Decomposition of gini and multivariate gini indices

Abstract

A new type of decomposition by population subgroup is proposed for the Gini inequality index. The decomposition satisfies the completely identical distribution (CID) condition, whereby the between-group inequality is null if and only if the distribution within each subgroup is identical to all the others. Thus, this decomposition contrasts strikingly with the subgroup decomposition of the generalized entropy measures, which satisfy the condition that the between-group inequality is null if the mean within each subgroup equals those of all the others. The new decomposition can be generalized to the distance-Gini index and the volume-Gini index, two multivariate Gini indices introduced by Koshevoy and Mosler, with some modification of the index definition and a somewhat loosened CID condition in the latter case. The source decomposition is also generalized to these multi-dimensional indices. Interaction terms appear among sources of different attributes in the decomposition for the modified volume–Gini index.

Appendix

1.1 Proofs

1.1.1 Proof of Eq. 16

On the assumption that F _i is continuous, by applying integration by parts, cv(F _i ,F) can be expanded as follows:

$$ \begin{aligned} & {\text{cv}}{\left( {F_{i} ,F} \right)} = {\int {{\left( {F_{i} - F} \right)}^{2} {\text{d}}y} } = - 2{\int {y{\left( {F_{i} - F} \right)}{\text{d}}F_{i} } } + 2{\int {y{\left( {F_{i} - F} \right)}{\text{d}}F} }{\text{ }} = 2{\left( {{\int {{\left( {y - \mu _{i} } \right)}F{\text{d}}F_{i} } } - {\int {{\left( {y - \mu _{i} } \right)}F_{i} {\text{d}}F_{i} } }} \right)} + 2\mu _{i} {\left( {{\int {F{\text{d}}F_{i} - \frac{1}{2}} }} \right)} + 2{\left( {{\int {{\left( {y - \mu } \right)}F_{i} {\text{d}}F} } - {\int {{\left( {y - \mu } \right)}F{\text{d}}F} }} \right)} + 2\mu {\left( {{\int {F_{i} {\text{d}}F} } - \frac{1}{2}} \right)}. \\ & \\ \end{aligned} $$

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Note that ∫F _i dF _i =∫FdF=1/2. Then, using the equality ${\int {F{\text{d}}F_{i} - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2 = - {\int {F_{i} {\text{d}}F + 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} }} }$ and the notation in section 2.3, Eq. 16 is obtained.

1.1.2 Proof of Eqs. 24 and 25

By applying the definition of cv_D(F _i ,F) in Eq. 22, the distance-Gini mean difference M _D(F) can be expanded as follows:

$$ M_{{\text{D}}} {\left( F \right)} = \frac{1}{2}{\int {{\int {{\left\| {x - y} \right\|}{\text{d}}F(x){\text{d}}F{\left( y \right)}} }} } = \frac{1}{2}{\sum {p_{i} {\int {{\int {{\left\| {x - y} \right\|}{\text{d}}F_{i} {\left( x \right)}{\text{d}}F{\left( y \right)}} }} }} } = \frac{1}{2}{\sum {p_{i} M_{{\text{D}}} {\left( {F_{i} } \right)}} } + \frac{1}{2}M_{{\text{D}}} {\left( F \right)} + \frac{1}{2}{\sum {p_{i} {\text{cv}}_{{\text{D}}} {\left( {F_{i} ,F} \right)}} }.$$

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Note that F = Σp _i F _i . Subtracting M _D(F) from both sides after doubling, decomposition 24 is obtained. Equation 25 is easily derived by replacing F with $\widetilde{F}$.

1.1.3 Range of the modified Torgersen index

Since

$$ z{\left( {\phi ,\widetilde{F}\left| \gamma \right.} \right)} = {\int {{\left( {\frac{y}{\mu } - \gamma } \right)}\phi {\left( y \right)}{\text{d}}F{\left( y \right)}} } = {\int {\frac{y}{\mu }\phi {\left( y \right)}dF{\left( y \right)}} } - \gamma {\int {\phi {\left( y \right)}dF{\left( y \right)}} } \in {\sum\limits_i {{\left[ {0,e_{i} } \right]}} } + {\left[ {0, - \gamma } \right]},$$

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where ${\sum\limits_i {{\left[ {0,e_{i} } \right]}} } + {\left[ {0, - \gamma } \right]}$ is the Minkowski sum of the line segments [0,e _i ] (i = 1,...,d) and [0,−γ], and e _i  = {e _ij } is a unit vector consisting of elements e _ij  = 0 if j ≠ i, or =1 if j = i, then, the volume of $Z_{{\text{T}}} {\left( {\widetilde{F}\left| \gamma \right.} \right)}$ is less than or equal to the volume of ${\sum\limits_i {{\left[ {0,e_{i} } \right]}} } + {\left[ {0, - \gamma } \right]}$, which is calculated at 1 + Σγ _i by using the following formula ${\text{vol}}{\left( {{\sum\limits_{i = 1}^n {{\left[ {0,a_{i} } \right]}} }} \right)} = {\sum\limits_{1 \leqslant i_{1} < \cdots < i_{d} < n} {{\left| {\det {\left( {a_{{i_{1} }} , \cdots ,a_{{i_{d} }} } \right)}} \right|}} }$ (e.g. [20]). Thus, the modified Torgersen index R _T(F|γ) is less than or equal to unity.

Assume that F is an n-point d-variate distribution in which there is a different monopolist for each variable (attribute). Then, R _T(F|γ) → 1 if n → ∞.

1.1.4 Proof of Eq. 41

By applying Eq. 38, Eq. 58 is derived.

$$ {\text{cv}}_{{\text{T}}} {\left( {F_{i} ,G{\left( \varepsilon \right)}\left| \gamma \right.} \right)} = \frac{{M_{{\text{T}}} {\left( {G{\left( \varepsilon \right)}\left| \gamma \right.} \right)}^{{1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-\nulldelimiterspace} d}} - \varepsilon M_{{\text{T}}} {\left( {F_{i} \left| \gamma \right.} \right)}^{{1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-\nulldelimiterspace} d}} - (1 - \varepsilon )M_{{\text{T}}} {\left( {F\left| \gamma \right.} \right)}^{{1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-\nulldelimiterspace} d}} }}{\varepsilon } - {\left( {1 - \varepsilon } \right)}\frac{{{\text{cv}}_{{\text{T}}} {\left( {F,G{\left( \varepsilon \right)}\left| \gamma \right.} \right)}}}{\varepsilon },$$

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where $G{\left( \varepsilon \right)} = \varepsilon F_{i} + {\left( {1 - \varepsilon } \right)}F$. The modified Torgersen mean difference M _T(G(ε)|γ) and the mixed volume MV _d−1(F,G(ε)|γ) have the following derivatives at ε = 0:

$$ \frac{d}{{d\varepsilon }}\frac{{MV_{{d - 1}} {\left( {F,G{\left( \varepsilon \right)}\left| \gamma \right.} \right)}}}{{1 + {\sum {\gamma _{i} } }}} = \frac{{MV_{{d - 1}} {\left( {F_{i} ,F\left| \gamma \right.} \right)}}}{{1 + {\sum {\gamma _{i} } }}} - M_{{\text{T}}} {\left( {F\left| \gamma \right.} \right)} = M_{{\text{T}}} {\left( {F\left| \gamma \right.} \right)}^{{{{\left( {d - 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {d - 1} \right)}} d}} \right. \kern-\nulldelimiterspace} d}} \frac{d}{{d\varepsilon }}M_{{\text{T}}} {\left( {G(\varepsilon )\left| \gamma \right.} \right)}^{{1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-\nulldelimiterspace} d}} .$$

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The derivative of ${\text{cv}}_{{\text{T}}} {\left( {F,G{\left( \varepsilon \right)}\left| \gamma \right.} \right)}$ at ε = 0 equals zero from its definition and Eq. 59. Thus, the second term on the right-hand side of Eq. 58 → 0 if ε → 0. Since the left-hand side → ${\text{cv}}_{{\text{T}}} {\left( {F_{i} ,F\left| \gamma \right.} \right)}$ if ε→0, Eq. 41 is derived.

1.1.5 Proof of the equality condition in Eq. 45 and the condition for null between-inequality in Eq. 47

It is trivial that ${\text{cv}}_{{\text{V}}} {\left( {G,F\left| \gamma \right.} \right)} = 0{\text{ }}if{\text{ }}G^{s} = F^{s}$ for any coordinate axis, and $Z_{{\text{T}}} {\left( {G\left| \gamma \right.} \right)} = Z_{{\text{T}}} {\left( {F\left| \gamma \right.} \right)}$If ${\text{cv}}_{{\text{V}}} {\left( {G,F\left| \gamma \right.} \right)} = 0$, all terms constituting cv_V(G,F|γ) in Eq. 45 should be zero, including cv_T(G,F|γ). On the assumption that M_T(F|γ)>0, this means that $Z_{{\text{T}}} {\left( {G\left| \gamma \right.} \right)} = \alpha Z_{{\text{T}}} {\left( {F\left| \gamma \right.} \right)}$, where α is a positive constant, according to Minkowski’s first inequality concerning the mixed volume. Since cv(G ^s, F ^s) should also be zero at the same time, G ^s=F ^s for any coordinate axis. This forces α to be unity. Thus, the equality condition in Eq. 45 is proved.

If M _T(F|γ)>0, the equality condition in Eq. 45 can be applied to the proof of the condition for the null between-inequality in Eq. 47. If M _T(F|γ)=0, the Brunn-Minkowski inequality (e.g. [17]) asserts that Z _T(F _i |γ) is on the same hyperplane as Z _T(F|γ). Let k ₁,...,k _t be the highest-dimensional sub-coordinate axes under the condition that $M_{{\text{T}}} {\left( {F^{{j_{1} \cdots j_{s} }} \left| \gamma \right.} \right)} > 0$. The equality condition in Eq. 45 asserts that $Z_{{\text{T}}} {\left( {F^{{k_{1} \cdots k_{t} }}_{i} \left| \gamma \right.} \right)} = Z_{{\text{T}}} {\left( {F^{{k_{1} \cdots k_{t} }} \left| \gamma \right.} \right)}$ (Note that ${\text{cv}}_{{\text{V}}} {\left( {F^{{k_{1} \cdots k_{t} }}_{i} ,F^{{k_{1} \cdots k_{t} }} \left| \gamma \right.} \right)} = 0$ if ${\text{cv}}_{{\text{V}}} {\left( {F_{i} ,F\left| \gamma \right.} \right)} = 0$). From this, Z_T(F _i |γ) should be identical to Z _T(F|γ). G ^s = F ^s is true for any coordinate axis, irrespective of the value of M _T(F|γ). Thus, the condition for null between-inequality in Eq. 47 is proved.

1.1.6 Proof of Eqs. 20 and 50

First, the proof of Eq. 50 is given inductively. Without loss of generality, we can assume that i = 1 and x={x ₁,x ₂,0,...,0} because of invariance for rotation on axis i and axis permutation. If d = 2, Eq. 50 can be proved as follows:

$$ {\rm I}_{2} {\left( x \right)} = {\int_{S^{1} } {\operatorname{sgn} {\left( {a \cdot x} \right)}a_{1} d\upsilon {\left( a \right)}} } = {\int_{ - \pi }^\pi {\operatorname{sgn} {\left( {a \cdot x} \right)}\cos \theta d\theta } } = {\int_{{\varphi - \pi } \mathord{\left/ {\vphantom {{\varphi - \pi } 2}} \right. \kern-\nulldelimiterspace} 2}^{{\varphi + \pi } \mathord{\left/ {\vphantom {{\varphi + \pi } 2}} \right. \kern-\nulldelimiterspace} 2} {\cos \theta d\theta } } - {\int_{{\varphi + \pi } \mathord{\left/ {\vphantom {{\varphi + \pi } 2}} \right. \kern-\nulldelimiterspace} 2}^{{\varphi + 3\pi } \mathord{\left/ {\vphantom {{\varphi + 3\pi } 2}} \right. \kern-\nulldelimiterspace} 2} {\cos \theta d\theta } } = 4\cos \varphi = \frac{1} {{C_{2} }}\frac{{x_{1} }} {{{\left\| x \right\|}}}. $$

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where a={cosθ,sinθ}, and ${\bf x} = {\left\{ {x_{1} ,x_{2} } \right\}} = {\left\{ {{\left\| \bf x \right\|}\cos \varphi ,{\text{ }}{\left\| \bf x \right\|}\sin \varphi } \right\}}$. Assume that Eq. 50 is proved if the dimension is d−1. Then,

$$ {\rm I}_{d} {\left( x \right)} = {\int_{S^{{d - 1}} } {\operatorname{sgn} {\left( {a \cdot x} \right)}a_{1} d\upsilon {\left( a \right)}} } = {\rm I}_{{d - 1}} {\left( x \right)}{\int_{ - 1}^1 {{\sqrt {1 - a^{2}_{d} } }^{{d - 2}} da_{d} } } = \frac{1} {{C_{d} }}\frac{{x_{1} }} {{{\left\| x \right\|}}}. $$

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Equation 50 is derived by summing Eq. 50 over i after multiplying by x _i .