  We present a Gaussian regression method for time series with missing data and
stationary residuals of unknown power spectral density (PSD). The missing data
are efficiently estimated by their conditional expectation as in universal
Kriging, based on the circulant approximation of the complete data covariance.
After initialization with an autoregessive fit of the noise, a few iterations
of estimation/reconstruction steps are performed until convergence of the
regression and PSD estimates, in a way similar to the
expectation-conditional-maximization algorithm. The estimation can be performed
for an arbitrary PSD provided that it is sufficiently smooth. The algorithm is
developed in the framework of the MICROSCOPE space mission whose goal is to
test the weak equivalence principle (WEP) with a precision of $10^{-15}$. We
show by numerical simulations that the developed method allows us to meet three
major requirements: to maintain the targeted precision of the WEP test in spite
of the loss of data, to calculate a reliable estimate of this precision and of
the noise level, and finally to provide consistent and faithful reconstructed
data to the scientific community.
