  This article derives an optimal (i.e., unbiased, minimum variance) estimator
for the pseudo-detector strain for a pair of co-located gravitational wave
interferometers (such as the pair of LIGO interferometers at its Hanford
Observatory), allowing for possible instrumental correlations between the two
detectors. The technique is robust and does not involve any assumptions or
approximations regarding the relative strength of gravitational wave signals in
the detector pair with respect to other sources of correlated instrumental or
environmental noise. An expression is given for the effective power spectral
density of the combined noise in the pseudo-detector. This can then be
introduced into the standard optimal Wiener filter used to cross-correlate
detector data streams in order to obtain an optimal estimate of the stochastic
gravitational wave background. In addition, a dual to the optimal estimate of
strain is derived. This dual is constructed to contain no gravitational wave
signature and can thus be used as on "off-source" measurement to test
algorithms used in the "on-source" observation.
