A PAGE TEST WITH NUISANCE PARAMETER-ESTIMATION

The detection of the onset of a signal is a common and relevant proble m in signal processing, The Page test [1] using the log-likelihood rat io is optimal for minimizing the worst case average delay before detec tion ((D) over bar) while constraining the average time between false alarms ((T) over bar). Realistic problems typically include unknown pa rameters 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 inher ent signal-absent decision making of the Page test is exploited to ide ntify signal-free data used to estimate the nuisance parameters, Due t o the independence of this data and the current Page test statistic, a nalysis is feasible, Wald- and Siegmund-based approximations to (D) ov er bar and (T) over bar 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-ba sed approximation provides more accuracy, It is seen that the linear a symptotic (in the sense of a large threshold) relationship between the threshold and (D) over bar is preserved when nuisance parameters are estimated, However, the exponential asymptotic relationship between th e threshold and (T) over bar becomes a pou er law approximating an exp onential.