Stacked lensing estimators and their covariance matrices: Excess surface
  mass density vs. Lensing shear
  Stacked lensing is a powerful means of measuring the average mass
distribution around large-scale structure tracers. There are two stacked
lensing estimators used in the literature, denoted as $\Delta\Sigma$ and
$\gamma_+$, which are related as $\Delta\Sigma=\Sigma_{\rm cr}\gamma_+$, where
$\Sigma_{\rm cr}(z_l,z_s)$ is the critical surface mass density for each
lens-source pair ($z_l$ and $z_s$ are lens and source redshifts, respectively).
In this paper we derive a formula for the covariance matrix of
$\Delta\Sigma$-estimator focusing on `weight' function to improve the
signal-to-noise ($S/N$). We assume that the lensing fields and the distribution
of lensing objects obey the Gaussian statistics. With this formula, we show
that, if background galaxy shapes are weighted by an amount of $\Sigma_{\rm
cr}^{-2}(z_l,z_s)$, the $\Delta\Sigma$-estimator maximizes the $S/N$ in the
shot noise limited regime. We also show that the $\Delta\Sigma$-estimator with
the weight $\Sigma_{\rm cr}^{-2}$ gives a greater $(S/N)^2$ than that of the
$\gamma_+$-estimator by about 5--25\% for lensing objects at redshifts
comparable with or higher than the median of source galaxy redshifts for
hypothetical Subaru HSC and DES surveys. However, for low-redshift lenses such
as $z_l<0.3$, the $\gamma_+$-estimator has higher $(S/N)^2$ than
$\Delta\Sigma$. We also discuss that the $(S/N)^2$ for $\Delta\Sigma$ at large
separations in the sample variance limited regime can be boosted, by up to a
factor of 1.5, if one adopts a weight of $\Sigma_{\rm cr}^{-\alpha}$ with
$\alpha>2$. Our formula allows one to explore how the combination of the
different estimators can approach an optimal estimator in all regimes of
redshifts and separation scales.
