![T = rSym = (Theil_T[(a,b)] + Theil_T[(a,b)]) / 2
H = zHoover = Hoover inequality ("Robin Hood index")
Software to compute theses measures:
in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf
The original source: TheilHooverPlatoGini.hs (Haskell code).
I additionally generated TheilHooverPlatoGini.py and TheilHooverPlatoGini.lua using ChatGPT.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:mxbe4sjpwj7pk547n2w3aiki/bafkreicoratowddgdcam3uhyh3nqxeydoa5nk4owe5tnf4n4xy2e6girse)
T = rSym = (Theil_T[(a,b)] + Theil_T[(a,b)]) / 2 H = zHoover = Hoover inequality ("Robin Hood index") Software to compute theses measures: in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf The original source: TheilHooverPlatoGini.hs (Haskell code). I additionally generated TheilHooverPlatoGini.py and TheilHooverPlatoGini.lua using ChatGPT.
Plato_Inequality_Indicator.hs (Haskell code) is here: in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf You can use e.g. GhatGPT to convert the code to other languages.
Two experimental inequality measures:
- zPlato
- rAccept and zAccept
The acceptance measure surely is disputable.
#ResearchProposal #TheilIndex #TheilRedundancy #HooverInequality #HooverIndex #RobinHoodIndex #Gini #GiniIndex #GiniCoefficient #PlatoInequality #inequality
See also ALT-text
![Graph depicting the "Distribution of Household Wealth in the U.S. by income group"
groups:
[0%,50%), group size 50%
[0%,90%), group size 40%
[90%,99%), group size 39%
[99%,99.9%), group size 0.9%
[99%.9%,100%], group size: 0.1%
time range:
1990-2020 (or later?)
Source: The Federal Reserve
-----------------------------------------------------------
Processed in Power Shell:
X:\Haskell> .\IneqComps.exe "50,2.36|40,37.8|9,41.7|0.9,12.6|0.1,5.51"
IneqComps V16 2025-06-25
rSym = 1.24162 (symmetric Theil redundancy)
zSym = 71.108% (symmetric Atkinson inequality, zSym=1-exp(-rSym))
zPlato = 70.605% (Plato inequality, Theil & Atkinson & Plato equivalent ratio: 85.303:14.697)
zGini = 69.040% (Gini inequality, Gini equivalent ratio: 84.520:15.480)
zHoover = 49.828% (Hoover inequality, Hoover equivalent ratio: 74.914:25.086)
1-Median = 92.121%
zAccept = -0.74334 (experimental: zAccept=zHoover-rSym)
X:\Haskell> .\IneqComps.exe --CSV "50,2.36|40,37.8|9,41.7|0.9,12.6|0.1,5.51"
1.241620979176E+00;711.084487868371E-03;706.052940093059E-03;690.396018805642E-03;498.279483845153E-03;921.205035971223E-03;-743.341495331143E-03;+1.000000000000E+00](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:xfzrs7vpxjepuj6kwyjceoqf/bafkreidocdgmrgdws672ewuoit5vevciodmql7umi4dz7fto6w4gyfspma)
![# Example in Python for how to compute the Plato inequality indicator:
# Goetz Kluge, 2025-06-25
import math
# Compute rSym (symmetrized Theil redundancy) from a table of pairs (a,b).
def compute_theil_redundancy(pairs):
a_total = sum(a for a, _ in pairs)
b_total = sum(b for _, b in pairs)
return 0.5 * sum(math.log(a / b) * ((a / a_total) - (b / b_total)) for a, b in pairs)
def theil2atkinson(redundancy): return 1 - math.exp(-redundancy)
# Compute zSym (symmetrized Atkinson inequality indicator)
def plato2atkinson(a): return 1 - ((1 - a) / (1 + a)) ** a
# zPlato: Convert zSym to zPlato using the Newton-Raphson iteration.def atkinson2plato(z_sym)
def deriv(a): d = (1 - a) / (1 + a)
return -d ** a * (math.log(d) - 2 * a / (1 - a ** 2))
def func(a, b): return plato2atkinson(a) - b
def newton_raphson(a, b, iter, tol):
if iter >= 100 or abs(func(a, b)) < tol: return a, iter
else: return newton_raphson(a - func(a, b) / deriv(a), b, iter + 1, tol)
if z_sym >= 0.99999999999999: return 1, 0
elif z_sym <= 0: return 0, 0
elif z_sym < 0.001: return math.sqrt(z_sym * 0.5), 0
else: initial_guess = 0.18992 + 0.72791 * z_sym
return newton_raphson(initial_guess, z_sym, 0, 1e-12)
# -- Usage example:
def main():
data_points = [(50, 2.5), (40, 47.5), (9, 27), (1, 23)]
r_sym = compute_theil_redundancy(data_points)
z_sym = theil2atkinson(r_sym)
z_plato, steps = atkinson2plato(z_sym)
print(f"rSym: {r_sym}")
print(f"zSym: {z_sym}")
print(f"zPlato: {z_plato}")
print(f"Newton-Raphson iterations: {steps}")
if __name__ == "__main__":
main()
# rSym: 1.161710269336159
# zSym: 0.6870495067042661
# zPlato: 0.6880377297113732
# Newton-Raphson iterations: 3](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:xfzrs7vpxjepuj6kwyjceoqf/bafkreiexmzulpmc6ic35nnp6s4g3v3wokszzjvcofibztxb5etkgjt72bm)
![-- Example in Haskell for how to compute the Plato inequality indicator
-- Goetz Kluge, 2025-06-25
-- Compute rSym (symmetrized Theil redundancy) from a table of pairs (a,b).
type PairsTable = [(Double,Double)]
type RSym = Double
computeTheilRedundancy :: PairsTable -> RSym
computeTheilRedundancy pairs = 0.5 * sum [log (a / b) * ((a / aTotal) - (b / bTotal)) | (a, b) <- pairs]
where
aTotal = sum [a | (a, _) <- pairs]
bTotal = sum [b | (_, b) <- pairs]
-- Compute zSym (symmetrized Atkinson inequality indicator)
theil2Atkinson :: Double -> Double
theil2Atkinson redundancy = 1 - exp (-redundancy)
-- zPlato: Convert zSym to zPlato using the Newton-Raphson iteration.
atkinson2Plato :: Double -> (Double, Int)
atkinson2Plato zSym
| zSym >= 0.99999999999999 = (1,0)
| zSym <= 0 = (0,0)
| zSym < 0.001 = (sqrt (zSym * 0.5), 0)
| otherwise =
let initialGuess = 0.18992 + 0.72791 * zSym
in newtonRaphson initialGuess zSym 0 1e-12
newtonRaphson :: Double -> Double -> Int -> Double -> (Double, Int)
newtonRaphson a b iter tol
| iter >= 100 || abs (func a b) < tol = (a, iter)
| otherwise = newtonRaphson (a - func a b / deriv a) b (iter + 1) tol
deriv :: Double -> Double
deriv a =
let d = (1 - a) / (1 + a)
in -d ** a * (log d - 2 * a / (1 - a ** 2))
func :: Double -> Double -> Double
func a b = plato2atkinson a - b
plato2atkinson :: Double -> Double
plato2atkinson a = 1 - ((1 - a) / (1 + a)) ** a
-- Usage example:
main :: IO ()
main = do
let dataPoints = [(50,2.5),(40,47.5),(9,27),(1,23)]
let rSym = computeTheilRedundancy dataPoints
let zSym = theil2Atkinson rSym
let (zPlato, steps) = atkinson2Plato zSym
putStrLn $ "rSym: " ++ show rSym
putStrLn $ "zSym: " ++ show zSym
putStrLn $ "zPlato: " ++ show zPlato
putStrLn $ "Newton-Raphson iterations: " ++ show steps](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:xfzrs7vpxjepuj6kwyjceoqf/bafkreifgidolrztns2d3qm2fxbluupipocz74zh6hakbhmnuxfvso2jtdi)
![Three examples for applications of IneqComps V07 2025-06-19:
Source data:
- Income after taxation, 1995, Germany (western part)
- Source: section 20.10.4 in "Statistisches Jahrbuch 1999"
- Sequence in Quantile: [taxpayers per quantile, avg.income (DM) per taxpayer]
(1) --------------------------------------------------------------------------
IneqComps.exe " 1145008,2954120640| 1274868,9680072724| 1489169,18586318289|1309984,22809441408| 1227877,27624776746|1333681,36713570568|3136635,110400142095| 3619401,162869425599| 3105688,170061263504| 3252768,217925697696| 3383398,291368085566| 3126897,420417555444|207672,68629365840| 49031,32752266721|13820,18659363220| 5249,15461396661| 1247,8458445892|686,14801378844"
IneqComps V07 2025-06-19
rSym = 0.39068 (symmetric Theil redundancy)
zSym = 32.340% (symmetric Atkinson index, zSym = 1-exp(-rSym))
zHoover = 29.691% (Hoover index)
zPlato = 42.754% (Plato inequality, Theil equivalent ratio: 71.377:28.623)
zGini = 42.162% (Gini index, Gini equivalent ratio: 71.081:28.919)
(2) --------------------------------------------------------------------------
IneqComps2ratios.exe "71.377,28.623|28.623,71.377"
IneqComps V07 2025-06-19
rSym = 0.39067 (symmetric Theil redundancy)
zSym = 32.340% (symmetric Atkinson index, zSym = 1-exp(-rSym))
zHoover = 42.754% (Hoover index)
zPlato = 42.754% (Plato inequality, Theil equivalent ratio: 71.377:28.623)
zGini = 42.754% (Gini index, Gini equivalent ratio: 71.377:28.623)
(3) --------------------------------------------------------------------------
IneqComps.exe "71.081,28.919|28.919,71.081"
IneqComps V07 2025-06-19
rSym = 0.37917 (symmetric Theil redundancy)
zSym = 31.557% (symmetric Atkinson index, zSym = 1-exp(-rSym))
zHoover = 42.162% (Hoover index)
zPlato = 42.162% (Plato inequality, Theil equivalent ratio: 71.081:28.919)
zGini = 42.162% (Gini index, Gini equivalent ratio: 71.081:28.919)](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:xfzrs7vpxjepuj6kwyjceoqf/bafkreiajy7wcuqp4fokzq5sz75mqo73bsnnc4mjntgw2doilnm6unybg3e)