  We present our recent high precision calculations (Phys. Rev. D102 (2020)
no.5, 054512 and JHEP 04 (2021) 044, JHEP 21 (2020) 004) of the first moment of
nucleon isovector polarized, unpolarized and transversity distributions, i.e.,
momentum fraction, helicity and transversity moment, respectively. We use the
standard method for the calculation of these moments (via matrix elements of
twist two operators), and carry out a detailed analysis of the sources of
systematic uncertainty, in particular of excited state contributions. Our
calculations have been performed using two different lattice setups
(Clover-on-HISQ and Clover-on-Clover), each with several ensembles. They give
consistent results that are in agreement with global fit analyses.
