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Asymptotic theory for extreme value generalized additive models

Takuma Yoshida

Abstract

The classical approach to analyzing extreme value data is the generalized Pareto distribution (GPD). When the GPD is used to explain a target variable with the large dimension of covariates, the shape and scale function of covariates included in GPD are sometimes modeled using the generalized additive models (GAM). In contrast to many results of application, there are no theoretical results on the hybrid technique of GAM and GPD, which motivates us to develop its asymptotic theory. We provide the rate of convergence of the estimator of shape and scale functions, as well as its local asymptotic normality.

Asymptotic theory for extreme value generalized additive models

Abstract

The classical approach to analyzing extreme value data is the generalized Pareto distribution (GPD). When the GPD is used to explain a target variable with the large dimension of covariates, the shape and scale function of covariates included in GPD are sometimes modeled using the generalized additive models (GAM). In contrast to many results of application, there are no theoretical results on the hybrid technique of GAM and GPD, which motivates us to develop its asymptotic theory. We provide the rate of convergence of the estimator of shape and scale functions, as well as its local asymptotic normality.
Paper Structure (14 sections, 14 theorems, 205 equations)

This paper contains 14 sections, 14 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme value theory
  4. Extreme value theory in regression
  5. Extreme Value Generalized Additive Models
  6. Peak over threshold
  7. Generalized additive model
  8. Asymptotic Theory
  9. General condition
  10. Main results
  11. Conclusion
  12. Outline of proof of theorems
  13. Lemmas
  14. Proof of Theorems

Key Result

Theorem 1

Suppose that (C1)--(C5). In each scenario (S1), (S2) or (S3), as $N\rightarrow \infty$, Under the optimal rate of number of knots $K= O(\{N p_N\}^{1/(2m+1)})$, If we take $p_N=O(N^{-1/(1-2\rho+1/m)})$, then the optimal convergence rates of the estimators are

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Theorem 2
  • Remark 5
  • Theorem 3
  • Theorem 4
  • Remark 6
  • ...and 23 more