  Inspiraling binaries of compact objects are primary targets for current and
future gravitational-wave observatories. Waveforms computed in General
Relativity are used to search for these sources, and will probably be used to
extract source parameters from detected signals. However, if a different theory
of gravity happens to be correct in the strong-field regime, source-parameter
estimation may be affected by a fundamental bias: that is, by systematic errors
induced due to the use of waveforms derived in the incorrect theory. If the
deviations from General Relativity are not large enough to be detectable on
their own and yet these systematic errors remain significant (i.e., larger than
the statistical uncertainties in parameter estimation), fundamental bias cannot
be corrected in a single observation, and becomes stealth bias. In this article
we develop a scheme to determine in which cases stealth bias could be present
in gravitational-wave astronomy. For a given observation, the answer depends on
the detection signal-to-noise ratio and on the strength of the modified-gravity
correction. As an example, we study three representative stellar-mass binary
systems that will be detectable with second-generation ground-based
observatories. We find that significant systematic bias can occur whether or
not modified gravity can be positively detected, for correction strengths that
are not currently excluded by any other experiment. Thus, stealth bias may be a
generic feature of gravitational-wave detections, and it should be considered
and characterized, using expanded models such as the parametrized post-Einstein
framework, when interpreting the results of parameter-estimation analyses.
