Sequential change detection in the presence of unknown parameters

It is commonly required to detect change points in sequences of random variables. In the most difficult setting of this problem, change detection must be performed sequentially with new observations being constantly received over time. Further, the parameters of both the pre- and post- change distributions may be unknown. In Hawkins and Zamba (Technometrics 47(2):164–173, 2005), the sequential generalised likelihood ratio test was introduced for detecting changes in this context, under the assumption that the observations follow a Gaussian distribution. However, we show that the asymptotic approximation used in their test statistic leads to it being conservative even when a large numbers of observations is available. We propose an improved procedure which is more efficient, in the sense of detecting changes faster, in all situations. We also show that similar issues arise in other parametric change detection contexts, which we illustrate by introducing a novel monitoring procedure for sequences of Exponentially distributed random variable, which is an important topic in time-to-failure modelling.