## Bernoulli

• Bernoulli
• Volume 24, Number 4B (2018), 3469-3493.

### Statistical inference for the doubly stochastic self-exciting process

#### Abstract

We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter $T^{-1}\int_{0}^{T}\theta_{t}^{*}\,dt$, where $\theta_{t}^{*}$ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-naïve estimator which is constructed as the naïve one when applying a first-order bias reduction to the local MLE. We show the associated central limit theorem. Monte Carlo simulations show the importance of the bias correction and that the method performs well in finite sample, whereas the empirical study discusses the implementation in practice and documents the stochastic behavior of the parameters.

#### Article information

Source
Bernoulli, Volume 24, Number 4B (2018), 3469-3493.

Dates
Revised: June 2017
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.bj/1524038760

Digital Object Identifier
doi:10.3150/17-BEJ966

Mathematical Reviews number (MathSciNet)
MR3788179

Zentralblatt MATH identifier
06869882

#### Citation

Clinet, Simon; Potiron, Yoann. Statistical inference for the doubly stochastic self-exciting process. Bernoulli 24 (2018), no. 4B, 3469--3493. doi:10.3150/17-BEJ966. https://projecteuclid.org/euclid.bj/1524038760

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#### Supplemental materials

• Supplement to “Statistical inference for the doubly stochastic self-exciting process”. The Appendix can be found in the Appendix referenced as [8] below. This material is comprised of one Appendix, which contains all the proofs.