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Estimation of the Spectral Measure from ConvexCombinations of Regularly Varying RandomVectors

Marco Oesting, Olivier Wintenberger

Abstract

The extremal dependence structure of a regularly varying random vector Xis fully described by its limiting spectral measure. In this paper, we investigate how torecover characteristics of the measure, such as extremal coefficients, from the extremalbehaviour of convex combinations of components of X. Our considerations result in aclass of new estimators of moments of the corresponding combinations for the spectralvector. We show asymptotic normality by means of a functional limit theorem and, focusingon the estimation of extremal coefficients, we verify that the minimal asymptoticvariance can be achieved by a plug-in estimator using subsampling bootstrap. We illustratethe benefits of our approach on simulated and real data.

Estimation of the Spectral Measure from ConvexCombinations of Regularly Varying RandomVectors

Abstract

The extremal dependence structure of a regularly varying random vector Xis fully described by its limiting spectral measure. In this paper, we investigate how torecover characteristics of the measure, such as extremal coefficients, from the extremalbehaviour of convex combinations of components of X. Our considerations result in aclass of new estimators of moments of the corresponding combinations for the spectralvector. We show asymptotic normality by means of a functional limit theorem and, focusingon the estimation of extremal coefficients, we verify that the minimal asymptoticvariance can be achieved by a plug-in estimator using subsampling bootstrap. We illustratethe benefits of our approach on simulated and real data.

Paper Structure

This paper contains 26 sections, 18 theorems, 190 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Extremes of convex combinations of regularly varying random vectors
  3. Asymptotic normality
  4. Asymptotic normality for deterministic thresholds
  5. Asymptotic normality for random thresholds via order statistics
  6. Application: Estimation of extremal coefficients
  7. Known margins
  8. The inefficiency of the benchmark
  9. Variance minimization
  10. Asymptotic normality of the plug-in estimator
  11. Unknown margins
  12. The benchmark and the moment-based estimator
  13. Variance minimization
  14. Asymptotic normality of the plug-in estimator
  15. Numerical illustrations
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\bm X^*$, $Y$ and $\bm \Theta$ be as above. Furthermore, let $g: (0,\infty) \times S_{d-1}^+ \to \mathbb{R}$ be a continuous bounded function. Then, we have

Figures (4)

  • Figure 1: The asymptotic standard deviations $\sqrt{V((v,1-v))}$ (solid lines) and $\sqrt{\widetilde{V}((v,1-v))}$ (dashed line) of the moment-based estimators with known and unknown margins, respectively, for each of the three scenarios reflecting strong, moderate and weak dependence (from left to right).
  • Figure 2: The results of the moment-based estimators, $1 / \widehat{\tau}_{I,n}^{MK}$ (left) and $1 / \widehat{\tau}_{I,n}^{MU}$ (right), as a function of $k$ for ten realizations of the max-stable Hüsler--Reiss model with moderate dependence ($\Gamma_{12}=2$) and $n=5\,000$.
  • Figure 3: The empirical pairwise tail dependence coefficients $\widehat{\chi}_{ij}(u_i,u_j)$, $1 \leq i,j \leq j$ based on the empirical $95\,\%$-quantiles as thresholds. Here stations 1--3 belong to the region in the northwest, stations 4--6 to the region in the south and stations 7--9 to the region in the northeast of France.
  • Figure 4: The estimated extremal coefficients $1/\widehat{\tau}_{I,n}^{MU}$ for the three regions in the northwest, the south and the northeast of France and different choices of $k$.

Theorems & Definitions (35)

  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • Corollary 6
  • Remark 7
  • ...and 25 more