  Long-baseline optical interferometry uses the power spectrum and bispectrum
constructs as fundamental observables. Noise arising in the detection of the
fringe pattern gives rise to both variance and biases in the power spectrum and
bispectrum. Previous work on correcting the biases and estimating the variances
for these quantities typically includes restrictive assumptions about the
sampling of the interferogram and/or about the relative importance of Poisson
and Gaussian noise sources. Until now it has been difficult to accurately
compensate for systematic biases in data which violates these assumptions. We
seek a formalism to allow the construction of bias-free estimators of the
bispectrum and power spectrum, and to estimate their variances, under less
restrictive conditions which include both unevenly-sampled data and
measurements affected by a combination of noise sources with Poisson and
Gaussian statistics. We used a method based on the moments of the noise
distributions to derive formulae for the biases introduced to the power
spectrum and bispectrum when the complex fringe amplitude is derived from an
arbitrary linear combinations of a set of discrete interferogram measurements.
We simulated interferograms with different combinations of photon noise and
read noise and with different fringe encoding schemes to illustrate the effects
of these biases. We have derived formulae for bias-free estimators of the power
spectrum and bispectrum which can be used with any linear estimator of the
fringe complex amplitude. We have demonstrated the importance of bias-free
estimators for the case of the detection of faint companions (for example
exoplanets) using closure phase nulling. We have derived formulae for the
variance of the power spectrum and have shown how the variance of the
bispectrum could be calculated.
