  It is well known that the power spectrum is not able to fully characterize
the statistical properties of non-Gaussian density fields. Recently, many
different statistics have been proposed to extract information from
non-Gaussian cosmological fields that perform better than the power spectrum.
The Fisher matrix formalism is commonly used to quantify the accuracy with
which a given statistic can constrain the value of the cosmological parameters.
However, these calculations typically rely on the assumption that the
likelihood of the considered statistic follows a multivariate Gaussian
distribution. In this work we follow Sellentin & Heavens (2017) and use two
different statistical tests to identify non-Gaussianities in different
statistics such as the power spectrum, bispectrum, marked power spectrum, and
wavelet scatering transform (WST). We remove the non-Gaussian components of the
different statistics and perform Fisher matrix calculations with the
\textit{Gaussianized} statistics using Quijote simulations. We show that
constraints on the parameters can change by a factor of $\sim 2$ in some cases.
We show with simple examples how statistics that do not follow a multivariate
Gaussian distribution can achieve artificially tight bounds on the cosmological
parameters when using the Fisher matrix formalism. We think that the
non-Gaussian tests used in this work represent a powerful tool to quantify the
robustness of Fisher matrix calculations and their underlying assumptions. We
release the code used to compute the power spectra, bispectra, and WST that can
be run on both CPUs and GPUs.
