  We develop a new multi-detector signal-based discriminator to improve the
sensitivity of searches for gravitational waves from compact binary
coalescences. The new statistic is the traditional $\chi^2$ computed on a
null-stream synthesized from the gravitational-wave detector strain time-series
of three detectors. This null-stream-$\chi^2$ statistic can be extended to
networks involving more than three detectors as well. The null-stream itself
was proposed as a discriminator between correlated unmodeled signals in
multiple detectors, such as arising from a common astrophysical source, and
uncorrelated noise transients. It can be useful even when the signal model is
known, such as for compact binary coalescences. The traditional $\chi^2$, on
the other hand, is an effective discriminator when the signal model is known
and lends itself to the matched-filtering technique. The latter weakens in its
effectiveness when a signal lacks enough cycles in band; this can happen for
high-mass black hole binaries. The former weakens when there are concurrent
noise transients in different detectors in the network or the detector
sensitivities are substantially different. Using simulated binary black hole
signals, noise transients and strain for Advanced LIGO (in Livingston and
Hanford) and Advanced Virgo detectors, we compare the performance of the
null-stream-$\chi^2$ statistic with that of the traditional $\chi^2$ statistic
using receiver-operating characteristics. The new statistic may form the basis
for better signal-noise discriminators in multi-detector searches in the
future.
