• Honam Mathematical Journal
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• v.46 no.2
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• pp.224-236
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• 2024

This study inspects the structure of CR-product of a holomorphic statistical manifold. Findings concerning geodesic submanifolds and totally geodesic foliations in the context of dual connections have been demonstrated. The integrability of distributions in CR-statistical submanifolds has been characterized. The statistical version of CR-product in the holomorphic statistical manifold has been researched. Additionally, some assertions for curvature tensor field of the holomorphic statistical manifold have been substantiated.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.432-446
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• 2022

This article introduces the structure of GCR-lightlike submanifolds of an indefinite Kaehler statistical manifold and derives their geometric properties. The characterizations on totally geodesic, mixed geodesic, D-geodesic and D'-geodesic GCR-lightlike submanifolds have also been obtained.

• Korean Journal of Mathematics
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• v.31 no.2
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• pp.133-137
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• 2023

Consider a Riemannian manifold M equipped with a metric g. In this article, we study a notion for statistical manifolds (M, g, ∇), which can have a nonzero torsion, abbreviated to SMT. Then it turns out that the tensor fields ∇g and ${\tilde{\nabla}}g$ obtained from two different SMT-connections are different.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.45-48
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• 2002

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of Stochastic Differential Equation(SDE) for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng(1995). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.

• Communications for Statistical Applications and Methods
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• v.29 no.6
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• pp.721-733
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• 2022

In this paper we compare parameter estimation by Grassmann manifold optimization and sequential candidate set algorithm in a structured principal fitted component (PFC) model. The structured PFC model extends the form of the covariance matrix of a random error to relieve the limits that occur due to too simple form of the matrix. However, unlike other PFC models, structured PFC model does not have a closed form for parameter estimation in dimension reduction which signals the need of numerical computation. The numerical computation can be done through Grassmann manifold optimization and sequential candidate set algorithm. We conducted numerical studies to compare the two methods by computing the results of sequential dimension testing and trace correlation values where we can compare the performance in determining dimension and estimating the basis. We could conclude that Grassmann manifold optimization outperforms sequential candidate set algorithm in dimension determination, while sequential candidate set algorithm is better in basis estimation when conducting dimension reduction. We also applied the methods in real data which derived the same result.

• Communications for Statistical Applications and Methods
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• v.22 no.4
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• pp.377-387
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• 2015

The area under the ROC curve can be represented by both Mann-Whitney and Wilcoxon rank sum statistics. Consider an ROC surface and manifold equal to three dimensions or more. This paper finds that the volume under the ROC surface (VUS) and the hypervolume under the ROC manifold (HUM) could be derived as functions of both conditional Mann-Whitney statistics and conditional Wilcoxon rank sum statistics. The nullhypothesis equal to three distribution functions or more are identical can be tested using VUS and HUM statistics based on the asymptotic large sample theory of Wilcoxon rank sum statistics. Illustrative examples with three and four random samples show that two approaches give the same VUS and $HUM^4$. The equivalence of several distribution functions is also tested with VUS and $HUM^4$ in terms of conditional Wilcoxon rank sum statistics.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.487-493
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• 2011

The Fisher information matrix plays a significant role in statistical inference in connection with estimation and properties of variance of estimators. In this paper, we define the parameter space of the t-manifold using its Fisher's matrix and characterize the t-manifold from the viewpoint of information geometry. The ${\alpha}$-scalar curvatures to the t-manifold are calculated.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1321-1336
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• 2018

In this paper, we obtain some necessary and sufficient conditions for the Reeb vector field of a trans-Sasakian 3-manifold to be minimal or harmonic. We construct some examples to illustrate main results. As applications of the above results, we obtain some new characteristic conditions under which a compact trans-Sasakian 3-manifold is homothetic to either a Sasakian or cosymplectic 3-manifold.

• Proceedings of the Korean Statistical Society Conference
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• 2003.05a
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• pp.37-42
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• 2003

We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can he expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.793-805
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• 2017

In this paper, we prove that the Ricci tensor of a three-dimensional almost Kenmotsu manifold satisfying ${\nabla}_{\xi}h=0$, $h{\neq}0$, is ${\eta}$-parallel if and only if the manifold is locally isometric to either the Riemannian product $\mathbb{H}^2(-4){\times}\mathbb{R}$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.