Hawkes Identification with a Prescribed Causal Basis: Closed-Form Estimators and Asymptotics

Abstract

Driven by the recent surge in neural-inspired modeling, point processes have gained significant traction in systems and control. While the Hawkes process is the standard model for characterizing random event sequences with memory, identifying its unknown kernels is often hindered by nonlinearity. Approaches using prescribed basis kernels have emerged to enable linear parameterization, yet they typically rely on iterative likelihood methods and lack rigorous analysis under model misspecification. This paper justifies a closed-form Least Squares identification framework for Hawkes processes with prescribed kernels. We guarantee estimator existence via the almost-sure positive definiteness of the empirical Gram matrix and prove convergence to the true parameters under correct specification, or to well-defined pseudo-true parameters under misspecification. Furthermore, we derive explicit Central Limit Theorems for both regimes, providing a complete and interpretable asymptotic theory. We demonstrate these theoretical findings through comparative numerical simulations.

Figures (4)

• Figure 1: Plots of the HIR estimation error $\Delta\phi(t)$: $\Delta\phi(t)$ shrinks as the Hawkes-Laguerre model order $P$ increases with the relative errors $\frac{\int_0^\infty\Delta\phi(t)^2dt}{\int_0^\infty\phi_0(t)^2dt}\times100\%=5.1\%, 3.58\%, 0.40\%, 0.23\%, 0.033\%$, under $P=1,2,3,4,5$, respectively.
• Figure 2: Plots of $h^\alpha(t)$: $h^\alpha(t)$ shrinks as $P$ increases, suggesting that $\mathop{\mathrm{E}}\nolimits[\lambda_0(0)\xi_h(0)\xi_h(0)^\top]$ stays close to its correctly specified counterpart $\mathop{\mathrm{E}}\nolimits[\lambda_0(0)\xi(0)\xi(0)^\top]$.
• Figure 3: Quantiles of the scaled estimation error $\sqrt{T}(\hat{\theta}-\theta_0)$ under correct model specification. The empirical quantiles of the LS estimates (blue) closely match the theoretical asymptotic quantiles (black).
• Figure 4: Quantiles of the scaled estimation error $\sqrt{T}(\hat{\theta}-\theta_*)$ under model misspecification. The empirical quantiles of the LS estimates (blue) closely match the theoretical asymptotic quantiles (black). The error variance under misspecification is much higher than that under correct specification.

Theorems & Definitions (2)

• Remark 1
• Remark 2