## Building unbiased estimators from non-gaussian likelihoods with application to shear estimation

## Abstract

We develop a general framework for generating estimators of a given quantity which are unbiased to a given order in the difference between the true value of the underlying quantity and the fiducial position in theory space around which we expand the likelihood. We apply this formalism to rederive the optimal quadratic estimator and show how the replacement of the second derivative matrix with the Fisher matrix is a generic way of creating an unbiased estimator (assuming choice of the fiducial model is independent of data). Next we apply the approach to estimation of shear lensing, closely following the work of Bernstein and Armstrong (2014). Our first order estimator reduces to their estimator in the limit of zero shear, but it also naturally allows for the case of non-constant shear and the easy calculation of correlation functions or power spectra using standard methods. Both our first-order estimator and Bernstein and Armstrong’s estimator exhibit a bias which is quadratic in true shear. Our third-order estimator is, at least in the realm of the toy problem of Bernstein and Armstrong, unbiased to 0.1% in relative shear errors Δg/g for shears up to |g| = 0.2.

- Authors:

- Stony Brook Univ., NY (United States)
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Publication Date:

- Research Org.:
- Brookhaven National Laboratory (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1183831

- Report Number(s):
- BNL-107853-2015-JA

Journal ID: ISSN 1475-7516; KA2301020; TRN: US1500515

- Grant/Contract Number:
- SC00112704

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2015; Journal Issue: 1; Journal ID: ISSN 1475-7516

- Publisher:
- Institute of Physics (IOP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS

### Citation Formats

```
Madhavacheril, Mathew S., McDonald, Patrick, Sehgal, Neelima, and Slosar, Anze. Building unbiased estimators from non-gaussian likelihoods with application to shear estimation. United States: N. p., 2015.
Web. doi:10.1088/1475-7516/2015/01/022.
```

```
Madhavacheril, Mathew S., McDonald, Patrick, Sehgal, Neelima, & Slosar, Anze. Building unbiased estimators from non-gaussian likelihoods with application to shear estimation. United States. doi:10.1088/1475-7516/2015/01/022.
```

```
Madhavacheril, Mathew S., McDonald, Patrick, Sehgal, Neelima, and Slosar, Anze. Thu .
"Building unbiased estimators from non-gaussian likelihoods with application to shear estimation". United States. doi:10.1088/1475-7516/2015/01/022. https://www.osti.gov/servlets/purl/1183831.
```

```
@article{osti_1183831,
```

title = {Building unbiased estimators from non-gaussian likelihoods with application to shear estimation},

author = {Madhavacheril, Mathew S. and McDonald, Patrick and Sehgal, Neelima and Slosar, Anze},

abstractNote = {We develop a general framework for generating estimators of a given quantity which are unbiased to a given order in the difference between the true value of the underlying quantity and the fiducial position in theory space around which we expand the likelihood. We apply this formalism to rederive the optimal quadratic estimator and show how the replacement of the second derivative matrix with the Fisher matrix is a generic way of creating an unbiased estimator (assuming choice of the fiducial model is independent of data). Next we apply the approach to estimation of shear lensing, closely following the work of Bernstein and Armstrong (2014). Our first order estimator reduces to their estimator in the limit of zero shear, but it also naturally allows for the case of non-constant shear and the easy calculation of correlation functions or power spectra using standard methods. Both our first-order estimator and Bernstein and Armstrong’s estimator exhibit a bias which is quadratic in true shear. Our third-order estimator is, at least in the realm of the toy problem of Bernstein and Armstrong, unbiased to 0.1% in relative shear errors Δg/g for shears up to |g| = 0.2.},

doi = {10.1088/1475-7516/2015/01/022},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 1,

volume = 2015,

place = {United States},

year = {2015},

month = {1}

}

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