{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T01:46:39Z","timestamp":1778636799733,"version":"3.51.4"},"reference-count":66,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,13]],"date-time":"2020-05-13T00:00:00Z","timestamp":1589328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u201csingle\u201d state-of-the-art models in terms of their accuracy and the stability.<\/jats:p>","DOI":"10.3390\/e22050545","type":"journal-article","created":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T10:27:19Z","timestamp":1589452039000},"page":"545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Weighted Quantile Regression Forests for Bimodal Distribution Modeling: A Loss Given Default Case"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3606-1182","authenticated-orcid":false,"given":"Micha\u0142","family":"Gostkowski","sequence":"first","affiliation":[{"name":"Department of Econometrics and Statistics, Institute of Economics and Finance, Warsaw University of Life Sciences-SGGW, 02-776 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6953-8907","authenticated-orcid":false,"given":"Krzysztof","family":"Gajowniczek","sequence":"additional","affiliation":[{"name":"Department of Artificial Intelligence, Institute of Information Technology, Warsaw University of Life Sciences-SGGW, 02-776 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,13]]},"reference":[{"key":"ref_1","unstructured":"Basel Committee on Banking Supervision (2005). An Explanatory Note on the Basel Iiirb Risk Weight Functions, Basel Committee on Banking Supervision."},{"key":"ref_2","unstructured":"Basel Committee on Banking Supervision (2011). Basel III Counterparty Credit Risk Frequently Asked Questions, Basel Committee on Banking Supervision."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.jbankfin.2017.03.001","article-title":"Downturn LGD modeling using quantile regression","volume":"79","year":"2017","journal-title":"J. Bank. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2842","DOI":"10.1016\/j.jbankfin.2011.03.011","article-title":"Comparison of modeling methods for Loss Given Default","volume":"35","author":"Qi","year":"2011","journal-title":"J. Bank. Financ."},{"key":"ref_5","unstructured":"Gupton, G.M., and Stein, R.M. (2005). LossCalc v2: Dynamic prediction of LGD. Moodys KMV Invest. Serv., Available online: http:\/\/www.defaultrisk.com\/_pdf6j4\/LCv2_DynamicPredictionOfLGD_fixed.pdf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1080\/14697688.2015.1059952","article-title":"Predicting recovery rates using logistic quantile regression with bounded outcomes","volume":"16","author":"Siao","year":"2015","journal-title":"Quant. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gajowniczek, K., Grzegorczyk, I., Z\u0105bkowski, T., and Bajaj, C. (2020). Weighted Random Forests to Improve Arrhythmia Classification. Electronics, 9.","DOI":"10.3390\/electronics9010099"},{"key":"ref_8","first-page":"983","article-title":"Quantile regression forests","volume":"7","author":"Meinshausen","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"23","DOI":"10.21314\/JRMV.2015.139","article-title":"Loss given default modeling: An application to data from a Polish bank","volume":"9","author":"Gostkowski","year":"2015","journal-title":"J. Risk Model Valid."},{"key":"ref_10","first-page":"108","article-title":"Application of mixed models and families of classifiers to estimation of financial risk parameters","volume":"16","author":"Grzybowska","year":"2015","journal-title":"Quant. Methods Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.econmod.2014.10.006","article-title":"Modeling loss given default with stochastic collateral","volume":"44","author":"Frontczak","year":"2015","journal-title":"Econ. Model."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hamerle, A., Knapp, M., and Wildenauer, N. (2006). Modelling Loss Given Default: A \u201cPoint in Time\u201d-Approach. Basel II Risk Parameters, 127\u2013142.","DOI":"10.1007\/3-540-33087-9_7"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Schuermann, T. (2004). What do We Know about Loss Given Default?. SSRN Electron. J.","DOI":"10.2139\/ssrn.525702"},{"key":"ref_14","unstructured":"Calabrese, R. (2012). Estimating bank loans loss given default by generalized additive models. UCD Geary Institute Discussion Paper Series, University College Dublin. WP2012\/24."},{"key":"ref_15","unstructured":"Chalupka, R., and Kopecsni, J. (2008). Modelling bank loan LGD of corporate and SME segments: A case study (No. 27\/2008). IES Working Paper, Charles University."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"25","DOI":"10.21314\/JRMV.2013.101","article-title":"Loss given default modeling: A comparative analysis","volume":"7","author":"Yashkir","year":"2013","journal-title":"J. Risk Model Valid."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1016\/j.jbankfin.2005.05.005","article-title":"Bank loan losses-given-default: A case study","volume":"30","author":"Dermine","year":"2006","journal-title":"J. Bank. Financ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"45","DOI":"10.21314\/JCR.2011.150","article-title":"Generalized beta regression models for random loss-given-default","volume":"7","author":"Huang","year":"2011","journal-title":"J. Credit Risk"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.ijforecast.2010.08.005","article-title":"Loss given default models incorporating macroeconomic variables for credit cards","volume":"28","author":"Bellotti","year":"2012","journal-title":"Int. J. Forecast."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.ejor.2018.01.020","article-title":"Loss functions for Loss Given Default model comparison","volume":"268","author":"Hurlin","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2510","DOI":"10.1016\/j.jbankfin.2010.04.011","article-title":"Forecasting bank loans loss-given-default","volume":"34","author":"Bastos","year":"2010","journal-title":"J. Bank. Financ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.jbankfin.2013.12.006","article-title":"Loss given default for leasing: Parametric and nonparametric estimations","volume":"40","author":"Miller","year":"2014","journal-title":"J. Bank. Financ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.ejor.2014.06.043","article-title":"Support vector regression for loss given default modelling","volume":"240","author":"Yao","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1023\/A:1017934522171","article-title":"Random forests","volume":"45","author":"Breiman","year":"2001","journal-title":"Mach. Learn."},{"key":"ref_25","unstructured":"Ho, T.K. (1995, January 14\u201316). Random decision forests. Proceedings of the 3rd International Conference on Document Analysis and Recognition, Montreal, QC, Canada."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Tanaka, K., Kinkyo, T., and Hamori, S. (2018). Financial Hazard Map: Financial Vulnerability Predicted by a Random Forests Classification Model. Sustainability, 10.","DOI":"10.3390\/su10051530"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Nafkha, R., Gajowniczek, K., and Z\u0105bkowski, T. (2018). Do Customers Choose Proper Tariff? Empirical Analysis Based on Polish Data Using Unsupervised Techniques. Energies, 11.","DOI":"10.3390\/en11030514"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Z\u0105bkowski, T., Gajowniczek, K., and Szupiluk, R. (2015, January 24\u201326). Grade analysis for energy usage patterns segmentation based on smart meter data. Proceedings of the 2015 IEEE 2nd International Conference on Cybernetics (CYBCONF), Gdynia, Poland.","DOI":"10.1109\/CYBConf.2015.7175938"},{"key":"ref_29","unstructured":"Sorzano, C.O.S., Vargas, J., and Montano, A.P. (2014). A survey of dimensionality reduction techniques. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Yazgana, P., and Kusakci, A.O. (2016). A Literature Survey on Association Rule Mining Algorithms. Southeast Eur. J. Soft Comput., 5.","DOI":"10.21533\/scjournal.v5i1.102"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"223","DOI":"10.3233\/IFS-151748","article-title":"Short term electricity forecasting based on user behavior from individual smart meter data","volume":"30","author":"Gajowniczek","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10522-017-9683-y","article-title":"A review of supervised machine learning applied to ageing research","volume":"18","author":"Fabris","year":"2017","journal-title":"Biogerontology"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gajowniczek, K., Nafkha, R., and Z\u0105bkowski, T. (2017, January 3\u20136). Electricity peak demand classification with artificial neural networks. Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, Prague, Czech Republic.","DOI":"10.15439\/2017F168"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Bakir, G., Hofmann, T., Sch\u00f6lkopf, B., Smola, A.J., Taskar, B., and Vishwanathan, S.V.N. (2007). Predicting Structured Data, MIT Press.","DOI":"10.7551\/mitpress\/7443.001.0001"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gajowniczek, K., Z\u0105bkowski, T., and Sodenkamp, M. (2018). Revealing Household Characteristics from Electricity Meter Data with Grade Analysis and Machine Learning Algorithms. Appl. Sci., 8.","DOI":"10.3390\/app8091654"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Tripoliti, E.E., Fotiadis, D.I., and Manis, G. (2010, January 2\u20135). Dynamic construction of Random Forests: Evaluation using biomedical engineering problems. Proceedings of the 10th IEEE International Conference on Information Technology and Applications in Biomedicine, Corfu, Greece.","DOI":"10.1109\/ITAB.2010.5687796"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.econlet.2016.09.024","article-title":"Random forests-based early warning system for bank failures","volume":"148","author":"Tanaka","year":"2016","journal-title":"Econ. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF00058655","article-title":"Bagging predictors","volume":"24","author":"Breiman","year":"1996","journal-title":"Mach. Learn."},{"key":"ref_39","unstructured":"Freund, Y., and Schapire, R.E. (1996, January 3\u20136). Experiments with a new boosting algorithm. Proceedings of the ICML\u201996 Proceedings of the Thirteenth International Conference on International Conference on Machine Learning, Bari, Italy."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.jfs.2019.100693","article-title":"Does machine learning help us predict banking crises?","volume":"45","author":"Beutel","year":"2019","journal-title":"J. Financ. Stab."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"251","DOI":"10.23956\/ijarcsse\/V7I1\/01113","article-title":"Random forest: A review","volume":"7","author":"Goel","year":"2017","journal-title":"Int. J. Adv. Res. Comput. Sci. Softw. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/978-3-030-04648-4_29","article-title":"Refined Weighted Random Forest and Its Application to Credit Card Fraud Detection","volume":"11280","author":"Xuan","year":"2018","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10115-012-0586-6","article-title":"A weighted voting framework for classifiers ensembles","volume":"38","author":"Kuncheva","year":"2012","journal-title":"Knowl. Inf. Syst."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Pham, H., and Olafsson, S. (2019). On Cesaro averages for weighted trees in the random forest. J. Classif., 1\u201314.","DOI":"10.1007\/s00357-019-09322-8"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Byeon, H., Cha, S., and Lim, K. (2019). Exploring Factors Associated with Voucher Program for Speech Language Therapy for the Preschoolers of Parents with Communication Disorder using Weighted Random Forests. Int. J. Adv. Comput. Sci. Appl., 10.","DOI":"10.14569\/IJACSA.2019.0100503"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3651","DOI":"10.1016\/j.eswa.2013.12.009","article-title":"Automated trading with performance weighted random forests and seasonality","volume":"41","author":"Booth","year":"2014","journal-title":"Expert Syst. Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.knosys.2019.04.015","article-title":"A weighted random survival forest","volume":"177","author":"Utkin","year":"2019","journal-title":"Knowl. Based Syst."},{"key":"ref_48","first-page":"23","article-title":"Estimation of a Unimodal Density","volume":"31","author":"Rao","year":"1969","journal-title":"Sankhy\u0101 Indian J. Stat."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"G\u00f3mez, Y.M., G\u00f3mez-D\u00e9niz, E., Venegas, O., Gallardo, D.I., and G\u00f3mez, H.W. (2019). An Asymmetric Bimodal Distribution with Application to Quantile Regression. Symmetry, 11.","DOI":"10.3390\/sym11070899"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Rindskopf, D., and Shiyko, M. (2010). Measures of Dispersion, Skewness and Kurtosis. Int. Encycl. Educ., 267\u2013273.","DOI":"10.1016\/B978-0-08-044894-7.01344-0"},{"key":"ref_51","unstructured":"Chatterjee, S., Handcock, M.S., and Simonoff, J.S. (1995). A Casebook for a First Course in Statistics and Data Analysis, Wiley."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"85","DOI":"10.22237\/jmasm\/1083370200","article-title":"Beta-Normal Distribution: Bimodality Properties and Application","volume":"3","author":"Famoye","year":"2004","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"094303","DOI":"10.1063\/1.2710292","article-title":"InAs\/InP quantum dots with bimodal size distribution: Two evolution pathways","volume":"101","author":"Bansal","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_54","first-page":"379","article-title":"A bimodal exponential power distribution","volume":"26","author":"Hassan","year":"2010","journal-title":"Pak. J. Statist"},{"key":"ref_55","first-page":"59","article-title":"The modes of a mixture of two normal distributions","volume":"6","author":"Sitek","year":"2016","journal-title":"Sil. J. Pure Appl. Math."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jmva.2008.04.010","article-title":"Maximum likelihood estimation for multivariate skew normal mixture models","volume":"100","author":"Lin","year":"2009","journal-title":"J. Multivar Anal."},{"key":"ref_57","unstructured":"Borkowski, B., Dudek, H., and Szczesny, W. (2003). Wybrane Zagadnienia Ekonometrii, Wydawnictwo Naukowe PWN."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Breiman, L. (2017). Classification and Regression Trees, Routledge.","DOI":"10.1201\/9781315139470"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Friedman, J., Hastie, T., and Tibshirani, R. (2001). The Elements of Statistical Learning. Springer Series in Statistics, Springer.","DOI":"10.1007\/978-0-387-21606-5"},{"key":"ref_60","unstructured":"Mehrabi, N., Morstatter, F., Saxena, N., Lerman, K., and Galstyan, A. (2019). A survey on bias and fairness in machinelearning. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Roszkowska, E. (2013). Rank ordering criteria weighting methods\u2014A comparative overview. Optimum. Studia Ekon., 5.","DOI":"10.15290\/ose.2013.05.65.02"},{"key":"ref_62","unstructured":"(2019, July 29). R: A Language and Environment for Statistical Computing. Available online: https:\/\/www.gbif.org\/tool\/81287\/r-a-language-and-environment-for-statistical-computing."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.econlet.2015.12.022","article-title":"Bank overdraft pricing and myopic consumers","volume":"139","author":"Williams","year":"2016","journal-title":"Econ. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1093\/bioinformatics\/btr597","article-title":"MissForest--non-parametric missing value imputation for mixed-type data","volume":"28","author":"Stekhoven","year":"2011","journal-title":"Bioinformatics"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Chavent, M., Genuer, R., and Saracco, J. (2019). Combining clustering of variables and feature selection using random forests. Commun. Stat. Simul. Comput., 1\u201320.","DOI":"10.1080\/03610918.2018.1563145"},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Hinloopen, J., and van Marrewijk, C. (2005). Comparing Distributions: The Harmonic Mass Index. SSRN Electron. J.","DOI":"10.2139\/ssrn.873831"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/545\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:28:33Z","timestamp":1760174913000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,13]]},"references-count":66,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["e22050545"],"URL":"https:\/\/doi.org\/10.3390\/e22050545","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,13]]}}}