Moment-based ellipticity measurement as a statistical parameter estimation problem |
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| Institution: | 1. Astrophysics Research Group, Department of Physics, Meerut College, Meerut 250002, India;2. Uttrakhand Space Application Centre, Dehradun 248006, India;3. Department of Science and Math, Maharshi Dayanand H.S., Meerut 250626, India;1. Department of Physics, Ariel University, Ariel, POB 3, 40700, Israel;2. Ilia State University, E. Kharadze Abastumani Astrophysical Observatory, Kakutsa Cholokashvili ave. 3/5, Tbilisi, 0162, Georgia;1. Departamento de Ciencias Físicas, Universidad Andres Bello, Av. Republica 220, Santiago, Chile;2. Instituto Argentino de Radioastronomía, CIC, CONICET, C.C. 5 (1894) Villa Elisa, Pcia. de Buenos Aires, Argentina;1. Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China;2. University of Chinese Academy of Sciences, Beijing 100049, China;3. School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China;4. Qian Xuesen Laboratory of Space Technology, NO. 104, Youyi Road, Haidian District, Beijing 100094, China;5. National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China;6. School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550001, China |
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| Abstract: | We show that galaxy ellipticity estimation for weak gravitational lensing with unweighted image moments reduces to the problem of measuring a combination of the means of three independent normal random variables. Under very general assumptions, the intrinsic image moments of sources can be recovered from observations including effects such as the point-spread function and pixellation. Gaussian pixel noise turns these into three jointly normal random variables, the means of which are algebraically related to the ellipticity. We show that the random variables are approximately independent with known variances, and provide an algorithm for making them exactly independent. Once the framework is developed, we derive general properties of the ellipticity estimation problem, such as the signal-to-noise ratio, a generic form of an ellipticity estimator, and Cramér-Rao lower bounds for an unbiased estimator. We then derive the unbiased ellipticity estimator using unweighted image moments. We find that this unbiased estimator has a poorly behaved distribution and does not converge in practical applications, but demonstrates how to derive and understand the behaviour of new moment-based ellipticity estimators. |
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