Statistical Tests for the Detection of a Self-Regulation Property for Poisson-Type Point Processes

For an orderly point process on the positive real numbers let t(i) be the ith point event after t=0 and the complete intensity function be given by g(t:r,d,c)=d+r1(t-t(i-1),{t-t(i-1)>c}) for t in (t(i-1),t(i)];i=1,2,...,d;d,c>0;r non-negative;t(0)=0 and 1(x,A)=1 if x satisfies the condition A and 1(x,A)=0 otherwise. We present tests for the hypotheses H: r=0 and K: r>0 for known d and c based on the observation of the point process in the time period (0,t(m)] where m is a fixed natural number. We indicate a locally most powerful test and look then at the problem from the point of view of asymptotic optimality of the procedure (contiguous hypotheses). These results are extended to other self-regulation properties