Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Bernstein-von Mises theorem and efficiency for semiparametric inference in multivariate Hawkes processes

Mael Duverger, Judith Rousseau

Abstract

In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting distribution for a regular estimator of smooth functional. Then, in the Bayesian setting, we prove a semiparametric Bernstein-von Mises (BvM) theorem for nonparametric random series priors. We apply this result to histogram and wavelet based priors. Taken together, the convolution and BvM theorems show that, from a frequentist point of view, semiparametric Bayesian procedures have asymptotically the optimal behavior. Deriving the BvM property for random series priors led us to prove L2 posterior contraction, complementing for these priors the results of Donnet, Rivoirard and Rousseau (2020).

The Bernstein-von Mises theorem and efficiency for semiparametric inference in multivariate Hawkes processes

Abstract

In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting distribution for a regular estimator of smooth functional. Then, in the Bayesian setting, we prove a semiparametric Bernstein-von Mises (BvM) theorem for nonparametric random series priors. We apply this result to histogram and wavelet based priors. Taken together, the convolution and BvM theorems show that, from a frequentist point of view, semiparametric Bayesian procedures have asymptotically the optimal behavior. Deriving the BvM property for random series priors led us to prove L2 posterior contraction, complementing for these priors the results of Donnet, Rivoirard and Rousseau (2020).
Paper Structure (37 sections, 36 theorems, 370 equations)

This paper contains 37 sections, 36 theorems, 370 equations.

Table of Contents

  1. Introduction
  2. Hawkes processes
  3. Our contribution
  4. Setup and notations
  5. Semiparametric efficiency
  6. LAN expansion
  7. Convolution theorem
  8. Explicit formula of the information operator
  9. Main results for Bayesian inference
  10. Prior model
  11. Posterior contraction in $L_1$ and $L_2$ norms
  12. The BvM property
  13. Application on specific prior models
  14. Random histogram priors
  15. Wavelet priors
  16. ...and 22 more sections

Key Result

Lemma 2.1

The bilinear map $\langle\cdot{,}\cdot\rangle_L$ is an inner product on $\mathbb{R}^K\times L_{2,\mathbf{h}^0}$ and induces the LAN norm $\Vert .\Vert_L$. In addition, $\Vert.\Vert_L$ is equivalent to the canonical norm $\Vert .\Vert_{(2,\mathbf{h}^0)}$ over $\mathbb{R}^K\times L_{2,\mathbf{h}^0}$.

Theorems & Definitions (62)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 3.2
  • ...and 52 more