Criteria for selection of response variables and the asymptotic properties in a multivariate calibration

  • Ryuei Nishii


Let a set ofp responsesy=(y1,...yp)′ has a multivariate linear regression on a set ofq explanatory variablesx=(x1,...xq)′. Our aim is to select the most informative subset of responses for making inferences about an unknownx from an observedy. Under normality ony, two selection methods, based on the asymptotic mean squared error and on the Akaike's information criterion, are proposed by Fujikoshi and Nishii (1986,Hiroshima Math. J.,16, 269–277). In this paper, under a mild condition we will derive the cross-validation criterion and obtain the asymptotic properties of the three procedures.

Key words

AIC calibration cross-validation multivariate linear regression variable selection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle,2nd International Symposium on Information Theory, 267–281, (eds. B. N. Petrov and F. Czáki), Akademiai Kiadó, Budapest.zbMATHGoogle Scholar
  2. [2]
    Brown, P. J. (1982). Multivariate calibration,J. R. Statist. Soc., B,44, 287–321.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Fujikoshi, Y. (1983). A criterion for variable selection in multiple discriminant analysis,Hiroshima Math. J.,13, 203–214.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Fujikoshi, Y. and Nishii, R. (1986). Selection of variables in a multivariate inverse regression problem,Hiroshima Math. J.,16, 269–277.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Lwin, T. and Maritz, J. S. (1982). An analysis of the linear-calibration controversy from the perspective of compound estimation,Technometrics,24, 235–242.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Nishii, R. (1984). Asymptotic properties of criteria for selection of variables in multiple regression,Ann. Statist.,12, 2, 758–765.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Rao, C. R. (1965).Linear Statistical Inference and Its Applications, Wiley, New York.zbMATHGoogle Scholar
  8. [8]
    Shibata, R. (1976). Selection of the order of an autoregressive model by Akaike's information criterion,Biometrika,63, 117–126.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Shukla, G. K. (1972). On the problem of calibration,Technometrics,14, 547–553.CrossRefGoogle Scholar
  10. [10]
    Williams, E. J. (1959).Regression Analysis, Wiley, New York.zbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1986

Authors and Affiliations

  • Ryuei Nishii
    • 1
  1. 1.Hiroshima UniversityHiroshimaJapan

Personalised recommendations