[Paper Review] Strong mixing condition for Hawkes processes and application to Whittle estimation from count data

Félix Cheysson, Gabriel Lang|arXiv (Cornell University)|Mar 9, 2020

Point processes and geometric inequalities31 references4 citations

TL;DR

This paper establishes a strong mixing condition with polynomial decay for stationary Hawkes processes using their Poisson cluster structure, enabling a spectral Whittle estimation method for count data over fixed intervals. The approach yields consistent and asymptotically normal estimates of the reproduction kernel and mean, validated through simulations and a case study with large time intervals.

ABSTRACT

This paper focuses on the time series generated by the event counts of stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, existing methods of estimation are not applicable. We first establish a strong mixing condition with polynomial decay rate for Hawkes processes, from their Poisson cluster structure. This allows us to propose a spectral approach to the estimation of Hawkes processes, based on Whittle's method, which provides consistent and asymptotically normal estimates under common regularity conditions on their reproduction kernels. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the Hawkes reproduction mean and kernel when time intervals are relatively large.

Motivation & Objective

• To address the lack of estimation methods for Hawkes processes when only event counts over fixed intervals are observed, rather than exact event times.
• To establish a strong mixing condition with polynomial decay for Hawkes processes based on their underlying Poisson cluster structure.
• To develop a spectral estimation method—Whittle's approach—suitable for count data and applicable under standard regularity conditions.
• To ensure the proposed estimator is consistent and asymptotically normal for the reproduction kernel and mean.
• To demonstrate the method’s performance through simulations and a real-data case study, especially under large time intervals.

Proposed method

• Leverages the Poisson cluster representation of Hawkes processes to derive a strong mixing condition with polynomial decay rate.
• Applies Whittle's method to the periodogram of count data over fixed time intervals to estimate the spectral density.
• Uses the spectral density to infer the reproduction kernel and mean through a contrast function minimizing discrepancy.
• Imposes standard regularity conditions on the reproduction kernel to ensure theoretical properties of the estimator.
• Employs a spectral approach that bypasses the need for exact point process observations, relying only on aggregated counts.
• Validates the method via simulation studies and a case study to assess robustness under large interval sizes.

Experimental results

Research questions

• RQ1 Can a strong mixing condition with polynomial decay be established for stationary Hawkes processes using their cluster structure?
• RQ2 Is Whittle estimation applicable and statistically valid when only aggregated counts over fixed intervals are observed?
• RQ3 Do the resulting Whittle estimates of the reproduction kernel and mean remain consistent and asymptotically normal under such data limitations?
• RQ4 How does the performance of the method vary when time intervals are relatively large?
• RQ5 Can the proposed method effectively recover the underlying Hawkes process parameters in practical settings?

Key findings

• A strong mixing condition with polynomial decay is established for stationary Hawkes processes through their Poisson cluster representation.
• The Whittle estimation method is shown to produce consistent and asymptotically normal estimates of the reproduction kernel and mean under standard regularity conditions.
• The method remains effective even when time intervals are relatively large, as demonstrated in simulations and the case study.
• The spectral approach successfully bypasses the need for exact event time observations, enabling estimation from count data alone.
• Theoretical guarantees are derived using the cluster structure, supporting the validity of the estimation framework.
• Empirical results confirm the method's robustness and accuracy in recovering Hawkes process parameters from aggregated counts.