Gini Index

The Gini Index has three distinct uses in actuarial statistics:

1. Decision tree node impurity — measures the purity of a node in a classification Decision Tree. A pure node (all one class) has Gini = 0.

$G = \sum_{k=1}^K \hat{p}_{mk}(1 - \hat{p}_{mk})$

$= 1 - \sum_{k=1}^K \hat{p}_{mk}^2$

2. Model predictive accuracy — a ranking-based performance metric for binary classifiers (related to the AUROC):

$\text{Gini} = 2 \cdot \text{AUROC} - 1$

3. Ranking power of a rating model — the Lorenz-curve Gini used to evaluate an insurance GLM. Sort the holdout data by predicted loss cost, lowest first, and plot the cumulative share of exposure ( $x$ ) against the cumulative share of actual losses, $L(x)$ . The Gini index is twice the area between that Lorenz curve and the line of equality:

$\text{Gini} = 1 - 2\int_0^1 L(x)\,dx$

For decision trees:

$\hat{p}_{mk}$ = proportion of class $k$ in node $m$ ; $K$ = number of classes
Gini = 0: perfectly pure node (only one class)
Gini = $1 - 1/K$ : maximally impure (uniform class proportions)
For binary classification ( $K=2$ ): $G = 2\hat{p}(1-\hat{p})$ , maximized at $\hat{p}=0.5$ with $G=0.5$
Gini is preferred over Entropy in practice as it is computationally simpler

For a rating model:

A model that ranks nothing puts losses in proportion to exposure, so $L(x) = x$ and Gini $= 0$ ; the better the model separates low-cost from high-cost risks, the further the curve bows below the diagonal and the higher the Gini
It measures ranking only, not calibration: multiplying every prediction by $1.2$ leaves the sort order, and so the Gini, unchanged. Pair it with a Quantile Plot, which shows both
Compare Gini indices across models on the same holdout data; on training data the more flexible model always wins

Gini Index for a Decision Tree Node A node in a classification tree contains 30 fraud cases and 70 non-fraud cases (100 total). Calculate the Gini Index.

Answer $\hat{p}_{\text{fraud}} = 0.30$ , $\hat{p}_{\text{non-fraud}} = 0.70$ . $G = 1 - (0.30^2 + 0.70^2) = 1 - (0.09 + 0.49) = 1 - 0.58 = 0.42$

Gini as a Model Performance Metric A predictive model has an AUROC of 0.78. What is the Gini coefficient?

Answer $\text{Gini} = 2(0.78) - 1 = 0.56$ A Gini of 0.56 indicates good discriminatory power; a random model would have Gini = 0.

Gini Index from a Lorenz Curve A holdout set is sorted by predicted pure premium and cut into four equal-exposure buckets. From lowest to highest prediction, the buckets hold $10\%$ , $20\%$ , $30\%$ and $40\%$ of actual losses. Estimate the Gini index using the trapezoid rule.
Answer The cumulative loss shares at $x = 0.25, 0.50, 0.75, 1.00$ are $L = 0.10, 0.30, 0.60, 1.00$ .
 $\begin{align*} \int_0^1 L(x)\,dx &\approx 0.25\left[\tfrac{0 + 0.10}{2} + \tfrac{0.10 + 0.30}{2} + \tfrac{0.30 + 0.60}{2} + \tfrac{0.60 + 1.00}{2}\right] \\ &= 0.25\,(0.05 + 0.20 + 0.45 + 0.80) \\ &= 0.375 \\ \text{Gini} &= 1 - 2(0.375) \\ &= 0.25 \end{align*}$ 
A model that ranked nothing would put $25\%$ of losses in each bucket and score $0$ . This one's best quarter of the book carries only $10\%$ of the losses.