Estimating the likelihood ratio function in the problem of ''failure'' of a stochastic process (Q1821470)

The author considers detection of change in a memoryless process, in the presence of fixed sample size, when the distributions of the two processes involved are unknown. He then proposes estimation of the likelihood ratio, before maximum likelihood detection of the time when the change occurs. The approach in this paper has two problems: (1) If the change occurs after a relatively small observation sequence, the likelihood ratio estimate can not be reliable. For efficiency and convergence of the estimate, asymptotically long observation sequences are needed, which implies defeat of the posed problem itself. Namely, the proposed approach is reliable, only if the change basically does not occur. (2) Even in the presence of well known processes, detecting a change in the presence of a fixed sample size is not quite meaningful, since the decision on the size implies knowledge as to within how many observations the change may occur. In detection of change in stochastic process problems, sequential approaches are meaningful, as those proposed by Page. If the distributions of the processes are unknown, it is better to estimate them a priori, via nonmixed samples. If the distributions are ''almost'' known, instead, minimax robust techniques are then appropriate.