  We study the significance of non-Gaussianity in the likelihood of weak
lensing shear two-point correlation functions, detecting significantly non-zero
skewness and kurtosis in one-dimensional marginal distributions of shear
two-point correlation functions in simulated weak lensing data. We examine the
implications in the context of future surveys, in particular LSST, with
derivations of how the non-Gaussianity scales with survey area. We show that
there is no significant bias in one-dimensional posteriors of $\Omega_{\rm m}$
and $\sigma_{\rm 8}$ due to the non-Gaussian likelihood distributions of shear
correlations functions using the mock data ($100$ deg$^{2}$). We also present a
systematic approach to constructing approximate multivariate likelihoods with
one-dimensional parametric functions by assuming independence or more flexible
non-parametric multivariate methods after decorrelating the data points using
principal component analysis (PCA). While the use of PCA does not modify the
non-Gaussianity of the multivariate likelihood, we find empirically that the
one-dimensional marginal sampling distributions of the PCA components exhibit
less skewness and kurtosis than the original shear correlation
functions.Modeling the likelihood with marginal parametric functions based on
the assumption of independence between PCA components thus gives a lower limit
for the biases. We further demonstrate that the difference in cosmological
parameter constraints between the multivariate Gaussian likelihood model and
more complex non-Gaussian likelihood models would be even smaller for an
LSST-like survey. In addition, the PCA approach automatically serves as a data
compression method, enabling the retention of the majority of the cosmological
information while reducing the dimensionality of the data vector by a factor of
$\sim$5.
