  The mark weighted correlation function (MCF) $W(s,\mu)$ is a computationally
efficient statistical measure which can probe clustering information beyond
that of the conventional 2-point statistics. In this work, we extend the
traditional mark weighted statistics by using powers of the density field
gradient $|\nabla \rho/\rho|^\alpha$ as the weight, and use the angular
dependence of the scale-averaged MCFs to constrain cosmological parameters. The
analysis shows that the gradient based weighting scheme is statistically more
powerful than the density based weighting scheme, while combining the two
schemes together is more powerful than separately using either of them.
Utilising the density weighted or the gradient weighted MCFs with $\alpha=0.5,\
1$, we can strengthen the constraint on $\Omega_m$ by factors of 2 or 4,
respectively, compared with the standard 2-point correlation function, while
simultaneously using the MCFs of the two weighting schemes together can be
$1.25$ times more statistically powerful than using the gradient weighting
scheme alone. The mark weighted statistics may play an important role in
cosmological analysis of future large-scale surveys. Many issues, including the
possibility of using other types of weights, the influence of the bias on this
statistics, as well as the usage of MCFs in the tomographic Alcock-Paczynski
method, are worth further investigations.
