# 戴尔指数

## 数学公式

${\displaystyle T_{T}=T_{\alpha =1}={\frac {1}{N}}\sum _{i=1}^{N}{\frac {x_{i}}{\mu }}\ln \left({\frac {x_{i}}{\mu }}\right)}$

${\displaystyle T_{L}=T_{\alpha =0}={\frac {1}{N}}\sum _{i=1}^{N}\ln \left({\frac {\mu }{x_{i}}}\right)}$

${\displaystyle T_{T}=\sum _{k=0}^{W}\,f_{k}\,{\frac {k}{\mu }}\ln \left({\frac {k}{\mu }}\right)}$

${\displaystyle \mu =\sum _{k=0}^{W}kf_{k}}$

${\displaystyle T_{T}=\int _{0}^{\infty }f(k){\frac {k}{\mu }}\ln \left({\frac {k}{\mu }}\right)dk}$

${\displaystyle \mu =\int _{0}^{\infty }kf(k)\,dk}$

## 信息論推導

${\displaystyle S=k\sum _{i=1}^{N}\left(p_{i}\log {\frac {1}{p_{i}}}\right)=-k\sum _{i=1}^{N}\left(p_{i}\log {p_{i}}\right)}$

${\displaystyle S_{\text{Theil}}=\sum _{i=1}^{N}\left({\frac {x_{i}}{N{\overline {x}}}}\ln {\frac {N{\overline {x}}}{x_{i}}}\right)}$

${\displaystyle T}$ 为戴尔指数，${\displaystyle S}$ 夏農熵，则有

${\displaystyle T=\ln(N)-S}$

${\displaystyle N}$  字符數 人口數
${\displaystyle i}$  某個特定字符 某個特定人
${\displaystyle x_{i}}$  第i個字符 character 第i個人的收入
${\displaystyle N{\overline {x}}}$  總字符數 總收入
${\displaystyle T_{T}}$  未被使用的資訊空間 未使用潛在價格機制

## 可分解性

${\displaystyle T=\sum _{k=1}^{m}s_{k}T_{T_{k}}+\sum _{k=1}^{m}s_{k}\ln {\frac {{\overline {x}}_{k}}{\overline {x}}}}$

## 參考文獻

1. 徐淑娟. 中国经济发展中的行业收入差距问题研究. 西南財經大學出版社. 2018-02-01: 33–34 [2019-01-14]. ISBN 7550428530 （中文（中国大陆）‎）.
2. ^ Introduction to the Theil index from the University of Texas (PDF). [2006-01-15]. （原始内容存档 (PDF)于2005-11-18）.
3. ^ Diversity and Social Segregation. geodacenter.asu.edu. [2016-03-18]. （原始内容存档于2012-07-10）.
4. ^ Segregation Measures. www.urban.org. Urban Institute. [5 February 2018] （英语）.
5. Parker, Lauren. Racial and Ethnic Segregation: In the News and On PolicyMap. PolicyMap. 20 July 2015 [5 February 2018].
6. ^ Redundancy, Entropy and Inequality Measures. [2019-01-11] （英语）.