Average run length function of CUSUM test with independent but non-stationary log-likelihood ratios

The characteristics of Page´s CUSUM test are revealed by its average run length (ARL) function. It has been studied extensively under the assumption of independent and identically distributed (i.i.d.) log-likelihood ratios (LLR). In this case, it has been known that ARL is the solution of a Fredholm integral equation of the second kind (FIESK). However, few results are known when the LLR are independent but non-stationary (i.e., not identically distributed). This paper develops an inductive integral equation governing the ARL function in this setting. Unfortunately, in general it can not be solved analytically and the uniqueness of the solution is not guaranteed. However, for two frequently encountered cases-the LLR sequence converges in distribution and it has a periodic distribution, respectively-numerical solutions can be obtained by extending the system of linear algebraic equations method proposed for solving FIESK. Our solutions to these two special cases are compared with results of Monte Carlo simulations.