Hyun Suk Park

Asymptotic Behavior of the Kernel Density Estimator from a Geometric Viewpoint

From the view of a geometric approach, we consider the problem of density estimation on the m-dimensional unit sphere by using the kernel method. The definition of the kernel estimator is motivated from the concept of the exponential map. This article shows that the asymptotic behavior of the estimator contains a geometric quantity (the sectional curvature) on the unit sphere. This implies that the behavior depends on whether the sectional curvature is positive or negative. Using observed data on normals to the orbital planes of long-period comets, numerical examples on the two-dimensional unit sphere are given.

Optimal Berry-Esseen bound for statistical estimations and its application to SPDE

We consider asymptotically normal statistics of the form F n / G n , where F n and G n are functionals of Gaussian fields. For these statistics, we establish an optimal Berry-Esseen bound for the Central Limit Theorem (CLT) of the sequence F n / G n is ź ( n ) in the following sense: there exist constants 0 < c < C < ∞ such that c ź d Kol ( F n / G n , Z ) / ź ( n ) ź C , where d Kol ( F n , Z ) = sup z ź R ź Pr ( F n ź z ) - Pr ( Z ź z ) ź . As an example, we find an optimal Berry-Esseen bound for the CLT of the maximum likelihood estimators for parameters occurring in parabolic stochastic partial differential equations.

Estimation of Hurst Parameter in Longitudinal Data with Long Memory

This paper considers the problem of estimation of the Hurst parameter H (1/2, 1) from longitudinal data with the error term of a fractional Brownian motion with Hurst parameter H that gives the amount of the long memory of its increment. We provide a new estimator of Hurst parameter H using a two scale sampling method based on -Sahalia and Jacod (2009). Asymptotic behaviors (consistent and central limit theorem) of the proposed estimator will be investigated. For the proof of a central limit theorem, we use recent results on necessary and sufficient conditions for multi-dimensional vectors of multiple stochastic integrals to converges in distribution to multivariate normal distribution studied by Nourdin et al. (2010), Nualart and Ortiz-Latorre (2008), and Peccati and Tudor (2005).

Mean distance of Brownian motion on a Riemannian manifold

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 for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng (Ann. Probab. 23(1) (1995) 173). 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.

Evaluation of the Efficiency of an Inverse Exponential Kernel Estimator for Spherical Data

This paper deals with the relative efficiency of two kernel estimators ˆ fn and ˆ gn by using spherical data, as proposed by Park (2012), and Bai et al. (1988), respectively. For this, we suggest the computing flows for the relative efficiency on the 2-dimensional unit sphere. An evaluation procedure between two estimators (given the same kernels) is also illustrated through the observed data on normals to the orbital planes of long-period comets.

The Expansion of Mean Distance of Brownian Motion on Riemannian Manifold

Abstract 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 be expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.