Conditionally specified models and dimension reduction in the exponential families

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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.

Original languageEnglish (US)
Pages (from-to)1574-1589
Number of pages16
JournalJournal of Multivariate Analysis
Volume99
Issue number8
DOIs
StatePublished - Sep 2008

Bibliographical note

Funding Information:
This research was supported by VA HSR&D Grant IIR 03-005.

Keywords

  • Conditional density ratios
  • Dimension reduction
  • Regression graphics
  • Sufficient summary

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