Y\right)$	$\frac{N-i}{i}$	$I_{p_{i}}\left( Y\right)$	$I_{p_{t}}\left( Y\right) -I_{p_{t}}\left( X^{\left( i\right) }\right)$	$L_{p_{t}}\left( X^{\left( i\right) }\right) -L_{p_{t}}\left( Y\right)$
1	6	6	9	0.8163	0.0313	0.0033
2	9	15	4	0.7701	0.0148	0.0033
3	17	32	2.333	0.7210	0.0098	0.0033
4	22	54	1.5	0.6707	0.0075	0.0033
5	26	80	1	0.6364	0.0062	0.0033
6	29	109	0.646	0.6140	0.0055	0.0033
7	34	143	0.428	0.6096	0.0053	0.0033
8	37	180	0.250	0.6250	0.0053	0.0033
9	41	221	0.111	0.6892	0.0054	0.0033
10	79	300	0.000	0.6202
Tot	300			$I\left( Y\right) =0.677$

For more details on this point see [21].

An important application of the inequality curves is that they can be used to define some orderings. Such orderings allow the comparison of distributions in terms of inequality. Below we report, in the case of continuous random variates, the definitions of the orderings based on the Lorenz and Zenga [40] curves.

(Partial order based on the Lorenz curve). Let X and Y be continuous non-negative random variables, with finite and positive expected values. Let $L_{p}\left( X\right)$ and $L_{p}\left( Y\right)$ denote their Lorenz curves. X is said to be more unequal than Y in the order based on the Lorenz curve (and is denoted by $X\ge _{L}Y$ ), if

$$\begin{aligned} L_{p}\left( X\right) \le L_{p}\left( Y\right) \qquad \forall p\in \left( 0,1\right) . \end{aligned}$$

The partial order based on the Zenga [40] curve was introduced in [25].

(Partial order based on the Zenga [40] curve). Let X and Y be continous non-negative random variables, with finite and positive expected values. Let $I_{p}\left( X\right)$ and $I_{p}\left( Y\right)$ denote their Zenga [40] inequality curves. X is said to be more unequal than Y in the order based on the Zenga [40] curve (and is denoted by $X\ge _{I}Y$ ), if

$$\begin{aligned} I_{p}\left( X\right) \ge I_{p}\left( Y\right) \qquad \forall p\in \left( 0,1\right) . \end{aligned}$$

The relationship between these two orderings is summarized in the following result (see [25]).

(Lemma of equivalence). Let X and Y be continuous non-negative random variables, with finite and positive expected values. Then:

$$\begin{aligned} X\ge _{L}Y \Leftrightarrow X\ge _{I}Y. \end{aligned}$$

The Lemma of equivalence has been extended to the partial order based on the Bonferroni [5] curve in [2].

Appendix 2: Brief comparison with other inequality index decomposition proposals

An inequality measure I is additively decomposable if : $I=\sum _{1}^{k}\, w_{g}I_{g}+I_{B},$ where $I_{g}$ is the index of subpopulation g, $I_{B}$ is the between part given by the value of the index evaluated on the distribution,

$\{ (M_{g}\left( Y\right) ;n_{.g});\, g=1,...,k\}$, $\sum _{1}^{k}\, w_{g}I_{g}$ is the within part and $w_{g}\ge 0$ is the weight attached to subpopulation g. By using the assumption that: (1) $I\left( y_{1},...,y_{N}\right) \ge 0,$ with equality if and only if $y_{1}=y_{2}=...=y_{N}$; 2) $I\left( y_{1},...,y_{N}\right)$ is a continuous function of $(y_{1},...,y_{N});$3) $I\left( y_{1},...,y_{N}\right)$ has continuous first and second derivatives; 4) $I\left( y_{1},...,y_{N}\right)$ fulfills the principle of transfers; 5) $I\left( y_{1},...,y_{N}\right)$ is scale invariant; 6) $I\left( y_{1},...,y_{N}\right)$ is invariant to the population replication; [7] and [33] have shown that there exist (and it is unique) a one parameter $\left( \alpha \right)$ family of inequality measures $\left\{ I^{\left( \alpha \right) }\right\}$ that is additively decomposable. The family is:

$$\begin{aligned} I^{\left( \alpha \right) }=\frac{1}{\alpha \left( \alpha -1\right) }\left[ \frac{1}{N}\sum _{i=1}^{N}\left( \frac{y_{i}}{M\left( Y\right) }\right) ^{\alpha }-1\right] ,\quad \left( \alpha \ne 0,\alpha \ne 1\right) \end{aligned}$$

$$\begin{aligned} I^{\left( 0\right) }=-\frac{1}{N}\sum _{i=1}^{N}log\frac{y_{i}}{M\left( Y\right) } \end{aligned}$$

$$\begin{aligned} I^{\left( 1\right) }=\frac{1}{N}\sum _{i=1}^{N}\frac{y_{i}}{M\left( Y\right) }log\frac{y_{i}}{M\left( Y\right) } \end{aligned}$$

where $I^{\left( 0\right) }$ and $I^{\left( 1\right) }$ are the Theil’s measures. It is important to remark that these indexes are not normalized, and as [33] observed their normalization is achieved at the cost of giving up the property of population replication. For more detail on this points see [37].

The Dagum [8, 9] decomposition of the Gini concentration index is essentially based on the following decomposition of Gini’s mean difference with replacement:

$$\begin{aligned} \Delta \left( Y\right) =\frac{1}{N^{2}}\sum _{l=1}^{k}\sum _{g=1}^{k}\sum _{h=1}^{r}\sum _{j=1}^{r}|y_{h}-y_{j}|\cdot n_{hl}\cdot n_{jg}=\sum _{l=1}^{k}\sum _{g=1}^{k}\Delta _{lg}\left( Y\right) \frac{n_{.l}}{N}\cdot \frac{n_{.g}}{N}, \end{aligned}$$

$$\begin{aligned} \text{ where },\,\Delta _{lg}\left( Y\right) =\frac{\sum _{h=1}^{r}\sum _{j=1}^{r}|y_{h}-y_{j}|\cdot n_{hl}\cdot n_{jg}}{n_{.l}\cdot n_{.g}} \end{aligned}$$

is the Gini’s mean difference between the distribution

$\left\{ \left( y_{h},n_{hl}\right) :h=1,\ldots ,r\right\}$ of the subpopulation l and the distribution $\left\{ \left( y_{h},n_{hg}\right) :h=1,\right. \left. \ldots ,r\right\}$of the subpopulation g. Finally, from the above representation of $\Delta \left( Y\right)$ we obtain:

$$\begin{aligned} \Delta \left( Y\right) =\Delta _{W}\left( Y\right) +\Delta _{B}\left( Y\right) ,\,\text{ where } \end{aligned}$$

$$\begin{aligned} \Delta _{W}\left( Y\right) =\sum _{l=1}^{k}\Delta _{ll}\left( Y\right) \left( \frac{n_{.l}}{N}\right) ^{2}\,\text{ and }\,{\Delta _{B}\left( Y\right) =\sum _{l=1}^{k}\sum _{g\ne l}\Delta _{lg}\left( Y\right) \frac{n_{.l}}{N}\cdot \frac{n_{.g}}{N}} \end{aligned}$$

are the within and the between part of the Gini mean difference with replacement.Then, from $G\left( Y\right) =\frac{\Delta \left( Y\right) }{2M\left( \left( Y\right) \right) }$ we obtains:

$$\begin{aligned} G\left( Y\right) =G_{W}\left( Y\right) +G_{B}\left( Y\right) ,\,\text{ where } \end{aligned}$$

$$\begin{aligned} G_{W}\left( Y\right) =\sum _{l=1}^{k}\frac{\Delta _{ll}\left( Y\right) }{2M\left( Y\right) }\left( \frac{n_{.l}}{N}\right) ^{2}\,\text{ and }\, G_{B}\left( Y\right) =\sum _{l=1}^{k}\sum _{g \ne l}\frac{\Delta _{lg}\left( Y\right) }{2M\left( Y\right) }\frac{n_{.l}}{N}\cdot \frac{n_{.g}}{N} \end{aligned}$$

are respectively the within and the between part of the Gini concentration ratio.

We illustrate now, the decomposition (13):

$$\begin{aligned} I_{h}\left( Y\right) =\sum _{l=1}^{k}\sum _{g=1}^{k}\left[ \frac{\overset{+}{M}_{hg}\left( Y\right) -\bar{M}_{hl}\left( Y\right) }{\overset{+}{M}_{h.}\left( Y\right) }\right] p\left( l|h\right) \cdot a\left( g|h\right) =I_{hW}\left( Y\right) +I_{hB}\left( Y\right) , \end{aligned}$$

where

$$\begin{aligned} I_{hW}\left( Y\right) =\sum _{l=1}^{k}\left[ \frac{\overset{+}{M}_{hl}\left( Y\right) -\bar{M}_{hl}\left( Y\right) }{\overset{+}{M}_{h.}\left( Y\right) }\right] p\left( l|h\right) \cdot a\left( l|h\right) \,\text{ and } \end{aligned}$$

$$\begin{aligned} I_{hB}\left( Y\right) =\sum _{l=1}^{k}\sum _{g\ne 1}\left[ \frac{\overset{+}{M}_{hg}\left( Y\right) -\bar{M}_{hl}\left( Y\right) }{\overset{+}{M}_{h.}\left( Y\right) }\right] p\left( l|h\right) \cdot a\left( g|h\right) \end{aligned}$$

are respectively the within and the between part of the point index $I_{h}\left( Y\right) .$

The decompositions in a within and a between parts of $G\left( Y\right) ,$ and $I_{h}\left( Y\right)$ are such that in the within part are compared incomes of the same subpopulation, while in the between part are compared only incomes of units of different subpopulations. It is important to point out that the decomposition of $I\left( Y\right)$ are obtained putting the decompositions of $I_{h}\left( Y\right)$ into the expression $I\left( Y\right) =\sum _{h=1}^{r}\, I_{h}\left( Y\right) \cdot \left( n_{h.}/N\right) .$

Ebert [11] proposed an “alternative” to the decomposition method proposed by [7] and [33]. The Ebert proposal decomposes the inequality index into the usual within-group and a new simple between-group term.In particular, the representation of the inequality between two subpopulations is obtained comparing the income of each unit in the first population with the income of each units in the second population. The between-group term is then formed by the sum of all these inequality values. The overall inequality is equal to the sum of the within-group term and the between-group term.

Using the notation of the present paper the Ebert proposal may be illustrated as follows. $N^{2}$ can be interpreted as the number of samples with size 2 (with replacement: w.r.) from the population composed of N distinct unit; $n_{.g}^{2}$ is the number of samples of size 2 (w.r.) from the subpopulation g;  $n_{.l}\cdot n_{.g}$ is the number of samples of size 2 with one unit from the subpopulation l and one unit from subpopulation g; $n_{h.}^{2}$is the number of samples of size 2 (w.r.) from the group composed of the $n_{h.}$unit with $Y=y_{h};$ in the same way can be interpreted: $n_{s.}\cdot n_{t.},$ $n_{sl}\cdot n_{tg},$ etc. Thus,

$$\begin{aligned} N^{2}=N\cdot N=(n_{1.}+ \cdots +n_{s.}+\cdots +n_{r.})\cdot (n_{1.}+\cdots +n_{t.}+\cdots +n_{r.}) \end{aligned}$$

$$\begin{aligned} =\sum _{s=1}^{r}\sum _{t=1}^{r}n_{s.}\cdot n_{t.}=\sum _{s=1}^{r}\sum _{t=1}^{r}\left[ \sum _{l=1}^{k}n_{sl}\sum _{g=1}^{k}n_{tg}\right] =\sum _{s=1}^{r}\sum _{t=1}^{r}\left[ \sum _{l=1}^{k}\sum _{g=1}^{k}n_{sl}\cdot n_{tg}\right] \end{aligned}$$

$$\begin{aligned} =\sum _{s=1}^{r}\sum _{t=1}^{r}\left[ \sum _{l=1}^{k}n_{sl}\cdot n_{tl}+\sum _{l=1}^{k}\sum _{g\ne l}^{k}n_{sl}\cdot n_{tg}\right] \end{aligned}$$

$$\begin{aligned} =\sum _{l=1}^{k}\sum _{s=1}^{r}\sum _{t=1}^{r}n_{sl}\cdot n_{tl}+\sum _{l=1}^{k}\sum _{g\ne l}^{k}\sum _{s=1}^{r}\sum _{t=1}^{r}n_{sl}\cdot n_{tg}=\sum _{l=1}^{k}n_{.l}^{2}+\sum _{l=1}^{k}\sum _{g\ne l}^{k}n_{.l}\cdot n_{.g}. \end{aligned}$$

Let D be a bounded non negative function, defined on all the $r\times r$ couples $\left\{ (y_{s},y_{t}):s=1,\right. \left. ...,r;t=1,...,r\right\}$, such that $D(y_{s},y_{t})=0$ if and only if $y_{s}=y_{t}$.

The value $D(y_{s},y_{t})$ represents the “inequality” between the income $y_{s}$ and the income $y_{t}$. Thus:

$$\begin{aligned} \sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot n_{s.}\cdot n_{t.}= & {} \sum _{l=1}^{k}\sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot n_{sl}\cdot n_{tl}\\&+\sum _{l=1}^{k}\sum _{g\ne l}^{k}\sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot n_{sl}\cdot n_{tg}\cdot \end{aligned}$$

Let $M_{..}\left( D\right)$, $M_{ll}\left( D\right)$ and $M_{lg}\left( D\right)$ be the following arithmetic means:

$$\begin{aligned} M_{..}\left( D\right) =\sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot \left( \frac{n_{s.}}{N}\right) \cdot \left( \frac{n_{t.}}{N}\right) , \end{aligned}$$

$$\begin{aligned} M_{ll}\left( D\right) =\sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot \left( \frac{n_{sl}}{n_{.l}}\right) \cdot \left( \frac{n_{tl}}{n_{.l}}\right) , \end{aligned}$$

$$\begin{aligned} M_{lg}\left( D\right) =\sum _{s=1}^{r}\sum _{t=1}^{r}D(y_{s},y_{t})\cdot \left( \frac{n_{sl}}{n_{.l}}\right) \cdot \left( \frac{n_{tg}}{n_{.g}}\right) . \end{aligned}$$

Thus:

$$\begin{aligned} M_{..}\left( D\right) =M_{W}\left( D\right) +M_{B}\left( D\right) ,\,\text{ where } \end{aligned}$$

$$\begin{aligned} M_{W}\left( D\right) =\sum _{l=1}^{k}M_{ll}\left( D\right) \cdot \left( \frac{n_{.l}}{N}\right) ^{2}\,\text{ and }\, M_{B}\left( D\right) =\sum _{l=1}^{k}\sum _{g\ne l}^{k}M_{lg}\left( D\right) \left( \frac{n_{.l}}{N}\right) \cdot \left( \frac{n_{.g}}{N}\right) \end{aligned}$$

are respectively the within and the between parts of the mean of all the $N^{2}$ inequalities. In other words we obtains a “decomposition” very similar to Dagum’s decomposition of the Gini mean difference with replacement. Note that $M_{..}\left( D\right)$ fulfills the property of invariance to the population replication and that for $r=1$, absence of inequality, $M_{..}\left( Y\right) =0.$ Among the functions analysed by Ebert we are interested in those whose values are $D(y_{s},y_{t})=|y_{s}-y_{t}|^{\epsilon },$ $\epsilon >0.$ In this way Ebert obtains the following single parameter family of inequality indexes:

$$\begin{aligned} M_{..}^{\left( e\right) }\left( D\right) =\sum _{s=1}^{r}\sum _{t=1}^{r}|y_{s}-y_{t}|^{\epsilon }\cdot \left( \frac{n_{s.}}{N}\right) \cdot \left( \frac{n_{t.}}{N}\right) . \end{aligned}$$

To compare Ebert’s proposal with the decomposition for $I_{h}\left( Y\right)$ and $I\left( Y\right)$ proposed in the present paper, we set $\epsilon =1.$ In this case $M_{..}^{\left( 1\right) }\left( D\right)$ is the Gini’s mean difference with replacement $\Delta \left( Y\right) .$

By the relations (7) and (8), for ${h=1,\ldots ,r-1}$, the difference

$[\overset{+}{M}_{h.}\left( Y\right) -\bar{M}_{h.}\left( Y\right) ]$ can be represented as follows:

$$\begin{aligned} \overset{+}{M}_{h.}\left( Y\right) -\bar{M}_{h.}\left( Y\right)= & {} \frac{\sum _{t=h+1}^{r}\, y_{t}\cdot n_{t.}}{N-P_{h.}}-\frac{\sum _{s=1}^{h}\, y_{s}\cdot n_{s.}}{P_{h.}}\\= & {} \frac{\left( \sum _{t=h+1}^{r}\, y_{t}\cdot n_{t.}\right) \cdot P_{h.}-\sum _{s=1}^{h}\, y_{s}\cdot n_{s.}\cdot \left( N-P_{h.}\right) }{\left( N-P_{h.}\right) \cdot P_{h.}}. \end{aligned}$$

Then, by the relations $P_{h.}=\sum _{s=1}^{h}\, n_{s.}$ and $\left( N-P_{h.}\right) =\sum _{t=h+1}^{r}\, n_{t.}$ we have

$$\begin{aligned}{}[\overset{+}{M}_{h.}\left( Y\right) -\bar{M}_{h.}\left( Y\right) ]= & {} \frac{1}{\left( N-P_{h.}\right) \cdot P_{h.}}\left[ \sum _{t=h+1}^{r}\, y_{t}\cdot n_{t.}\sum _{s=1}^{h}\, n_{s.}-\sum _{s=1}^{h}\, y_{s}\cdot n_{s.}\sum _{t=h+1}^{r}\, n_{t.}\right] \\= & {} \left[ \sum _{s=1}^{h}\sum _{t=h+1}^{r}\left( y_{t}-y_{s}\right) \cdot \left( \frac{n_{s.}}{P_{h.}}\right) \cdot \left( \frac{n_{t.}}{N-P_{h.}}\right) \right] =M_{h..}\left( D\right) , \end{aligned}$$

where the values of D are now given by $D\left( y_{s},y_{t}\right)$=$\left( y_{t}-y_{s}\right)$ and the mean $M_{h..}\left( D\right)$ is evaluated on the $h\times \left( r-h\right)$ couples $\left\{ \left( y_{s},y_{t}\right) :s=1,...,h;\, t=h+1,...,r\right\} .$

The decomposition of $[\overset{+}{M}_{h.}\left( Y\right) -\bar{M}_{h.}\left( Y\right) ]=M_{h..}\left( D\right)$ is obtained by the following decomposition of $n_{s.}\cdot n_{t.}:$

$$\begin{aligned} n_{s.}\cdot n_{t.}=\left[ \sum _{l=1}^{k}n_{sl}\sum _{g=1}^{k}n_{tg}\right] =\left[ \sum _{l=1}^{k}\sum _{g=1}^{k}n_{sl}\cdot n_{tg}\right] =\sum _{l=1}^{k}n_{sl}\cdot n_{tl}+\sum _{l=1}^{k}\sum _{g\ne l}^{k}n_{sl}\cdot n_{tg}. \end{aligned}$$

Putting this latter relation in the previous expression for $M_{h..}\left( D\right)$ furnishes:

$$\begin{aligned} M_{h..}\left( D\right) =M_{hW}\left( D\right) +M_{hB}\left( D\right) , \end{aligned}$$

where

$$\begin{aligned} M_{hW}\left( D\right) =\frac{1}{\left( N-P_{h.}\right) \cdot P_{h.}}\sum _{l=1}^{k}\sum _{s=1}^{h}\sum _{t=h+1}^{r}\left( y_{t}-y_{s}\right) n_{sl}\cdot n_{tl}, \end{aligned}$$

$$\begin{aligned} M_{hB}\left( D\right) =\frac{1}{\left( N-P_{h.}\right) \cdot P_{h.}}\sum _{l=1}^{k}\sum _{g\ne l}^{k}\sum _{s=1}^{h}\sum _{t=h+1}^{r}\left( y_{t}-y_{s}\right) n_{sl}\cdot n_{tg} \end{aligned}$$

are respectively the within and the between part of $M_{h..}\left( D\right)$. It is possible to show, by the use of (6), (10), and (11), that

$$\begin{aligned} \sum _{s=1}^{h}\sum _{t=h+1}^{r}\left( y_{t}-y_{s}\right) \frac{n_{sl}}{P_{h.}}\cdot \frac{n_{tg}}{N-P_{h.}}=[\overset{+}{M}_{hg}\left( Y\right) -\bar{M}_{hl}\left( Y\right) ]p\left( l|h\right) \cdot a\left( g|h\right) . \end{aligned}$$

Consequently,

$$\begin{aligned} M_{hW}\left( D\right) =\sum _{l=1}^{k}[\overset{+}{M}_{hl}\left( Y\right) -\bar{M}_{hl}\left( Y\right) ]p\left( l|h\right) \cdot a\left( l|h\right) ,\,\text{ and } \end{aligned}$$

$$\begin{aligned} M_{hB}\left( D\right) =\sum _{l=1}^{k}\sum _{g\ne l}^{k}[\overset{+}{M}_{hg}\left( Y\right) -\bar{M}_{hl}\left( Y\right) ]p\left( l|h\right) \cdot a\left( g|h\right) . \end{aligned}$$

The within part can also be written as:

$$\begin{aligned} M_{hW}\left( D\right) =\sum _{l=1}^{k}\left[ I_{hl}\left( Y\right) \right] \cdot \overset{+}{M}_{hl}\left( Y\right) \cdot p\left( l|h\right) \cdot a\left( l|h\right) ,\,\text{ where } \end{aligned}$$

$$\begin{aligned} I_{hl}\left( Y\right) =\frac{\overset{+}{M}_{hl}\left( Y\right) -\bar{M}_{hl}\left( Y\right) }{\overset{+}{M}_{hl}\left( Y\right) } \end{aligned}$$

is the point index evaluated on the subpopulation l. Finally, dividing both sides of $M_{h..}\left( D\right) =M_{hW}\left( D\right) +M_{hB}\left( D\right)$ for $\overset{+}{M}_{h.}\left( Y\right)$ gives

$$\begin{aligned} M_{h..}\left( \frac{D}{\overset{+}{M}_{h.}\left( Y\right) }\right) =M_{hW}\left( \frac{D}{\overset{+}{M}_{h.}\left( Y\right) }\right) +M_{hB}\left( \frac{D}{\overset{+}{M}_{h.}\left( Y\right) }\right) \end{aligned}$$

or,

$$\begin{aligned} I_{h}\left( Y\right)= & {} \sum _{l=1}^{k}\left[ I_{hl}\left( Y\right) \right] \cdot \frac{\overset{+}{M}_{hl}\left( Y\right) }{\overset{+}{M}_{h.}\left( Y\right) }\cdot p\left( l|h\right) \cdot a\left( l|h\right) \\&+\sum _{l=1}^{k}\sum _{g\ne l}^{k}\left[ \frac{\overset{+}{M}_{hg}\left( Y\right) -\bar{M}_{hl}\left( Y\right) }{\overset{+}{M}_{h.}\left( Y\right) }\right] p\left( l|h\right) \cdot a\left( g|h\right) . \end{aligned}$$

Appendix 3: Numerical illustration of the decomposition by subpopulations of the point $I_{h}\left( Y\right)$ and the synthetic $I\left( Y\right)$ inequality indexes

In this section, for the example introduced in Sect. 2.1, we show the decompositions by subpopulations of $I_{2}(Y)=0.9184$ and of $I(Y)=0.7805.$ See Tables 17, 18, 19 and 20.

From Table 19 we may obtain many important information. For example the contribution of the subpopulation 1 to $I_{2}(Y)$ is $B_{21.}(Y)=0.45918$. According to formula (14), this value can be expressed as the product of:

• the relative variation $\frac{\overset{+}{M}_{2.}(Y)-\bar{M}_{21}(Y)}{\overset{+}{M}_{2.}(Y)}$ =$\frac{12.25-1}{12.25}$=0.918367 and

• the relative frequency $p(1|2)=\frac{P_{21}}{P_{2.}}$=$\frac{2}{4}=0.5.$

Table 17 Relative frequencies: $p\left( l|h\right) =\frac{P_{hl}}{P_{h.}}$ $\forall \left( h\right)$; $a\left( g|h\right) =\frac{n_{.g}-P_{hg}}{N-P_{h.}}$ for $h=1,\ldots ,r-1$, and $a\left( g|h\right) =\frac{n_{rg}}{n_{r.}},$ for $h=r$

Table 18 Calculus of the contributions $B_{2lg}\left( Y\right) =[\frac{\overset{+}{M}_{2g}\left( Y\right) -\bar{M}_{2l}\left( Y\right) }{\overset{+}{M}_{2.}\left( Y\right) }]p\left( l|2\right) \cdot a\left( g|2\right)$

Table 19 Decomposition of $I_{2}\left( Y\right) =0.9184$ into the contributions: $B_{2lg}(Y)$; $B_{2l.}(Y)$ ;$B_{2lW}\left( Y\right) ,B_{2lB}\left( Y\right)$; $B_{2.W}\left( Y\right) , B_{2.B}\left( Y\right)$

Table 20 Decomposition of $I\left( Y\right) =0.7852$ into the contributions:$B_{.lg}(Y)$; $B_{.l.}(Y)$; $B_{.lW}\left( Y\right) , B_{.lB}\left( Y\right)$; $B_{..W}\left( Y\right) , B_{..B}\left( Y\right)$