AN APPROXIMATION METHOD OF THE STATISTICAL TEST FOR DETECTING CHANGE-POINT IN THE ANALYSIS OF RECURRENT EVENTS

Abstract

We investigate a null distribution of the test statistics when the counting process of recurrent events is modeled as a homogenous Poisson process. We give lower bound and upper bound theoretically and consider to approximate the null distribution by linear combinations of lower bound and upper bound, since it is difficult to derive exact distribution of the test statistics which is a distribution of maximum value of dependent chi-square distributions with 1 degrees of freedom (χ^2_1 distribution) in case of n-recurrent events. The lower bound and the upper bound are given by the distribution of a maximum value of independent n-2 χ^2_1 distributions and χ^2_1 distribution, respectively. Furthermore, we investigate approximation by the empirical distribution since the approximations using the bounds cannot provide good approximation of the null distribution. Our simulation study suggests that the null distribution of the test statistics can be well approximated by gamma distribution corresponding to the number of events over the entire region. The regression formula is also developed to determine the parameters of gamma distribution corresponding to the number of events. Our study shows that the gamma distribution is more appropriate approximation than the approximation using the bounds since it could be possible to keep significant level as a nominal level in the hypothesis testing.