• Journal of the military operations research society of Korea
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• v.17 no.1
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• pp.43-60
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• 1991

Consider N queues without arrivals and with m identical servers. All jobs are independent and service requirements of jobs in a queue are i.i.d. random variables. At any time only one server may be assigned to a queue and switching between queues are allowed. A unit cost is imposed per job per unit time. The objective is to minimized the expected total cost. An flow approximation model is considered and an upperbound for the percentage error of best nonswitching policies to an optimal policy is found. It is shown that the best nonswitching policy is not worse than $11\%$ of an optimal policy For the stochastic model, we consider the case in which the service requirements of all jobs are i.i.d. with an exponential distribution. A longest first policy is shown to be optimal and a worst case analysis shows that the nonswitching policy which starts with the longest queues is not worse than $11\%$ of the optimal policy.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.243-247
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• 2014

Let {$X_n$, $n{\geq}1$} be a sequence of i.i.d. random variables with absolutely continuous cumulative distribution function(cdf) F(x) and the corresponding probability density function(pdf) f(x). In this paper, we give characterizations of Pareto and Weibull distribution by considering conditional expectations of record values.

• Journal of the Korean Statistical Society
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• v.14 no.2
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• pp.87-94
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• 1985

This paper extends the results of Ellis and Rosen (1982 a) to some more general integrals and applies our main theorem to compute the specific free energy of some models in statistical mechanics. The general integrals of this paper mean the integrals with respect to the probability measures induced by the sample mean of n i.i.d. random variables taking values in a separable Banach space.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.235-249
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• 2012

In the paper by Liu and Lin (Statist. Probab. Lett. 76 (2006), no. 16, 1787-1799), a new kind of precise asymptotics in the law of large numbers for the sequence of i.i.d. random variables, which includes complete convergence as a special case, was studied. This paper is devoted to the study of this new kind of precise asymptotics in the law of large numbers for moving average process under $\phi$-mixing assumption and some results of Liu and Lin [6] are extended to such moving average process.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.2
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• pp.47-54
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• 2001

Let {Xn, n = 1,2,${\cdots}$} be i.i.d. random variables with the only unknown parameters mean ${\mu}$ and variance a ${\sigma}^2$. We consider a sequential confidence interval C1 for the mean with coverage probability 1-${\alpha}$ and expected length of confidence interval $E_{\theta}$(Length of CI)/${\mid}{\mu}{\mid}{\leq}k$ (k : constant) and give some asymptotic properties of the stopping time in various limiting situations.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.859-869
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• 2006

Let {$X,\;X_n;n\;{\geq}\;1$} be a sequence of ${\imath}.{\imath}.d.$ random variables which belong to the attraction of the normal law, and $X^{(1)}_n,...,X^{(n)}_n$ be an arrangement of $X_1,...,X_n$ in decreasing order of magnitude, i.e., $\|X^{(1)}_n\|{\geq}{\cdots}{\geq}\|X^{(n)}_n\|$. Suppose that {${\gamma}_n$} is a sequence of constants satisfying some mild conditions and d'($t_{nk}$) is an appropriate truncation level, where $n_k=[{\beta}^k]\;and\;{\beta}$ is any constant larger than one. Then we show that the conditionally trimmed sums obeys the self-normalized law of the iterated logarithm (LIL). Moreover, the self-normalized LIL for conditionally censored sums is also discussed.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.429-437
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• 2013

Let $X_1$, $X_2$, ${\cdots}$, $X_n$ be independent and identically distributed random variables having common exponential density with unknown mean ${\mu}$. In the sequential confidence interval estimation for the exponential hazard rate ${\theta}=1/{\mu}$, when the loss function is strictly convex, the following stopping rule is proposed with the half length d of prescribed confidence interval $I_n$ for the parameter ${\theta}$; ${\tau}$ = smallest integer n such that $n{\geq}z^2_{{\alpha}/2}\hat{\theta}^2/d^2+2$, where $\hat{\theta}=(n-1)\bar{X}{_n}^{-1}/n$ is the minimum risk estimator for ${\theta}$ and $z_{{\alpha}/2}$ is defined by $P({\mid}Z{\mid}{\leq}{\alpha}/2)=1-{\alpha}({\alpha}{\in}(0,1))$ Z ~ N(0, 1). For the confidence intervals $I_n$ which is required to satisfy $P({\theta}{\in}I_n){\geq}1-{\alpha}$. These estimated intervals $I_{\tau}$ have the asymptotic consistency of the sequential procedure; $$\lim_{d{\rightarrow}0}P({\theta}{\in}I_{\tau})=1-{\alpha}$$, where ${\alpha}{\in}(0,1)$ is given.

• Journal of the Korean Operations Research and Management Science Society
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• v.19 no.1
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• pp.85-98
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• 1994

The distribution of random sum of i. i. d. exponential random variables is called GHP (Generalized Phase-Type) distribution. The class of GPH distributions is large enough to include PH (Phase-Type) distributions and has several properties which can be applied conveniently for computational purposes. In this paper, we show that any distribution difined on R$^{+}$ can be app-roximated by the GPH distribution and demonstrate the accuracy of the approximation through various numerical examples. Also, we introduce an efficient way to compute the delay and waiting various numerical examples. Also, we introduce an efficient way to compute the delay and waiting time distributions of the GPH/GPH/1 queueing system which can be used as an approximation model for the GI/G/1 system, and validate its accuracy through numerical examples. The theoretical and experimental results of this paper help us accept the usefulness of the approximations based on GPH distribution.n.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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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.

• Journal of the Korean Statistical Society
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• v.24 no.2
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• pp.337-347
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• 1995

In the problem of some sequential estimation, the stopping times may be written in the form $N(c) = inf{n \geq n_0; n \geq c^2 S^2_n/\delta^2 (\bar{X}_n)}$ where ${s^2_n}$ and ${\bar{X}_n}$ are the sequences of sample variance and sample mean of the independently and identically distributed (i.i.d.) random variables with distribution $F_{\theta}(x), \theta \in \Theta$, respectively, and $\delta$ is either constant or any given positive real valued function. We obtain some asymptotic normality and asymptotic expectation of the N(c) in various limiting situations. Specially, uniform asymptotic normality and uniform asymptotic expectation of the N(c) are given.