**M. Perone Pacifico, C. Genovese, I. Verdinelli, L. Wasserman**

This paper extends False Discovery Rates
to random fields, where there are uncountably many hypothesis tests.
This provides a method for finding local regions in the field
where there is a significant signal while
controlling either the proportion of area or the number
of clusters in which false rejections occur.
We develop
confidence envelopes for the proportion of false
discoveries as a function of the rejection threshold.
This yields algorithms for constructing
a confidence superset for the locations
of the true nulls.
From this we derive rejection
thresholds that control
the mean and quantiles of the proportion of false discoveries.
We apply this method to scan statistics and
functional neuroimaging.

*Keywords:* false discovery rates, multiple hypothesis test, random fields

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