The classic Gini index (GI) is generalized recently by using nonlinear weight sequences as sparsity measures for sparse quantification and machine condition monitoring. The generalized GIs with different weight parameters are more robust to random transients. However, they show insufficient performance in discriminating repetitive transients under noise contamination. To overcome this shortage, this paper proposes a two-parameter generalization method to tune not only the weight parameter but also the norm order, allowing for a full generalization of the classic GI to quantify transient features and leading to new statistical indicators which are named fully generalized GIs (FGGIs). Mathematical derivations show that FGGIs satisfy at least four of the six typical attributes of sparsity measures and that those with weight parameter equal to one satisfy at least five sparse attributes, proving that they are a new family of sparsity measures. Numerical simulations demonstrate that FGGIs can monotonically evaluate the sparseness of the signals and that the FGGIs with appropriate parameters exhibit improved performance in resisting random transient interferences and discriminating noise-contaminated repetitive transients compared to traditional sparsity measures. The performance of FGGIs in the condition monitoring of rolling element bearings is validated using two different run-to-failure experiment datasets, including a gradual failure and a sudden failure. The results show that increasing the norm order can improve the capability of FGGIs to characterize transient fault features, allowing more accurate trending of bearing health conditions, and therefore achieving better condition monitoring performance than the traditional sparsity measures.