# Hawkes processes: interpretation of maximum likelihood estimates disagreeing with moment estimator estimates?

So, I have some data.. and a parameterized Hawkes process which I estimate parameters for via maximum likelihood... the residuals ( the compensator aka the dual-predictable projection) are good in that they both have nearly 1 for mean and variance and have almost no remaining autocorrelation. The odd thing is that even though the residuals fit so well, the theoretical moments are extremely far away from the empirical moments. The opposite happens if I estimate to match the moments... then the moments are closer to matching but the compensator is extremely far from Poissonian.

Is this some sort of paradox? How do I reconcile these facts? Does it imply that the parameterization needs to be modified somehow since it doesn't fit well enough?

• I have not heard of the Hawkes process, but it sounds like something one might encounter with heavy tailed distributions. Did you test for this? – Sid Oct 11 '17 at 22:24
• Yes.. it is indeed a heavy-tailed distribution I am working with.. diverging integral of the autocorrelation of the waiting time sequence – crow Oct 12 '17 at 2:51

## 1 Answer

The problem could be arising because of the heavy tailed nature of the distributions you are dealing with. Section 2.2.1.3 and the remark preceding it in this text discuss the problem you seem to be encountering.

Be careful while dealing with MLEs of parameters for heavy tailed distribution. As stated in the remark

In particular, the empirical mean can always be computed for a sample from the Cauchy distribution, however, it cannot have the interpretation of an estimate of the mean in that case

• Ah hah, thank you. this makes sense to me. I am familiar with the Cauchy distribution and I'm dealing with a similiar thing here.. so it seems that the mean/variance/etc of the sample data I have is not actually representative of the mean and variance of the generating process... in that case, it seems that as long as I get satisfactory residuals from the fitting process then I should be able to use them for filtering/predicting the intensity of the process.. I think – crow Oct 12 '17 at 16:30
• Well, thats not entirely true. It does not mean that the mean and variances are lost causes. It depends on how the tail is dying out. There might be some workarounds to estimate the mean and the variance if they do exist. See section 2.2.1 of the text. – Sid Oct 12 '17 at 18:55
• L-moments and Laguerre polynomials... interesting, I shall investigate – crow Oct 13 '17 at 15:30
• does this hold true even if the moments of the distribution in question exist? The closed-form for the moments of the theoretical process i'm talking does exist – crow Oct 13 '17 at 16:18