• Communications for Statistical Applications and Methods
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• v.24 no.6
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• pp.663-671
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• 2017

Some genetic association tests include an unidentifiable nuisance parameter under the null hypothesis of no association. When the mode of inheritance (MOI) is not specified in a case-control design, the Cochran-Armitage (CA) trend test contains an unidentifiable nuisance parameter. The transmission disequilibrium test (TDT) in a family-based association study that includes the unaffected also contains an unidentifiable nuisance parameter. The hypothesis tests that include an unidentifiable nuisance parameter are typically performed by taking a supremum of the CA tests or TDT over reasonable values of the parameter. The p-values of the supremum test statistics cannot be obtained by a normal or chi-square distribution. A common method is to use a Davies's upper bound of the p-value instead of an exact asymptotic p-value. In this paper, we provide a unified sine-cosine process expression of the CA trend test that does not specify the MOI and the TDT that includes the unaffected. We also present a closed form expression of the exact asymptotic formulas to calculate the p-values of the supremum tests when the score function can be written as a linear form in an unidentifiable parameter. We illustrate how to use the derived formulas using a pharmacogenetics case-control dataset and an attention deficit hyperactivity disorder family-based example.

• 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 Biomedical Engineering Research
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• v.40 no.2
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• pp.62-67
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• 2019

The purpose of this study was to evaluate the performance of deep neural network model in order to determine whether there is a risk factor for coronary artery disease based on the cardiac variation parameter. The study used unidentifiable 297 data to evaluate the performance of the model. Input data consists of heart rate parameters, which are SDNN (standard deviation of the N-N intervals), PSI (physical stress index), TP (total power), VLF (very low frequency), LF (low frequency), HF (high frequency), RMSSD (root mean square of successive difference) APEN (approximate entropy) and SRD (successive R-R interval difference), the age group and sex. Output data are divided into normal and patient groups, and the patient group consists of those diagnosed with diabetes, high blood pressure, and hyperlipidemia among the various risk factors that can cause coronary artery disease. Based on this, a binary classification model was applied using Deep Neural Network of deep learning techniques to classify normal and patient groups efficiently. To evaluate the effectiveness of the model used in this study, Kernel SVM (support vector machine), one of the classification models in machine learning, was compared and evaluated using same data. The results showed that the accuracy of the proposed deep neural network was train set 91.79% and test set 85.56% and the specificity was 87.04% and the sensitivity was 83.33% from the point of diagnosis. These results suggest that deep learning is more efficient when classifying these medical data because the train set accuracy in the deep neural network was 7.73% higher than the comparative model Kernel SVM.