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Computational Statistics

, Volume 22, Issue 4, pp 599–617 | Cite as

Pooled marginal slicing approach via SIR α with discrete covariables

  • Benoît Liquet
  • Jérôme SaraccoEmail author
Original Paper

Abstract

In this paper, we consider a semiparametric regression model involving both p-dimensional quantitative covariable X and categorical predictor Z, and including a dimension reduction of X via K indices Xβ k . The dependent variable Y can be real or q-dimensional. We propose an approach based on SIR α and pooled marginal slicing methods in order to estimate the space spanned by the β k ’s. We establish \(\sqrt{n}\) -consistency of the proposed estimator. Simulation studies show the numerical qualities of our estimator.

Keywords

Discrete predictor Semiparametric regression model Sliced inverse regression (SIR) Pooled marginal slicing (PMS) 

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References

  1. Aragon Y (1997) A gauss implementation of multivariate sliced inverse regression. Comput Stat 12:355–372zbMATHMathSciNetGoogle Scholar
  2. Aragon Y, Saracco J (1997) Sliced inverse regression (SIR): an appraisal of small sample alternatives to slicing. Comput Stat 12:109–130MathSciNetGoogle Scholar
  3. Barreda L, Gannoun A, Saracco J (2003) Some extensions of multivariate SIR. J Stat Comput Simul 77(1):1–17CrossRefMathSciNetGoogle Scholar
  4. Bura E, Cook RD (2001a) Estimating the structural dimension of regressions via parametric inverse regression. J R Stat Soc Ser B 63:393–410zbMATHCrossRefMathSciNetGoogle Scholar
  5. Bura E, Cook RD (2001b) Extending sliced inverse regression: the weighted chi-squared test. J Am Stat Assoc 96:996–1003zbMATHCrossRefMathSciNetGoogle Scholar
  6. Carroll RJ, Li KC (1992) Measurement error regression with unknown link: dimension reduction and data visualization. J Am Stat Assoc 87:1040–1050zbMATHCrossRefMathSciNetGoogle Scholar
  7. Carroll RJ, Li KC (1995) Binary regressors in dimension reduction models: a new look at treatment comparisons. Stat Sin 5:667–688zbMATHMathSciNetGoogle Scholar
  8. Chiaromonte F, Cook RD, Li B (2002) Sufficient dimension reduction in regressions with categorical predictors. Ann Stat 30:475–497zbMATHCrossRefMathSciNetGoogle Scholar
  9. Cook RD, Weisberg S (1991) Discussion of “sliced inverse regression for dimension reduction”. J Am Stat Assoc 86:328–332CrossRefGoogle Scholar
  10. Duan N, Li KC (1991) Slicing regression: a link-free regression method. Ann Stat 19:505–530zbMATHMathSciNetGoogle Scholar
  11. Ferré L (1998) Determining the dimension in sliced inverse regression and related methods. J Am Stat Assoc 93:132–140zbMATHCrossRefGoogle Scholar
  12. Gannoun A, Saracco J (2003a) An asymptotic theory for SIRα method. Stat Sin 13:297–310zbMATHMathSciNetGoogle Scholar
  13. Gannoun A, Saracco J (2003b) Two cross validation criteria for SIRα and PSIRα methods in view of prediction. Comput Stat 18:585–603zbMATHMathSciNetGoogle Scholar
  14. Gather U, Hilker T, Becker C (2002) A note on outlier sensitivity of sliced inverse regression. Statistics 36:271–281zbMATHMathSciNetGoogle Scholar
  15. Hall P, Li KC (1993) On almost linearity of low-dimensional projections from high-dimensional data. Ann Stat 21:867–889zbMATHMathSciNetGoogle Scholar
  16. Hsing T, Carroll RJ (1992) An asympotic theory for sliced inverse regression. Ann Stat 20:1040–1061zbMATHMathSciNetGoogle Scholar
  17. Hsing T (1999) Nearest neighbor inverse regression. Ann Stat 27:697–731zbMATHCrossRefMathSciNetGoogle Scholar
  18. Kötter T (1996) An asymptotic result for sliced inverse regression. Comput Stat 11:113–136zbMATHGoogle Scholar
  19. Kötter T (2000) Sliced inverse regression. In: Schimek MG (ed) Smoothing and regression. Approaches, computation, and application. Wiley, New York, pp 497–512Google Scholar
  20. Li KC (1991) Sliced inverse regression for dimension reduction (with discussion). J Am Stat Assoc 86: 316–342zbMATHCrossRefGoogle Scholar
  21. Li KC, Aragon Y, Shedden K, Thomas Agnan C (2003) Dimension reduction for multivariate response data. J Am Stat Assoc 98:99–109zbMATHCrossRefMathSciNetGoogle Scholar
  22. Saracco J (1997) An asymptotic theory for sliced inverse regression. Commun Stat Theory Methods 26:2141–2171zbMATHCrossRefMathSciNetGoogle Scholar
  23. Saracco J (1999) Sliced inverse regression under linear constraints. Commun Stat Theory Methods 28(10):2367–2393zbMATHCrossRefMathSciNetGoogle Scholar
  24. Saracco J (2001) Pooled slicing methods versus slicing methods. Commun Stat Simul Comput 30:489–511zbMATHCrossRefMathSciNetGoogle Scholar
  25. Saracco J (2005) Asymptotics for pooled marginal slicing estimator based on SIRα. J Multivariate Anal 96:117–135zbMATHCrossRefMathSciNetGoogle Scholar
  26. Schott JR (1994) Determining the dimensionality in sliced inverse regression. J Am Stat Assoc 89:141–148zbMATHCrossRefMathSciNetGoogle Scholar
  27. Zhu LX, Fang KT (1996) Asymptotics for kernel estimate of sliced inverse regression. Ann Stat 24: 1053–1068zbMATHCrossRefMathSciNetGoogle Scholar
  28. Zhu LX, Ng KW (1995) Asymptotics of sliced inverse regression. Stat Sin 5:727–736zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.ISPED, INSERM U875Université Victor Segalen Bordeaux 2Bordeaux CedexFrance
  2. 2.GRETha, UMR CNRS 5113Université de Montesquieu Bordeaux 4Pessac CedexFrance

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