Tag: Hoover index

Hoover index is one of the simplest inequality indices to measure the deviation from an ideal equal distribution. It can be interpreted as the maximum vertical deviation of the Lorenz curve from the 45 degree line.

Theil index is an inequality measure related to the Shannon entropy. It is often used to measure economic  inequality.

Like the Shannon entropy, Theil index can be decomposed in two independent components, for example to descbribe inequality “within” and “in between” subgroups. Low Theil or Hoover index means low inequality, high values stand for a high deviation from an equal distribution.

With
E_i – Effect in group i, i = 1 to N
E_t – Total sum of effects in all N groups
A_i – Number of items in class i
A_t – Total number of items in all N groups

Theil Index:

Eq. 1a     T_T = ln (A_t/E_t) – ∑[ E_i/E_t ln (A_i/E_i)]
Eq. 1b      T_L = ln (A_t/E_t) – ∑[ A_i/A_t ln (E_i/A_i)]

Taking relative (proportional) variables
p_i = E_i/E_t
w_i = A_i/A_t we get

Eq. 2a      T_T = ∑[ p_i ln (p_i/w_i)]
Eq. 2b      T_L = ∑[ w_i ln (w_i/p_i)]

The symmetric Theil index T_s = ½ ( T_T + T_L) can be expressed as:

Eq. 3      T_s = ½ ∑[ (p_i –w_i) ln (p_i/w_i)]

Comparing the symmetric Theil index with the

Hoover index

Eq. 4      Hv = ½ ∑ |p_i – w_i|

we see that for the symmetric Theil index the difference (pi – w_i) is weighted with the logarithm of p_i/w_i.

The normalized Theil index ranges from 0 to 1:

Eq. 5     T_norm = 1 – e^–T

How does the Theil index relate to Shannon entropy?

For w_i = 1/N (same number of items in all groups) we get with Shannon entropy
H = – ∑ p_i ln p_i and true diversity D = exp (H):

Eq. 6a      T_T = ln (N) – H
Eq. 6b      T_Tnorm = 1 – D/N

and with
MLD = (1/N) ∑ ln (1/p_i)
(MLD = mean logarithmic deviation)

Eq. 7      T_L = MLD – ln (N)

For the symmetric Theil index:

Eq. 8     T_s = ½ (MLD – H)

The symmetric Theil index is simply half of the difference between mean log deviation and Shannon entropy.

Decomposition

The Theil index can be decomposed to find “within group” (w) and “between group” (b) components:

Eq. 9      T = T_w + T_b

For j subgroups (j = 1 to K) with individual Theil index T_j

Eq. 10a   T_T = ∑ s_j T_Tj +  ∑ s_j ln (s_j/w_j)
Eq. 10b   T_L = ∑ w_j T_Lj + ∑ w_j ln (w_j/s_j)

s_j is the share of E in group j (E_j/E_tot); w_j the relative number of items in subgroup j (N_j/N_tot). The first term in (10) gives the “within group” component, the second the “between group” component.