Fast estimation of aperture mass statistics - I. Aperture mass variance and an application to the CFHTLenS data

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. 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 yields an answer that is in close agreement with the correlation function approach. We then apply this approach to the CFHTLenS as a pilot scheme and find that our method recovers to high accuracy the official result for the variance of both the E- and B-mode signal. 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 efficient methods for recovering higher order aperture mass statistics in linear order operations.