On Some Conservative Bounds for the Barrier Crossing Probability
Résumé
For some safety-critical applications, it is important to calculate the probability that a discrete time scalar (or vector) autoregressive (AR) process leaves an open interval (or an open ball) during a certain period of time. Let us consider the two following scenarios where the conservative bounds for the barrier crossing probability are useful. The first possible scenario is related to the control charts: the Shewhart chart and the Geometric Moving Average (GMA) chart, which are used for detecting the abrupt changes. Traditionally, the Average Run Length (ARL) to a false alarm and the worst-case mean detection delay are used as statistical performance measures of the Shewhart and GMA charts. The disadvantage of the ARL criterion for safety-critical applications consists in the existence of the right ``tail'' of the detection delay distribution. For safety-critical applications, it is more convenient to use the probability of missed detection and the probability of false alarm defined with respect to some periods. These probabilities are reduced to the barrier crossing probability for the AR process. The second scenario is related to the barrier crossing probability for a certain risk indicator, which is the AR process generated by some estimation errors. The safety of the system is compromised if the probability that this risk indicator leaves a given confidence zone at least once during a certain period becomes too important. Sometimes, we are also interested in the calculation of the instantaneous risk probability. In practice, the main difficulty for the above-mentioned scenarios is that the Cumulative Distribution Functions (CDFs) (with infinite support) of the innovation noise in the above-mentioned AR model and its initial state are unknown and only their upper and lower bounds are available. Numerical methods to compute the conservative bounds for the above-mentioned barrier crossing probability are considered in the presentation.