Bootstrap resamples can be used to investigate the tail of empirical distributions as well as return value estimates from the extremal behaviour of the sample. Specifically, the confidence intervals on return value estimates or bounds on in-sample tail statistics can be obtained using bootstrap techniques. However, non-parametric bootstrapping from the entire sample is expensive. It is shown here that it suffices to bootstrap from a small subset consisting of the highest entries in the sequence to make estimates that are essentially identical to bootstraps from the entire sample. Similarly, bootstrap estimates of confidence intervals of threshold return estimates are found to be well approximated by using a subset consisting of the highest entries. This has practical consequences in fields such as meteorology, oceanography and hydrology where return values are calculated from very large gridded model integrations spanning decades at high temporal resolution or from large ensembles of independent and identically distributed model fields. In such cases the computational savings are substantial.

Bootstrap resamples of time series are commonly used to obtain non-parametric
confidence intervals (CIs) on return values

We will present a simple argument for why it is sufficient to retain only a
small subset

This paper is organized as follows. Section

Consider the sample

The probability of drawing one of the highest

A full bootstrap resample

Denote a short bootstrap resample from the

The number

The length

Figure

The ratio

Condition (2) can be handled by randomly perturbing the size of the
resamples,

Here we present worked examples of how the two conditions
presented above can reduce the problem of estimating CIs on tail statistics
for a data set of independent ensemble forecasts at long lead time (

Histogram of the significant wave height from archived ensemble
forecasts in the central North Sea (Ekofisk, 56.5

Consider as an example the problem of how to calculate in-sample return
estimates from the sample of independent forecasts presented above. These
forecasts can be considered iid (as they are not from correlated time
series). An in-sample return estimate is calculated directly from the tail of
the empirical distribution rather than by applying extreme value analysis. As
explained by

A quantile–quantile comparison of 10 000 bootstrapped direct
100-year return estimates of significant wave height taken from a forecast
ensemble

A quantile–quantile (QQ) comparison of

Condition (2) given above states that the size of the reduced resamples

A similar problem to the estimation of CIs for in-sample return values is how
to obtain the CI for the highest percentiles, e.g. the 99th percentile
(

Mean and standard deviation on 100-year in-sample return estimates
based on

Mean and standard deviation on 100-year in-sample return estimates
with a threshold

Bootstrapping the 99th percentile,

Consider now the problem of estimating CIs for return estimates
from threshold exceedances from a data set of independent forecasts.
This differs from a peaks-over-threshold approach which is how correlated
time series must be handled to estimate return levels

A brute-force approach would be to make

This problem arises when estimating GPD return values from the independent
ensemble forecasts (Fig.

Bootstrapping the upper percentiles

The upper and lower 95 % CIs and the mean 100-year return estimates
(dashed) based on

CIs and other statistics of the extremes and the tail of
empirical distributions are commonly found using non-parametric bootstrap
techniques. Here we have shown that it is unnecessary to bootstrap from the
entire original sample. The actual number

The advantages of restricting resamples to a small subset

We have investigated the conditions that must be met to form a non-parametric
bootstrap for tail statistics such as return levels (which depend on all
three parameters of the generalized extreme value distribution or the GPD)
from a small subset of the highest entries in the original sample. As
mentioned in the Introduction, an important question is whether
non-parametric bootstraps yield CIs with sufficient coverage, i.e. CIs that
are wide enough. This has been extensively studied by

The data sets presented in this study are archived in the
MARS database of the European Centre for Medium-Range Weather Forecasts
(ECMWF); see

The ratio curves presented in Figs

The authors declare that they have no conflict of interest.

This study was carried out with support from the Research Council of Norway through the ExWaCli project (grant no. 226239) and the follow-up, ExWaMar (grant no. 256466). Edited by: N. Pinardi Reviewed by: S. Caires and R. Katz