  The determination of the covariance matrix and its inverse, the precision
matrix, is critical in the statistical analysis of cosmological measurements.
The covariance matrix is typically estimated with a limited number of
simulations at great computational cost before inversion into the precision
matrix; therefore, it can be ill-conditioned and overly noisy when the sample
size $n$ used for estimation is not much larger than the data vector dimension.
In this work, we consider a class of methods known as shrinkage estimation for
the precision matrix, which combines an empirical estimate with a target that
is either analytical or stochastic. These methods include linear and non-linear
shrinkage applied to the covariance matrix (the latter represented by the
so-called NERCOME estimator), and the direct linear shrinkage estimation of the
precision matrix which we introduce in a cosmological setting. Using Bayesian
parameter inference as well as metrics like matrix loss functions and the
eigenvalue spectrum, we compare their performance against the standard sample
estimator with varying sample size $n$. We have found the shrinkage estimators
to significantly improve the posterior distribution at low $n$, especially for
the linear shrinkage estimators either inverted from the covariance matrix or
applied directly to the precision matrix, with an empirical target constructed
from the sample estimate. Our results should be particularly relevant to the
analyses of Stage-IV spectroscopic galaxy surveys such as the Dark Energy
Spectroscopic Instrument (DESI) and Euclid, whose statistical power can be
limited by the computational cost of obtaining an accurate precision matrix
estimate.
