Bootstrap and empirical likelihood methods in extremes

Abstract

One of the major interests in extreme-value statistics is to infer the tail properties of the distribution functions in the domain of attraction of an extreme-value distribution and to predict rare events. In recent years, much effort in developing new methodologies has been made by many researchers in this area so as to diminish the impact of the bias in the estimation and achieve some asymptotic optimality in inference problems such as estimating the optimal sample fractions and constructing confidence intervals of various quantities. In particular, bootstrap and empirical likelihood methods, which have been widely used in many areas of statistics, have drawn attention. This paper reviews some novel applications of the bootstrap and the empirical likelihood techniques in extreme-value statistics.

Semi-parametric and Parametric Inference of Extreme Value Models for Rainfall Data

“…Equation 1 indicates that for all continuity points x of G, F ∈ D(G) (F is in the domain of attraction of the distribution G) and thus, [G(x)] n = G(a n x + b n ). For a complete review and details, see: Qi (2008), Fisher and Tippett (1928), Galambos (1978) and Zhou (2008).…”

“…For example, the Gumbel distribution is obtained by assuming a n = 1 whereas Fréchet and negative Weibull are derived by taking a n = 1. For derivations, readers are referred to Qi (2008) and Galambos (1978). The behavior of extreme values are commonly described by the above mentioned distributions whose cumulative distribution functions are displayed below:…”

“…In this study, the uncertainty of the extreme value index estimators is investigated using the bootstrap technique (Efron 1979(Efron , 1981(Efron , 1987Davison and Hinkley 1997;Carpenter and Bithell 2000;Shao and Tu 1995;Efron and Tibshirani 1997). This technique is a data-driven method that employs Monte Carlo simulations to draw random sub-samples from the data under consideration in order to generate an empirical estimate for the sampling distribution of a statistic (Qi 2008). The bootstrap has been frequently used in practical applications for determining the accuracy (or uncertainty) of statistical analyses (Dunn 2001;Kharin and Zwiers 2000;Dixon 2002;Khaliq et al 2005;Ferro and Segers 2003;Zwiers 1990;Zhang et al 2005;Wilks 1997;Zwiers and Ross 1991).…”

Efficient bootstrap estimates for tail statistics

“…Bootstrap resamples of time series are commonly used to obtain non-parametric confidence intervals (CIs) on return values (Naess and Clausen, 2001;Naess and Hungnes, 2002) and to investigate the behaviour of the tail of the empirical distribution (Coles, 2001;Beirlant et al, 2006;Qi, 2008). Although non-parametric CIs tend to be too narrow, see Kyselý (2008), the procedure itself is algorithmically and numerically straightforward to implement and is thus a convenient technique for rapidly assessing the width of CIs without having to assume a certain parametric distribution.…”

Bootstrapping endpoint

“…Instead, subsample bootstrap methods have been proposed to deal with extremes; see Swanepoel (1986), Deheuvels et al (1993), Athreya and Fukuchi (1997). More references can be found in Qi (2008), which reviewed some applications of bootstrap methods in analyzing extreme values.…”