Minimax bounds for the risk function of estimators of functionals ofthe spectral density of Gaussianfields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes.Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators aregiven for all dimensions. For non-linear integral functionals, theseestimators are constructed for the two and three dimensional problems.

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.

The aim of the current paper is twofold. Primarily we wish to extend some of our earlier results on excursions of random fields and introduce a more powerful technique for obtaining the mean value of a certain characteristic of these excursions. Since these results can also be used to tie together the scattered results of previous authors we also include a full review of earlier work in this field.