  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.
