# #StackBounty: #standard-deviation #auc #gini How to get approximative confidence interval for Gini and AUC?

### Bounty: 50

I found an interesting way to calculate a confidence interval for the Gini and respectively AUC coefficient for credit risk scoring.

Question: Can anyone explain me, why the sum

\$\$
AUC = frac{1}{n cdot m}sum_{i=1}^n sum_{j=1}^m S(X_i,y_j), text{with}
\$\$
\$\$
S(x_i,y_j) = begin{cases} 1 & text{if } x_i > y_j \ 0.5 & text{if } x_i = y_j \ 0 & text{if } x_i < y_j
end{cases}
\$\$
has got the standard deviation
\$\$
SE(AUC)= \$\$ \$\$sqrt{frac{AUC(1-AUC) + (m-1)left(frac{AUC}{2-AUC}- AUC^2right) + (n-1)left(frac{2AUC^2}{1+AUC}-AUC^2right)}{n cdot m}} \$\$
and especially why the AUC is accepted as standard normal distributed?

In the book “Kreditrisikomessung” written by Henking, Bluhm and Fahrmeier (ISBN-10 3-540-32145-4 on page 223) is then a confidence interval given by

\$\$
AUC pm z_{alpha/2} SE(AUC)
\$\$

Get this bounty!!!

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