• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.171-180
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• 2009

In this paper, we give three criteria for non-polynomial functions interpolated from the set of univariate polynomials of degree less than m over a finite field to be a permutation on the same set.

• Journal of the Korean Statistical Society
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• v.25 no.4
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• pp.499-514
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• 1996

Consider an additive regression model of Y on X = (X$_1$,X$_2$,. . .,$X_p$), Y = $sum_{j=1}^pf_j(X_j) + $\varepsilon$$, where $f_j$s are smooth functions to be estimated and $\varepsilon$ is a random error. If $X_j$s are fixed design points, we call it the fixed design additive model. Since the response variable Y is observed at fixed p-dimensional design points, the behavior of the nonparametric regression estimator depends on the design. We propose a fixed design called permutation fixed design, and fit the regression function by the kernel method. The estimator in the permutation fixed design achieves the univariate optimal rate of convergence in mean squared error for any p $\geq$ 2.

• Honam Mathematical Journal
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• v.9 no.1
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• pp.113-119
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• 1987

Random variables $X_1$, $X_1$, ..., $X_m$ are said to be weakly associated if whenever $\pi$ is a permutation of {1, 2,..., m}, $1{\leq}k<m$, and f: $R^{k}{\rightarrow}R$, g: $R^{m-k}{\rightarrow}R$ are coordinatewise nondecreasing functions then Cov $[f(X_{x(1)},...,\;X_{\pi(k)},\;g(X_{x(k+1)},...,\;X_{x(m)})]{\geq}0$, whenever the covariance is defined. An infinite collection of random variables is weakly associated if every finite subcollection is weakly associated. The basic properties of weak association and central limit theorem for weakly associated random variables are derived. We also extend this idea to point random fields and prove that a Cox process with a stationary weakly associated intensity rardom measure is weakly associated. Another inequalities and the fact that positive correlated normal random variables are weakly associated are also proved.