Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach. Issue 1 (2nd January 2016) - Chen, Feng; Hall, Peter
Abstract : There is ample evidence that in applications of self-exciting point-process models, the intensity of background events is often far from constant. If a constant background is imposed that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modeling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this article, we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy, in cases where there is not sufficient empirical evidence to suggest that the background intensity function is more complex than those models. This approach is seldom, if ever, used for nonparametric function estimation in other settings, not least because in those cases more data are typically available. However, our "ontological parsimony" argument is appropriate in the context of self-exciting point-process models. Supplementary materials are available online.