A Page test with nuisance parameter estimation

The detection of the onset of a signal is a common and relevant problem in signal processing. The Page test [1954] using the log-likelihood ratio is optimal for minimizing the worst case average delay before detection (D¯) while constraining the average time between false alarms (T¯). Realistic problems typically include unknown parameters having the same value under signal-absent and signal-present hypotheses, known as nuisance parameters. In this correspondence, the Page test is generalized to account for nuisance parameters. The inherent signal-absent decision making of the Page test is exploited to identify signal-free data used to estimate the nuisance parameters. Due to the independence of this data and the current Page test statistic, analysis is feasible. Wald- and Siegmund-based approximations to D¯ and T¯ are derived and shown to simplify to those of the standard Page test as the estimation becomes perfect. The results for a Gaussian shift in mean signal with unknown variance are derived and verified through simulation, where it is seen that the Siegmund-based approximation provides more accuracy. It is seen that the linear asymptotic (in the sense of a large threshold) relationship between the threshold and D¯ is preserved when nuisance parameters are estimated. However, the exponential asymptotic relationship between the threshold and T¯ becomes a power law approximating an exponential