  We present multivariate unbiased estimators for second, third, and fourth
order cumulants $C_2(x,y)$, $C_3(x,y,z)$, and $C_4(x,y,z,w)$. Many relevant new
estimators are derived for cases where some variables are average-free or pairs
of variables have a vanishing second order cumulant. The well-know Fisher
k-statistics is recovered for the single variable case. The variances of
several estimators are explicitly given in terms of higher order cumulants and
discussed with respect to random processes that are predominately Gaussian. We
surprisingly find that the frequently used third order estimator
$\overline{x^3}$ for $C_3(x,x,x)$ of a process $x$ with zero average is
outperformed by alternative estimators. The new (Gauss-optimal) estimator
$\overline{x^3} - 3 \overline{x^2}\overline{x}(m-1)/(m+1)$ improves the
variance by a factor of up to $5/2$. Similarly, the estimator $\overline{x^2
z}$ for $C_3(x,x,z)$ can be replaced by another Gauss-optimal estimator. The
known estimator $\overline{xyz}$ for $C_3(x,y,z)$ as well as previously known
estimators for $C_2$ and $C_4$ of one average-free variable are shown to be
Gauss-optimal. As a side result of our work we present two simple recursive
formulas for finding multivariate cumulants from moments and vice versa.
