[Paper Review] Consistency of full-sample bootstrap for estimating high-quantile, tail probability, and tail index

С.Г. Литвинова, Mervyn J. Silvapulle|arXiv (Cornell University)|Apr 27, 2020

Statistical Methods and Inference4 citations

TL;DR

This paper establishes the asymptotic validity of the full-sample nonparametric bootstrap for estimating high-quantiles, tail probabilities, and the extreme value index in univariate heavy-tailed distributions. It demonstrates that the bootstrap provides more reliable coverage rates than asymptotic methods, especially for light and heavy tails, resolving long-standing doubts about its consistency in extreme value inference.

ABSTRACT

We show that the full-sample bootstrap is asymptotically valid for constructing confidence intervals for high-quantiles, tail probabilities, and other tail parameters of a univariate distribution. This resolves the doubts that have been raised about the validity of such bootstrap methods. In our extensive simulation study, the overall performance of the bootstrap method was better than that of the standard asymptotic method, indicating that the bootstrap method is at least as good, if not better than, the asymptotic method for inference. This paper also lays the foundation for developing bootstrap methods for inference about tail events in multivariate statistics; this is particularly important because some of the non-bootstrap methods are complex.

Motivation & Objective

• To resolve longstanding doubts about the validity of full-sample bootstrap for tail parameters in univariate distributions.
• To establish asymptotic consistency of the full-sample bootstrap for high-quantiles, tail probabilities, and extreme value index estimation.
• To evaluate the finite-sample performance of bootstrap versus asymptotic methods through extensive simulations.
• To lay the foundation for extending bootstrap methods to multivariate and time series tail inference.
• To provide a rigorous theoretical justification using tail empirical processes and bootstrap theory.

Proposed method

• The method employs the full-sample nonparametric bootstrap, resampling with replacement from the original i.i.d. sample to estimate sampling distributions of tail statistics.
• Theoretical validity is established via convergence of the bootstrap distribution to the true sampling distribution of tail estimators, relying on tail empirical process theory.
• The bootstrap confidence intervals are constructed using percentile, basic, and t-methods, with performance evaluated across multiple k-values in the Hill estimator framework.
• Theoretical results are derived using techniques from Csörgő and Mason (1989) on bootstrapping tail empirical processes.
• Simulations compare bootstrap and asymptotic confidence intervals under varying tail heaviness, sample sizes, and tail probabilities.
• The method is applied to extreme value index estimation using the Hill estimator, with bootstrap resampling used to approximate the sampling distribution of the estimator.

Experimental results

Research questions

• RQ1 Is the full-sample bootstrap asymptotically valid for constructing confidence intervals for high-quantiles and tail probabilities in heavy-tailed distributions?
• RQ2 How does the bootstrap performance compare to the standard asymptotic method in terms of coverage probability and interval length?
• RQ3 Does the bootstrap method maintain reliable coverage across different tail types (light, heavy, bounded) and sample sizes?
• RQ4 Can the bootstrap be consistently applied to extreme value index estimation via the Hill estimator?
• RQ5 What is the finite-sample behavior of bootstrap confidence intervals for tail parameters when the true quantile exceeds the maximum observed value?

Key findings

• The full-sample bootstrap is asymptotically valid for estimating high-quantiles, tail probabilities, and the extreme value index in univariate distributions.
• In simulations, the bootstrap method achieved coverage rates closer to the nominal level than the asymptotic method across all distributions tested.
• The bootstrap outperformed the asymptotic method in terms of coverage for all distributions except for some Frechet(1) and Frechet(2) cases with less extreme tail probabilities.
• For lighter-tailed distributions (e.g., normal, exponential), the bootstrap showed a clear advantage in coverage reliability.
• Confidence interval lengths were comparable between the two methods, indicating that the bootstrap's improved performance was not due to narrower intervals.
• The results support the use of the bootstrap as a more reliable inference tool than asymptotic methods for tail parameters in practice.