The Annals of Applied Statistics

Photo-$z$ estimation: An example of nonparametric conditional density estimation under selection bias

Rafael Izbicki, Ann B. Lee, and Peter E. Freeman

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Abstract

Redshift is a key quantity for inferring cosmological model parameters. In photometric redshift estimation, cosmologists use the coarse data collected from the vast majority of galaxies to predict the redshift of individual galaxies. To properly quantify the uncertainty in the predictions, however, one needs to go beyond standard regression and instead estimate the full conditional density $f(z|\mathbf{x})$ of a galaxy’s redshift $z$ given its photometric covariates $\mathbf{x}$. The problem is further complicated by selection bias: usually only the rarest and brightest galaxies have known redshifts, and these galaxies have characteristics and measured covariates that do not necessarily match those of more numerous and dimmer galaxies of unknown redshift. Unfortunately, there is not much research on how to best estimate complex multivariate densities in such settings.

Here we describe a general framework for properly constructing and assessing nonparametric conditional density estimators under selection bias, and for combining two or more estimators for optimal performance. We propose new improved photo-$z$ estimators and illustrate our methods on data from the Sloan Data Sky Survey and an application to galaxy–galaxy lensing. Although our main application is photo-$z$ estimation, our methods are relevant to any high-dimensional regression setting with complicated asymmetric and multimodal distributions in the response variable.

Article information

Source
Ann. Appl. Stat., Volume 11, Number 2 (2017), 698-724.

Dates
Received: April 2016
Revised: January 2017
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1500537720

Digital Object Identifier
doi:10.1214/16-AOAS1013

Mathematical Reviews number (MathSciNet)
MR3693543

Zentralblatt MATH identifier
06775889

Keywords
Density estimation nonparametric statistics selection bias photometric redshift

Citation

Izbicki, Rafael; Lee, Ann B.; Freeman, Peter E. Photo-$z$ estimation: An example of nonparametric conditional density estimation under selection bias. Ann. Appl. Stat. 11 (2017), no. 2, 698--724. doi:10.1214/16-AOAS1013. https://projecteuclid.org/euclid.aoas/1500537720


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Supplemental materials

  • Supplement to “Photo-$z$ estimation: An example of nonparametric conditional density estimation under selection bias”. We provide the data and code used in the paper as supplementary material.