• Communications for Statistical Applications and Methods
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• 제21권5호
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• pp.423-433
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• 2014

Test statistics using cumulative sums of residuals have been widely used in various regression models including generalized linear models(GLM). Recently, Pan and Lin (2005) extended this testing procedure to the generalized linear mixed models(GLMM) having random effects, in which we encounter difficulties in computing the marginal likelihood that is expressed as an integral of random effects distribution. The Gaussian quadrature algorithm is commonly used to approximate the marginal likelihood. Many commercial statistical packages provide an option to apply this type of goodness-of-fit test in GLMs but available programs are very rare for GLMMs. We suggest a computational algorithm to implement the testing procedure in GLMMs by a freely accessible R package, and also illustrate through practical examples.

• 대한수학회논문집
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• 제33권2호
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• pp.605-618
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• 2018

The main goal of this paper is to study an extension of random summations of independent and identically distributed random variables when the number of summands in random summation is a partial sum of n independent, identically distributed, non-negative integer-valued random variables. Some characterizations of random summations are considered. The central limit theorems and weak law of large numbers for extended random summations are established. Some weak limit theorems related to geometric random sums, binomial random sums and negative-binomial random sums are also investigated as asymptotic behaviors of extended random summations.

• 대한수학회보
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• 제52권5호
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• pp.1729-1735
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• 2015

It is known that the partial sum of the Taylor series of an holomorphic function of one complex variable converges in norm on $H^p(\mathbb{D})$ for 1 < p < ${\infty}$. In this paper, we consider various type of partial sums of a holomorphic function of several variables which also converge in norm on $H^p(\mathbb{B}_n)$ for 1 < p < ${\infty}$. For the partial sums in several variable cases, some variables could be chosen slowly (fastly) relative to other variables. We prove that in any cases the partial sum converges to the original function, regardlessly how slowly (fastly) some variables are taken.

• 대한수학회보
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• 제46권4호
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• pp.617-626
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• 2009

Let {$Y_i$,-$\infty$ < i < $\infty$} be a doubly infinite sequence of i.i.d. random variables with E|$Y_1$| < $\infty$, {$a_{ni}$,-$\infty$ < i < $\infty$ n $\geq$ 1} an array of real numbers. Under some conditions on {$a_{ni}$}, we obtain necessary and sufficient conditions for $\sum\;_{n=1}^{\infty}\frac{1}{n}P(|\sum\;_{i=-\infty}^{\infty}a_{ni}(Y_i-EY_i)|$>$n{\epsilon})$<{\infty}$. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.

• 대한수학회보
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• 제51권1호
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• pp.29-41
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• 2014

In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)+${\cdots}$+f(nx) = 0, with $n{\geq}2$, which are related to the partial sums of the Riemann zeta function. These curves ${\alpha}$-densify a large class of compact sets of the plane for arbitrary small ${\alpha}$, extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the $n^{th}$ power of the density approaches the Jordan content of the compact set which the curve densifies.

• 대한수학회지
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• 제45권6호
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• pp.1601-1611
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• 2008

Let X, $X_1$, $X_2$, ... be i.i.d. random variables with zero means, variance one, and set $S_n\;=\;{\sum}^n_{i=1}\;X_i$, $n\;{\geq}\;1$. Gut and $Sp{\check{a}}taru$ [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P($|S_n|\;{\geq}\;{\varepsilon}{\sqrt{2n\;{\log}\;{\log}\;n}}$) is replaced by E{$|S_n|/{\sqrt{n}}-{\varepsilon}{\sqrt{2\;{\log}\;{\log}\;n}$}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.

• 대한수학회지
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• 제53권1호
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• pp.45-55
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• 2016

Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

• 대한수학회논문집
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• 제20권2호
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• pp.221-229
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• 2005

In this paper we consider the decompositions of subdirect sums and direct sums in bounded BCK-algebras. The main results are as follows. Given a bounded BCK-algebra X, if X can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ of a nonzero ideal family $\{A_i\;{\mid}\;i{\in}I\}$ of X, then I is finite, every $A_i$ is bounded, and X is embeddable in the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X is with condition (S), then it can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ if and only if it can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$, then it is isomorphic to the direct product $\prod_{i{\in}I}A_i$.

• Journal of the Korean Statistical Society
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• 제25권2호
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• pp.175-183
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• 1996

Let $S_n,n$ = 1, 2,... denote the partial sums of not necessarily in-dependent random variables. Let N(c) = min${ n ; S_n > c}$, c $\geq$ 0. Theorem 2 states that N (c), (suitably normalized), tends to 0 in p-mean, 1 $\leq$ p < 2, as c longrightarrow $\infty$ under mild conditions, which generalizes earlier result by Gut(1974). The proof follows by applying Theorem 1, which generalizes the known result $E$\mid$S_n$\mid$^p$ = o(n), 0 < p< 2, as n .rarw..inf. to randomly indexed partial sums.

• 한국언어정보학회지:언어와정보
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• 제12권2호
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• pp.115-137
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• 2008

We examine the meanings of two plural forms in Korean. Kwak (2003) and Jun (2004) claimed that unmarked (i.e. zero) plural forms denote groups and 'deul' plural forms denote sums, and Kang (2007) indicated problems in such claims. Soon after, Jun (2007) raised an objection to Kang's view by slightly modifying his former analysis, and now this paper is a reply to Jun (2007). Jun's revision of his former view is one step forward but still it is faced with many authentic counter examples. In the mean time, Noh (2008) raised an objection to Kang (2007) from a totally different direction, claiming that unmarked forms are not plural but only singular. This position is not sound, either. We have so many cases where unmarked forms are used with plural interpretation. All in all, our view that unmarked forms denote singular as well plural individuals(sums, groups) and 'deul' forms denote only plural individuals(sums, groups) is to be maintained.