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Measurement Error Models in Astronomy

Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 902)

Abstract

I discuss the effects of measurement error on regression and density estimation. I review the statistical methods that have been developed to correct for measurement error that are most popular in astronomical data analysis, discussing their advantages and disadvantages. I describe functional models for accounting for measurement error in regression, with emphasis on the methods of moments approach and the modified loss function approach. I then describe structural models for accounting for measurement error in regression and density estimation, with emphasis on maximum-likelihood and Bayesian methods. As an example of a Bayesian application, I analyze an astronomical data set subject to large measurement errors and a non-linear dependence between the response and covariate. I conclude with some directions for future research.

Keywords

Markov Chain Monte Carlo Active Galactic Nucleus Weighted Little Square Markov Chain Monte Carlo Algorithm Markov Chain Monte Carlo Sampler 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I would like to thank Anca Constantin for sharing her data set with me before publication, and Aneta Siemiginowska, Xiaohui Fan, and Tommaso Treu for helpful comments on an earlier version of this manuscript. I acknowledges support by NASA through Hubble Fellowship grant #HF-51243.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.

References

  1. 1.
    Akritas, M. G., & Bershady, M. A. 1996, T. Astrophys. J., 470, 706CrossRefGoogle Scholar
  2. 2.
    Andreon, S. 2006, Monthly Notic. of the Royal Astron. Soc., 369, 969Google Scholar
  3. 3.
    Bevington, P. R., & Robinson, D. K., Data Reduction and Error Analysis for the Physical Sciences, 3rd edn. (McGraw-Hill, New York, 2003)Google Scholar
  4. 4.
    Bovy, J., Hogg, D. W., & Roweis, S. T. 2009, arXiv:0905.2979Google Scholar
  5. 5.
    Carroll, R. J., Roeder, K., & Wasserman, L., 1999, Biometrics, 55, 44MATHCrossRefGoogle Scholar
  6. 6.
    Carroll, R. J., Ruppert, D., Stefanski, L. A., Crainiceanu, C. M., Measurement Error in Nonlinear Models: A Modern Perspective, 2nd edn. (Chapman & Hall/CRC, Boca Raton, 2006)MATHCrossRefGoogle Scholar
  7. 7.
    Cheng, C-L., & Van Ness, J. W., Statistical Regression with Measurement Error (Arnold, London, 1999)Google Scholar
  8. 8.
    Cheng, C-L., & Riu, J. 2006, Technometrics, 48, 511Google Scholar
  9. 9.
    Fuller, W. A., Measurement Error Models (John Wiley & Sons, New York, 1987)MATHCrossRefGoogle Scholar
  10. 10.
    Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B., Bayesian Data Analysis, 2nd edn. (Chapman & Hall/CRC, Boca Raton, 2004)MATHGoogle Scholar
  11. 11.
    Huang, X., Stefanski, L. A., & Davidian, M. 2006, Biometrika, 93, 53MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kashyap, V. L., van Dyk, D. A., Connors, A., Freeman, P. E., Siemiginowska, A., Xu, J., & Zezas, A. 2010, T. Astrophys. J., 719, 900CrossRefGoogle Scholar
  13. 13.
    Kelly, B.C., Fan, X. & Vestergaard, M. 2008, Astrophys. J., 682, 874CrossRefGoogle Scholar
  14. 14.
    Lee, H., et al. 2011, T. Astrophys. J., 731, 126CrossRefGoogle Scholar
  15. 15.
    Little, R. J. A., & Rubin, D. B. Statistical Analysis with Missing Data, 2nd ed. (John Wiley & Sons, Hoboken, 2002)MATHGoogle Scholar
  16. 16.
    Liu, J.S., Monte Carlo Strategies in Scientific Computing, (Springer, New York, 2004)Google Scholar
  17. 17.
    Patriota, A. G., & Bolfarine, H. 2008, T. Indian J. of Stat., 70, 267MathSciNetMATHGoogle Scholar
  18. 18.
    Patriota, A. G., Bolfarine, H., & de Castro, M. 2009, Statist. Method., 6, 408MATHCrossRefGoogle Scholar
  19. 19.
    Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P., Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge Unv. Press, New York, 2007)MATHGoogle Scholar
  20. 20.
    Robert, C. P., & Casella, G., Monte Carlo Statistical Methods, 2nd edn. (Springer, New York, 2004)MATHGoogle Scholar
  21. 21.
    Roy, S., Banerjee, T., 2006, Ann. Instit. Statist. Math., 58, 153MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Sprent, P. 1966, J. Royal Stat. Soc. Ser. B, 28, 278MathSciNetMATHGoogle Scholar
  23. 23.
    Tremaine, S., et al. 2002, T. Astrophys. J., 574, 740CrossRefGoogle Scholar
  24. 24.
    van Dyk, D. A., Connors, A., Kashyap, V. L., & Siemiginowska, A. 2001, T. Astrophys. J., 548, 224CrossRefGoogle Scholar
  25. 25.
    Van Dyk, D. A., DeGennaro, S., Stein, N., Jefferys, W. H., & von Hipple, T. 2009, T. Ann. of App. Stat., 3, 117MATHGoogle Scholar
  26. 26.
    Yu, Y., & Meng, X-L. 2011, to appear, J. of Comput. & Graph. Stat.,Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA

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