Boundary-crossing probabilities of some random fields related to likelihood ratio tests for epidemic alternatives

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.