The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey:
  Pairwise-Inverse-Probability and Angular Correction for Fibre Collisions in
  Clustering Measurements
  The completed eBOSS catalogues contain redshifts of 344080 QSOs over
0.8<z<2.2 covering 4808 deg$^2$, 174816 LRGs over 0.6<z<1.0 covering 4242
deg$^2$ and 173736 ELGs over 0.6<z<1.1 covering 1170 deg$^2$ in order to
constrain the expansion history of the Universe and the growth rate of
structure through clustering measurements. Mechanical limitations of the
fibre-fed spectrograph on the Sloan telescope prevent two fibres being placed
closer than 62", the fibre-collision scale, in a single pass of the instrument
on the sky. These `fibre collisions' strongly correlate with the intrinsic
clustering of targets and can bias measurements of the two-point correlation
function resulting in a systematic error on the inferred values of the
cosmological parameters. We combine the new techniques of
pairwise-inverse-probability weighting and the angular up-weighting to correct
the clustering measurements for the effect of fibre collisions. Using mock
catalogues we show that our corrections provide unbiased measurements, within
data precision, of both the projected correlation function $w_p$ and the
multipoles $\xi^l$ of the redshift-space correlation functions down to
0.1Mpc/h, regardless of the tracer type. We apply the corrections to the eBOSS
DR16 catalogues. We find that, on scales greater than s~20Mpc/h for $\xi^l$, as
used to make BAO and large-scale RSD measurements, approximate methods such as
Nearest-Neighbour up-weighting are sufficiently accurate given the statistical
errors of the data. Using the PIP method, for the first time for a
spectroscopic program of the Sloan Digital Sky Survey we are able to
successfully access the 1-halo term in the 3D clustering measurements down to
~0.1Mpc/h scales. Our results will therefore allow studies that use the
small-scale clustering measurements to strengthen the constraints on both
cosmological parameters and the halo-occupation distribution models.
