Weighted Quantile Regression Forests for Bimodal Distribution Modeling: A Loss Given Default Case

Due to various regulations (e.g., the Basel III Accord), banks need to keep a specified amount of capital to reduce the impact of their insolvency. This equity can be calculated using, e.g., the Internal Rating Approach, enabling institutions to develop their own statistical models. In this regard, one of the most important parameters is the loss given default, whose correct estimation may lead to a healthier and riskless allocation of the capital. Unfortunately, since the loss given default distribution is a bimodal application of the modeling methods (e.g., ordinary least squares or regression trees), aiming at predicting the mean value is not enough. Bimodality means that a distribution has two modes and has a large proportion of observations with large distances from the middle of the distribution; therefore, to overcome this fact, more advanced methods are required. To this end, to model the entire loss given default distribution, in this article we present the weighted quantile Regression Forest algorithm, which is an ensemble technique. We evaluate our methodology over a dataset collected by one of the biggest Polish banks. Through our research, we show that weighted quantile Regression Forests outperform “single” state-of-the-art models in terms of their accuracy and the stability.

Global and Geographically Weighted Quantile Regression for Modeling the Incident Rate of Children’s Lead Poisoning in Syracuse, NY, USA

Objective: The purpose of this study was to explore the full distribution of children’s lead poisoning and identify “high risk” locations or areas in the neighborhood of the inner city of Syracuse (NY, USA), using quantile regression models. Methods: Global quantile regression (QR) and geographically weighted quantile regression (GWQR) were applied to model the relationships between children’s lead poisoning and three environmental factors at different quantiles (25th, 50th, 75th, and 90th). The response variable was the incident rate of children’s blood lead level ≥ 5 µg/dL in each census block, and the three predictor variables included building year, town taxable values, and soil lead concentration. Results: At each quantile, the regression coefficients of both global QR and GWQR models were (1) negative for both building year and town taxable values, indicating that the incident rate of children lead poisoning reduced with newer buildings and/or higher taxable values of the houses; and (2) positive for the soil lead concentration, implying that higher soil lead concentration around the house may cause higher risks of children’s lead poisoning. Further, these negative or positive relationships between children’s lead poisoning and three environmental factors became stronger for larger quantiles (i.e., higher risks). Conclusions: The GWQR models enabled us to explore the full distribution of children’s lead poisoning and identify “high risk” locations or areas in the neighborhood of the inner city of Syracuse, which would provide useful information to assist the government agencies to make better decisions on where and what the lead hazard treatment should focus on.