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.
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References
R.J. Adler, The Geometry of Random Fields (Wiley, New York, 1981)
R.J. Adler, J.E. Taylor, Random Fields and Their Geometry (Springer, New York, 2007)
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. 716–723
H. Bozdogan, Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52, 345–370 (1987)
D. Dacunha-Castelle, E. Gassiat, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. 1, 285–317 (1997)
H.H. Harman, Modern Factor Analysis, 3rd edn. (The University of Chicago Press, Chicago, 1976)
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)
H. Hotelling, Tubes and spheres in n-space a class of statistical problems. Am. J. Math. 61, 440–460 (1939)
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)
H. Weyl, On the volume of tubes. Am. J. Math. 61, 461–472 (1939)
S.S. Wilks, The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9, 60–62 (1938)
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Ninomiya, Y. (2014). Signal Detection and Model Selection. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_18
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DOI: https://doi.org/10.1007/978-4-431-55060-0_18
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