• Journal of the Korean Statistical Society
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• v.30 no.2
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• pp.179-192
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• 2001

When a statistical model has a hierarchical structure such as multilayer perceptrons in neural networks or Gaussian mixture density representation, the model includes distribution with unidentifiable parameters when the structure becomes redundant. Since the exact structure is unknown, we need to carry out statistical estimation or learning of parameters in such a model. From the geometrical point of view, distributions specified by unidentifiable parameters become a singular point in the parameter space. The problem has been remarked in many statistical models, and strange behaviors of the likelihood ratio statistics, when the null hypothesis is at a singular point, have been analyzed so far. The present paper studies asymptotic behaviors of the maximum likelihood estimator and the Bayesian predictive estimator, by using a simple cone model, and show that they are completely different from regular statistical models where the Cramer-Rao paradigm holds. At singularities, the Fisher information metric degenerates, implying that the cramer-Rao paradigm does no more hold, and that he classical model selection theory such as AIC and MDL cannot be applied. This paper is a first step to establish a new theory for analyzing the accuracy of estimation or learning at around singularities.

• Journal of the Korean Institute of Telematics and Electronics
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• v.26 no.9
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• pp.1322-1331
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• 1989

This paper proposes a new adaptation algorithm with which a model reference adaptive control can give a local boundedness of the tracking error applied to a continuous-time linear time-invariant single-input single-output plant whose reduced-order model is of relative degree 1 and whose unmodeled dynamics may be represented in a sigular perturbation form. With the addition of an offset term and an extra adaptation structure, this algorithm is shown to have a robustness property in the sense that this gives zero residual tracking errors when the unmodeled dynamics are disappeared.