Asymptotically Optimal Sequential Change-Point Detection under Composite Hypotheses

The problem of sequential detection of a change- point in the density function of observations from a sequence of independent random variables is considered when both before and after a change-point this density function belongs to a certain family of distributions, i.e. in the general situation of composite hypotheses. A new quality criterion for change-point detection is proposed. The asymptotic a priori lower bound for this criterion is established for any method of change-point detection. A method of change-point detection is proposed for which this lower bound is attained asymptotically so that the method can be called asymptotically optimal. In particular, for the case of a simple hypothesis before a change-point, this method coincides with the generalized cumulative sums (CUSUM) method.