On a discrimination problem for a class of stochastic processes with ordered first-passage times

Consider the Bayes problem in which one has to discriminate if the random unknown initial state of a stochastic process is distributed according to either of two preassigned distributions, on the base of the observation of the first-passage time of the process through 0. For processes whose first-passage times to state 0 are increasing in the initial state according to the likelihood ratio order, such problem is solved by determining the Bayes decision function and the corresponding Bayes error. The special case of fixed initial values including a family of first-passage times with proportional reversed hazard functions is then studied. Finally, various applications to birth-and-death and to diffusion processes are discussed.