Abstract.
In 2008, Marsan and Lenglin\'e discovered a way to
estimate the triggering function of a Hawkes process nonparametrically.
Their method requires an iterative and computationally
intensive procedure which ultimately produces approximate, though not
exact, maximum likelihood estimates.
Here, we point out a mathematical curiosity that allows one to compute
exact maximum likelihood estimates of the nonparametric triggering function
directly and
extremely rapidly. The method here requires that the number p of intervals
on which the nonparametric estimate is sought equals the number n of
observed points. Extensions to the more typical case where n is much
greater than p are discussed.
Introduction.
One of the most exciting advances recently in the statistical analysis of
point processes was the discovery by Marsan and Lengline (2008) of a
method for estimating
the triggering function of a Hawkes process nonparametrically.
Their method, which uses a variant of the E-M algorithm for point
processes described by Veen et al. (2006), writes the triggering function
as a step function and then estimates the steps by approximate maximum
likelihood. The procedure thus does not rely on a parametric form for the
triggering function, and is extremely useful as a tool for a variety of
purposes including suggesting the
functional form of a triggering function, assessing the goodness of fit of
a particular proposed functional form, and simulating or forecasting the
process without relying on a particular and possibly mis-specified
functional form for the triggering function.
Unfortunately the method proposed by Marsan and Lenglin\'e (2008)
requires an iterative and computationally
intensive procedure. In addition, the method
ultimately produces {\sl approximate}
maximum likelihood estimates whose asymptotic properties are not well
understood.
Here, we point out a mathematical curiosity that allows one to compute
exact maximum likelihood estimates of the triggering function directly and
extremely rapidly. The method here requires that the number p of intervals
on which the nonparametric estimate is sought equals the number n of
observed points. Extensions to the more typical case where n is much
greater than p are discussed.
The resulting computation times are many times smaller than with the
iterative method of Marsan and Lengline (2008), and the proposed
estimator is shown to work well in simulations and with several actual
datasets involving earthquakes and wildfires.
The structure of this paper is as follows.
Following a brief review of point processes, Hawkes processes, and the
ingenious algorithm of Marsan and Lenglin\'e (2008) in Section 2, the
technique proposed here for the case $p=n$ is described in Section 3.
Extensions to the case where $n >> p$ and other details regarding
implementation are discussed in Section 4, followed
by simulations in Section 5 and applications to real data in Section 6.
Concluding remarks are given in Section 7.