Approximations of distributions of scan statistics of inhomogeneous Poisson processes

- The asymptotic distributions of scan statistics of inhomogeneous Poisson processes are derived. In the literature, very little work has been done for scan statistics of inhomogeneous Poisson processes.
- The classic scan statistics are extended to the weighted scan statistics for adjusting the inhomogeneity and their exact and asymptotic distributions are also derived.

In this paper, the distributions of scan statistics of inhomogeneous Poisson processes are studied. First, the distribution of the continuous scan statistic of an inhomogeneous Poisson process is approximated by the distribution of the discrete scan statistic for a sequence of Bernoulli trials with unequal probabilities of success. Next, we introduce a weighted scan statistic to adjust the inhomogeneity and the weighted continuous scan statistic is also approximated by the weighted discrete scan statistic. The finite Markov chain imbedding technique is used to obtain the exact distributions of weighted discrete scan statistics. An example using the weighted scan statistic of an inhomogeneous Poisson process for detecting DNA copy number variation is given. Numerical results and simulations are also given to illustrate our theoretical results.