  We explore an alternative method to the usual shear correlation function
approach for the estimation of aperture mass statistics in weak lensing survey
data. Our approach builds on the direct estimator method of Schneider (1998).
In this paper, to test and validate the methodology, we focus on the aperture
mass dispersion. After computing the signal and noise for a weighted set of
measured ellipticites we show how the direct estimator can be made into a
linear order algorithm that enables a fast and efficient computation. We then
investigate the applicability of the direct estimator approach in the presence
of a real survey mask with holes and chip gaps. For this we use a large
ensemble of full ray-tracing mock simulations. By using various weighting
schemes for combining information from different apertures we find that inverse
variance weighting the individual aperture estimates with an aperture
completeness greater than 70 per cent coverage yields an answer that is in
close agreement with the standard correlation function approach. We then apply
this approach to the CFHTLenS as pilot scheme and find that our method recovers
to high accuracy the Kilbinger (2013) result for the variance of both the E and
B mode signal, after we correct the catalogue for the shear bias in the lensfit
algorithm for pairs closer than 9". We then explore the cosmological
information content of the direct estimator using the Fisher information
approach. We show that there is a only modest loss in cosmological information
from the rejection of apertures that are of low completeness. This method
unlocks the door to fast and efficient methods for recovering higher order
aperture mass statistics in linear order operations.
