Signal Detection and Model Selection

Chapter
Part of the Mathematics for Industry book series (MFI, volume 5)

Abstract

Signal detection is a basic statistical problem in various fields including engineering, econometrics and psychometrics. It is performed by statistical testing or model selection, but we cannot apply conventional statistical theory to it. The reason is that the signal model, a statistical model for signal detection, has an irregularity, called non-identifiability. Because of this non-identifiability problem, the signal model needs to be shrunk in its geometrical representation. After drawing it, we prove there is an asymptotic property of the likelihood ratio statistics for the model, which is indicated by the geometrical representation. Then, on the basis of this asymptotic property, we introduce a criterion for model selection considering non-identifiability that is a reevaluated Akaike information criterion (AIC). We check the validity of the reevaluated AIC through simulation studies and real data analysis using a factor analysis model, which can be regarded as a kind of signal model.

Keywords

Factor analysis Information criterion Likelihood ratio Locally conic parameterization Non-identifiability 

References

  1. 1.
    R.J. Adler, The Geometry of Random Fields (Wiley, New York, 1981)MATHGoogle Scholar
  2. 2.
    R.J. Adler, J.E. Taylor, Random Fields and Their Geometry (Springer, New York, 2007)Google Scholar
  3. 3.
    H. Akaike, in Information Theory and an Extension of the Maximum Likelihood Principle, ed. by B.N. Petrov, F. Csaki. 2nd International Symposium on Information Theory (Akademiai Kiado, Budapest, 1973), pp. 716–723Google Scholar
  4. 4.
    H. Bozdogan, Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52, 345–370 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    D. Dacunha-Castelle, E. Gassiat, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. 1, 285–317 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H.H. Harman, Modern Factor Analysis, 3rd edn. (The University of Chicago Press, Chicago, 1976)Google Scholar
  7. 7.
    K.J. Holzinger, F. Swineford, A study in factor analysis: the stability of a bi-factor solution, in Supplementary Educational Monographs, vol. 48 (University Chicago Press, Chicago, 1939)Google Scholar
  8. 8.
    H. Hotelling, Tubes and spheres in n-space a class of statistical problems. Am. J. Math. 61, 440–460 (1939)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Ninomiya, H. Yanagihara, K.-H. Yuan, Selecting the number of factors in exploratory factor analysis via locally conic parameterization. ISM Research Memorandum, 1078 (2008)Google Scholar
  10. 10.
    H. Weyl, On the volume of tubes. Am. J. Math. 61, 461–472 (1939)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.S. Wilks, The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9, 60–62 (1938)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityNishikuJapan

Personalised recommendations