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ZAtkinson = 1-ZMacRae[1] ≥ 1-exp(Σi=1..N(Ei*ln(Ai/Ei))/Etotal)*Etotal/Atotal |
nosniktA inequality |
ZnosniktA ≥ 1-exp(Σi=1..N(Ai*ln(Ei/Ai))/Atotal)*Atotal/Etotal |
Theil redundancy |
RTheil |
= -ln(1-ZAtkinson) = -ln(ZMacRae) |
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≥ ln(Atotal/Etotal) - Σi=1..N(Ei*ln(Ai/Ei))/Etotal |
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liehT redundancy |
RliehT |
= -ln(1-ZnosniktA) |
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≥ ln(Etotal/Atotal) - Σi=1..N(Ai*ln(Ei/Ai))/Atotal |
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Symmetric redundancy |
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Rsym |
= -ln(1-Zsym) = 2*ZPlato*arctanh(ZPlato) |
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= (RTheil(E|A)+RTheil(A|E))/2 = (RTheil+RliehT)/2 |
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≥ Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal))/2 |
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Symmetric inequality |
Zsym |
= 1-exp(-Rsym) = 1-√((1-ZAtkinson)*(1-ZnosniktA)) |
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≥ 1-exp(Σi=1..N(ln(Ai/Ei)*(Ei/Etotal-Ai/Atotal))/2) |
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Hoover inequality |
ZHoover ≥ Σi=1..N|Ei/Etotal-Ai/Atotal|/2 |
Coulter inequality |
ZCoulter ≥ √(Σi=1..N(Ei/Etotal-Ai/Atotal)2/2) |
Gini inequality |
sort data: Ei/Ai>Ei-1/Ai-1
ZGini ≥ 1-Σi=1..N((2*Σk=1..i(Ek)-Ei)*Ai)/(Etotal*Atotal)
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EU inequality |
1:a = (1-ZGini)/(1+ZGini) is the SOEP "equality parameter"
therefore: ZEurope = 2*ZGini/(1+ZGini)
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Plato inequality |
ZPlato
≈ 1 - arcsin((1-Zsym)(0.06*Zsym+0.61))*2/π
(error < 0.002 for Zsym < 0.75)
Zsym
= 1-((1-ZPlato)/(1+ZPlato))ZPlato,
Rsym
= 2*ZPlato*arctanh(ZPlato)
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Redistributive Aggression |
RA |
= (RTheil+RliehT)/2 - ZHoover = Rsym - ZHoover |
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= Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal) - |Ei/Etotal-Ai/Atotal|)/2
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