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Our problem is to find a good approximation to the P-value of the maximum ofa random field of test statistics for a cone alternative at each point in asample of Gaussian random fields. These test statistics have been proposed inthe neuroscience literature for the analysis of fMRI data allowing for unknowndelay in the hemodynamic response. However the null distribution of the maximumof this 3D random field of test statistics, and hence the threshold used todetect brain activation, was unsolved. To find a solution, we approximate theP-value by the expected Euler characteristic (EC) of the excursion set of thetest statistic random field. Our main result is the required EC density,derived using the Gaussian Kinematic Formula.

Detecting sparse cone alternatives for Gaussian random fields, with an application to fMRI