Confronting missing observations with probability weights: Fourier space
  and generalised formalism
  Due to instrumental limitations, the nature of which vary from case to case,
spectroscopic galaxy redshift surveys usually do not collect redshifts for all
galaxies in the population of potential targets. Especially problematic is the
entanglement between this incompleteness and the true cosmological signal,
arising from the fact that the proportion of successful observations is
typically lower in regions with higher density of galaxies. The result is a
fictitious suppression of the galaxy clustering that, if not properly
corrected, can impact severely on cosmological-parameter inference. Recent
developments in the field have shown that an unbiased estimate of the 2-point
correlation function in the presence of missing observations can be obtained by
weighting each pair by its inverse probability of being targeted. In this work
we expand on the concept of probability weights by developing a more mature
statistical formalism, which provides us with a deeper understanding of their
fundamental properties. We take advantage of this novel perspective to handle
the problem of estimating the inverse probability, specifically, we discuss how
to efficiently determine the weights from a finite set of realisations of the
targeting and how to model exactly the resulting sampling effects. This allows
us to derive an inverse-probability-based estimator for the power spectrum,
which is the main result of this work, but also to improve robustness and
computational efficiency of the already existing configuration-space estimator.
Finally, we propose a strategy to further extend the concept of inverse
probability, providing examples of how traditional missing-observation
countermeasures can be included in this more general picture. The effectiveness
of the different models and weighting schemes discussed in this work is
demonstrated using realisations of an idealised simple survey strategy.
