income distribution from N, min,max and mediansalary, and total compensation My goal is to obtain a reasonable approximation of the Gini index of a company (UBS).
I need to obtain an estimate of the salaries distribution from publicly available data:


*

*Nuber of employees=60205 

*total compensation paid=15.182E9 CHF

*min salary=50000 CHF

*median salary = 100000 CHF

*max salary=11430000 CHF


I know it's very underdeterminated, but what's the best that can be obtained from this ?
 A: You can at least obtain upper and lower bounds.
The data you describe effectively impose constraints on the distribution. If $y_i$ is the income of some individual $i \in \{1,\dots,n\}$, $n=60205$, and $m = 30103$, the median employee, the constraints are:


*

*$y_{i} \geq 50 000$ for all $i$.

*$y_m = 100 000$.

*$y_n = 11 430 000$.

*$\sum_{i=1}^{n} y_i = 15.182 E9$

*$y_i \leq y_j$ for all $i<j$.


To obtain lower and upper bounds, you need to find the maximal and minimal value of the Gini index under constraints 1 to 4. You may need to prove it formally, but intuitively I think one can see that the Gini index will be minimized by making the lowest incomes as high as possible under the above constraints. In your case this would give 


*

*$y_i = 100 000$ for all $i \leq m$.

*$y_i = \frac{15.182 E9 - m*100000 - 11 430 000}{30102}$ for all $m<i<n$

*$y_n = 11 430 000$.


(this wage schedule satisfies constraint 5 because the median is lower than the mean so $y_i > y_j$ for $i \leq m$ and $m<j<n$.)
Conversely, the Gini index will be maximized when the lowest incomes are as low as can be (given the constraints). In your case this means setting


*

*$y_i = 50000$ for all $i \leq m$.

*$y_i = 100000$ for all $m\leq i<59264$

*$y_{59264} = 5170000$.

*$y_i = 11 430 000$ for all $59264 \leq i \leq n$.


If I made no computation mistakes, you can use these distribution to compute lower and upper bound for the Gini coefficient of your distribution.
I think anything better than upper and lower bounds would require you to make assumptions on the distribution beyond the data which is available to you. 
A: Another option is to calculate the maximum entropy distribution relative to your constraints. You can look up the method here. You will need to solve a constrained optimization problem using Lagrange Multipliers (or just brute-force it using a numerical package ;-).
BTW: The below method by Marten Van Der Linden is also great for getting bounds. Perhaps compare your results from Max Entropy with his bounds to see what that tells you.
