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

Abstract

To accommodate numerous practical scenarios, in this article 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 the insurance industry to model claim frequencies. Additionally, we propose a very flexible bootstrap-based approach for 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.

ACKNOWLEDGMENTS

The authors are very appreciative of valuable suggestions and useful comments provided by the referees and the co-editor Dr. Patrick Brockett, which helped to significantly improve the article.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research esd supported by the NSERC Alliance–MITACS Accelerate grant titled “New Order of Risk Management: Theory and Applications in the Era of Systemic Risk” from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the national research organization Mathematics of Information Technology and Complex Systems (MITACS) of Canada.