  Gravitational wave detectors will need optimal signal-processing algorithms
to extract weak signals from the detector noise. Most algorithms designed to
date are based on the unrealistic assumption that the detector noise may be
modeled as a stationary Gaussian process. However most experiments exhibit a
non-Gaussian ``tail'' in the probability distribution. This ``excess'' of large
signals can be a troublesome source of false alarms. This article derives an
optimal (in the Neyman-Pearson sense, for weak signals) signal processing
strategy when the detector noise is non-Gaussian and exhibits tail terms. This
strategy is robust, meaning that it is close to optimal for Gaussian noise but
far less sensitive than conventional methods to the excess large events that
form the tail of the distribution. The method is analyzed for two different
signal analysis problems: (i) a known waveform (e.g., a binary inspiral chirp)
and (ii) a stochastic background, which requires a multi-detector signal
processing algorithm. The methods should be easy to implement: they amount to
truncation or clipping of sample values which lie in the outlier part of the
probability distribution.
