Evaluations of likelihood ratio methods for surveillance. Differences and robustness.

Abstract

In many areas there is a need for continual observation of a time series, with the goal of detecting an important change in the underlying process as soon as possible after it has occurred. In recent years there have been a growing number of papers in economics, medicine, environmental control and other areas dealing with the need of methods for surveillance. Examples are given in Frisen (1992) and Frisen (1994a). The timeliness of decisions is taken into account in the vast literature on quality control charts where simplicity is often a major concern. Also, the literature on stopping rules is relevant. For an overview, see the textbook by Wetherill and Brown (1990) the surveys by Zacks (1983) or Lai (1995) and the bibliography by Frisen (1994b). Methods based on likelihood ratios are known to have several optimality properties. Evaluations are made of the full likelihood ratio (LR) method, which will be expressed as a certain combination of conditional likelihood ratios. In the cases studied here, the LR method has the Shiryaev optimality. Also, the Shiryaev-Roberts and the CUSUM methods are evaluated. These two methods combine conditional likelihood ratios in other ways. A comparison is also made with the Shewhart method that is a commonly used method. When control charts are used in practice, it is necessary to know several characteristics of the method. Asymptotic properties have been studied by Srivastava and Wu (1993) and Siegmund and Venkatraman (1995) and others. Here, properties for fInite time of change are studied. The usual ARLo and ARLl (which are the average nm lengths until an alarm under the hypothesis of no change and the hypothesis of immediate change, respectively) are used. Besides that, the probability of a false alarm, the expected delay, the probability of successful detection and the predictive value are used for evaluations. Since the methods have interesting optimality properties, the results also enlighten different criteria of optimality .