Measuring Discrete Risks on Infinite Domains: Theoretical Foundations, Conditional Five Number Summaries, and Data Analyses

Abstract

To accommodate numerous practical scenarios, in this paper we extend statistical inference for smoothed quantile estimators from finite domains to infinite domains. We accomplish the task with the help of a newly designed truncation methodology for discrete loss distributions with infinite domains. A simulation study illustrates the methodology in the case of several distributions, such as Poisson, negative binomial, and their zero inflated versions, which are commonly used in insurance industry to model claim frequencies. Additionally, we propose a very flexible bootstrap-based approach for the use in practice. Using automobile accident data and their modifications, we compute what we have termed the conditional five number summary (C5NS) for the tail risk and construct confidence intervals for each of the five quantiles making up C5NS, and then calculate the tail probabilities. The results show that the smoothed quantile approach classifies the tail riskiness of portfolios not only more accurately but also produces lower coefficients of variation in the estimation of tail probabilities than those obtained using the linear interpolation approach.

Keywords: Bootstrap, Claim Counts, Smoothed Quantiles, Value-at-Risk, Truncated Distributions