Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bootstrapping Estimators based on the Block Maxima Method

Axel Bücher, Torben Staud

Abstract

The block maxima method is a standard approach for analyzing the extremal behavior of a potentially multivariate time series. It has recently been found that the classical approach based on disjoint block maxima may be universally improved by considering sliding block maxima instead. However, the asymptotic variance formula for estimators based on sliding block maxima involves an integral over the covariance of a certain family of multivariate extreme value distributions, which makes its estimation, and inference in general, an intricate problem. As an alternative, one may rely on bootstrap approximations: we show that naive block-bootstrap approaches from time series analysis are inconsistent even in i.i.d.\ situations, and provide a consistent alternative based on resampling circular block maxima. As a by-product, we show consistency of the classical resampling bootstrap for disjoint block maxima, and that estimators based on circular block maxima have the same asymptotic variance as their sliding block maxima counterparts. The finite sample properties are illustrated by Monte Carlo experiments, and the methods are demonstrated by a case study of precipitation extremes.

Bootstrapping Estimators based on the Block Maxima Method

Abstract

The block maxima method is a standard approach for analyzing the extremal behavior of a potentially multivariate time series. It has recently been found that the classical approach based on disjoint block maxima may be universally improved by considering sliding block maxima instead. However, the asymptotic variance formula for estimators based on sliding block maxima involves an integral over the covariance of a certain family of multivariate extreme value distributions, which makes its estimation, and inference in general, an intricate problem. As an alternative, one may rely on bootstrap approximations: we show that naive block-bootstrap approaches from time series analysis are inconsistent even in i.i.d.\ situations, and provide a consistent alternative based on resampling circular block maxima. As a by-product, we show consistency of the classical resampling bootstrap for disjoint block maxima, and that estimators based on circular block maxima have the same asymptotic variance as their sliding block maxima counterparts. The finite sample properties are illustrated by Monte Carlo experiments, and the methods are demonstrated by a case study of precipitation extremes.
Paper Structure (26 sections, 17 theorems, 170 equations, 24 figures, 1 algorithm)

This paper contains 26 sections, 17 theorems, 170 equations, 24 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Basic model assumptions
  4. Empirical means of rescaled disjoint or sliding block maxima
  5. The circular block maxima sample
  6. Bootstrapping block maxima estimators
  7. Application: bootstrapping the pseudo-maximum likelihood estimator for the Fréchet distribution
  8. Simulation study
  9. Case study
  10. Specific time series models for three selected applications
  11. Precipitation data from a weather station
  12. Conclusion
  13. Appendix: Integrability and bias conditions
  14. Application: bootstrapping the pseudo-maximum likelihood estimator for the Fréchet distribution
  15. Proofs
  16. ...and 11 more sections

Key Result

Theorem 2.3

Under Conditions cond:mda and cond:ser_dep, for any finite set of real valued functions $h_1, \dots, h_q$ satisfying the integrability Condition cond:int_h(a) with $\nu > 2/\omega$ where $\omega$ is from Condition cond:ser_dep, we have, writing $\bm h=(h_1, \dots, h_q)^\top$, where with $\bm Z\sim G$ from Condition cond:mda and $(\bm Z_{1,\xi}, \bm Z_{2,\xi}) \sim G_\xi$ from eq:Gxi-prod. Moreov

Figures (24)

  • Figure 1: Empirical distribution of the estimation error and the bootstrap estimation error for 100-year return level estimation in Model \ref{['mod:armax-gpd']} with $\gamma = -0.2, \beta = 0.5, m = 80, r = 365$. Left: naive sliding approach. Right: circular bootstrap approach. See Section \ref{['sec:bootgen']} and Figure \ref{['fig:RlHists']} for further details.
  • Figure 2: Illustration of calculating the circular block maxima sample.
  • Figure 3: Return level estimation with fixed block size $r = 365$ and ‘annuity’ $T = 100$. Left: Mean squared error $\mathrm{MSE}({\hat{\mathrm{RL}}}^{[{ {\operatorname{m} }}]})$. Right: relative MSE with respect to the disjoint blocks method, i.e., $\mathrm{MSE}({\hat{\mathrm{RL}}}^{[{ {\operatorname{m} }}]}) / \mathrm{MSE}({\hat{\mathrm{RL}}}^{[{ {\operatorname{d} }}]})$.
  • Figure 4: Comparison of the error distributions from \ref{['eq:err']} for return level estimation with fixed block size $r = 365$ and ‘annuity’ $T = 100$ in Model \ref{['mod:armax-gpd']} with $\gamma = -0.2, \beta = .5, m = 80$.
  • Figure 5: Bootstrap-based estimation of the return level estimation variance $\sigma_{ {\operatorname{m} }}^2 = \operatorname{Var}({\hat{\mathrm{RL}}}^{[{ {\operatorname{m} }}]})$ with fixed block size $r=365$ and 'annuity' $T=100$. Left two columns: target parameter $\sigma_{ {\operatorname{s} }}^2$ (dashed line), with the two colored lines representing the empirical variance of the naive sliding and the circular bootstrap sample, respectively, averaged over $N=1,000$ simulation runs. Right two columns: the same with target parameter $\sigma_{ {\operatorname{d} }}^2$ (dashed line) and the colored line the empirical variance of the disjoint bootstrap sample.
  • ...and 19 more figures

Theorems & Definitions (37)

  • Theorem 2.3
  • Remark 2.4: Normal approximations for statistics depending on observable block maxima
  • Example 2.5
  • Proposition 3.1: Weak convergence of circular block maxima
  • Theorem 3.2
  • Theorem 4.1: Asymptotic validity of the circmax-resampling bootstrap
  • Remark 4.2: Bootstrapping the disjoint block maxima empirical process
  • Remark 4.3: Inconsistency of naive resampling of sliding block maxima
  • Proposition 4.4
  • Remark 9.4
  • ...and 27 more