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Conditionally specified models and dimension reduction in the exponential families

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  • Noorbaloochi, Siamak
  • Nelson, David

Abstract

We consider informative dimension reduction for regression problems with random predictors. Based on the conditional specification of the model, we develop a methodology for replacing the predictors with a smaller number of functions of the predictors. We apply the method to the case where the inverse conditional model is in the linear exponential family. For such an inverse model and the usual Normal forward regression model it is shown that, for any number of predictors, the sufficient summary has dimension two or less. In addition, we develop a test of dimensionality. The relationship of our method with the existing dimension reduction theory based on the marginal distribution of the predictors is discussed.

Suggested Citation

  • Noorbaloochi, Siamak & Nelson, David, 2008. "Conditionally specified models and dimension reduction in the exponential families," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1574-1589, September.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:8:p:1574-1589
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    References listed on IDEAS

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    1. Li K-C. & Aragon Y. & Shedden K. & Thomas Agnan C., 2003. "Dimension Reduction for Multivariate Response Data," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 99-109, January.
    2. Weisberg, Sanford, 2002. "Dimension Reduction Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 7(i01).
    3. Cook, R. Dennis & Ni, Liqiang, 2005. "Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 410-428, June.
    4. Francesca Chiaromonte & R. Cook, 2002. "Sufficient Dimension Reduction and Graphics in Regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 768-795, December.
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    Cited by:

    1. Nelson, David & Noorbaloochi, Siamak, 2013. "Information preserving sufficient summaries for dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 347-358.
    2. Noorbaloochi Siamak & Nelson David & Asgharian Masoud, 2010. "Balancing and Elimination of Nuisance Variables," The International Journal of Biostatistics, De Gruyter, vol. 6(2), pages 1-22, February.

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