  Transient phenomena are interesting and potentially highly revealing of
details about the processes under observation and study that could otherwise go
unnoticed. It is therefore important to maximize the sensitivity of the method
used to identify such events. In this article, we present a general procedure
based on the use of the likelihood function for identifying transients which is
particularly suited for real-time applications because it requires no grouping
or pre-processing of the data. The method makes use of all the information that
is available in the data throughout the statistical decision-making process,
and is suitable for a wide range of applications. Here we consider those most
common in astrophysics, which involve searching for transient sources, events
or features in images, time series, energy spectra, and power spectra, and
demonstrate the use of the method in the case of a weak X-ray flare in a time
series and a short-lived quasi-periodic oscillation in a power spectrum. We
derive a fit statistic that is ideal for fitting arbitrarily shaped models to a
power density distribution, which is of general interest in all applications
involving periodogram analysis.
