Tail models and the statistical limit of accuracy in risk assessment

摘要

Purpose - This paper aims to evaluate the accuracy of a quantile estimate. Especially when estimating high quantiles from a few data, the quantile estimator itself is a random number with its own distribution. This distribution is first determined and then it is shown how the accuracy of the quantile estimation can be assessed in practice. Design/methodology/approach - The paper considers the situation that the parent distribution of the data is unknown, the tail is modeled with the generalized pareto distribution and the quantile is finally estimated using the fitted tail model. Based on well-known theoretical preliminary studies, the finite sample distribution of the quantile estimator is determined and the accuracy of the estimator is quantified. Findings - In general, the algebraic representation of the finite sample distribution of the quantile estimator was found. With the distribution, all statistical quantities can be determined In particular, the expected value, the variance and the bias of the quantile estimator are calculated to evaluate the accuracy of the estimation process. Scaling laws could be derived and it turns out that with a fat tail and few data, the bias and the variance increase massively. Research limitations/implications - Currently, the research is limited to the form of the tail, which is interesting for the financial sector. Future research might consider problems where the tail has a finite support or the tail is over-fat. Practical implications - The ability to calculate error bands and the bias for the quantile estimator is equally important for financial institutions, as well as regulators and auditors. Originality/value - Understanding the quantile estimator as a random variable and analyzing and evaluating it based on its distribution gives researchers, regulators, auditors and practitioners new opportunities to assess risk.