Bootstrap Bandwidth Selection and Confidence Regions for Double Smoothed Default Probability Estimation

Abstract

For a fixed time, t, and a horizon time, b, the probability of default (PD) measures the probability that an obligor, that has paid his/her credit until time t, runs into arrears not later that time t+b. This probability is one of the most crucial elements that influences the risk in credits. Previous works have proposed nonparametric estimators for the probability of default derived from Beran’s estimator and a doubly smoothed Beran’s estimator of the conditional survival function for censored data. They have also found asymptotic expressions for the bias and variance of the estimators, but they do not provide any practical way to choose the smoothing parameters involved. In this paper, resampling methods based on bootstrap techniques are proposed to approximate the bandwidths on which Beran and smoothed Beran’s estimators of the PD depend. Bootstrap algorithms for the calculation of confidence regions of the probability of default are also proposed. Extensive simulation studies show the good behavior of the presented algorithms. The bandwidth selector and the confidence region algorithm are applied to a German credit dataset to analyze the probability of default conditional on the credit scoring.