Improving the local scoring algorithm using gradient sampling

We adapt the gradient sampling algorithm to the local scoring algorithm to solve complex estimation problems based on an optimization of an objective function. This overcomes non-differentiability and non-smoothness of the objective function. The new algorithm estimates the Clarke generalized subgradient used in the local scoring, thus reducing numerical instabilities. The method is applied to quantile regression and to the peaks-over-threshold method, as two examples. Real applications are provided for a retail store and temperature data analysis.