The Shark Fin Function - Asymptotic Behavior of the Filtered Derivative for Point Processes in Case of Change Points

A multiple filter test (MFT) for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process $G$ asymptotically takes the form \begin{align*} G \sim L, \end{align*} while $L$ is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is to describe the behavior of $G$ under the alternative hypothesis of rate changes and potential simultaneous variance changes. We derive the approximation \begin{align*} G \sim \Delta \cdot\left(\Lambda + L\right), \end{align*} with deterministic functions $\Delta$ and $\Lambda$. The function $\Lambda$ accounts for the systematic deviation of $G$ in the neighborhood of a change point. When only the rate changes, $\Lambda$ is hat shaped. When also the variance changes, $\Lambda$ takes the form of a shark's fin. In addition, the parameter estimates required in practical application are not consistent in the neighborhood of a change point. Therefore, we derive the factor $\Delta$ termed here the distortion function. It accounts for the lack in consistency and describes the local parameter estimating process relative to the true scaling of the filtered derivative process.