Confronting missing observations with probability weights: Fourier space and generalized formalism

Due to instrumental limitations, spectroscopic galaxy surveys usually do not collect redshifts for the whole 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 galaxy density. The result is a fictitious suppression of the galaxy clustering that can impact severely on cosmological parameter inference. Recent developments have shown that an unbiased estimate of the two-point correlation 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 realizations of the targeting and how to model exactly the resulting sampling effects. This allows us to derive an inverse-probability-based power-spectrum estimator, 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 inverse-probability prescription, providing examples of how traditional missing-observation countermeasures can be included in this more general picture. The effectiveness of models and weighting schemes discussed in this work is demonstrated using realizations of an idealized survey strategy.