• Communications for Statistical Applications and Methods
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• 제9권3호
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• pp.765-774
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• 2002

Suppose one is given a vector X of a finite set of quantities $X_i$ which are independent Poisson random variables. A null hypothesis $H_0$ about E(X) is to be tested against an alternative hypothesis $H_1$. A quantity $\sum\limits_{i}w_ix_i$ is to be computed and used for the test. The optimal values of $W_i$ are calculated for three cases: (1) signal to noise ratio is used in the test, (2) normal approximations with unequal variances to the Poisson distributions are used in the test, and (3) the Poisson distribution itself is used. The above three cases are considered to the situations that are without background noise and with background noise. A comparison is made of the optimal values of $W_i$ in the three cases for both situations.

• Journal of the Korean Statistical Society
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• 제12권1호
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• pp.10-17
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• 1983

Let $X^{(n)}_j, j=1,2,\cdots,n, n=1,2,\cdots$ be a triangular array of random variables which arise naturally in a study of ferromagnetism in statistical mechanics. This paper presents weak convergence of random function $W_n(t)$, an appropriately normalized partial sum process based on $S^{(n)}_n = X^{(n)}_i+\cdot+X^{(n)}_n$. The limiting process W(t) is shown to be Gaussian when weak dependence exists. At the critical point where the change form weak to strong dependence takes place, W(t) turns out to be non-Gaussian. Our results are direct extensions of work by Ellis and Newmam (1978). An example is considered and the relation of these results to critical phenomena is briefly explained.

• Communications for Statistical Applications and Methods
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• 제21권3호
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• pp.245-251
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• 2014

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called ${\psi}$-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The ${\psi}$-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

• 대한수학회논문집
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• 제26권1호
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• pp.151-161
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• 2011

Let {$X_n$, $n{\geq}1$} be a sequence of asymptotically almost negatively associated random variables and $S_n=\sum^n_{i=1}X_i$. In the paper, we get the precise results of H$\acute{a}$jek-R$\acute{e}$nyi type inequalities for the partial sums of asymptotically almost negatively associated sequence, which generalize and improve the results of Theorem 2.4-Theorem 2.6 in Ko et al. ([4]). In addition, the large deviation of $S_n$ for sequence of asymptotically almost negatively associated random variables is studied. At last, the Marcinkiewicz type strong law of large numbers is given.

• Communications for Statistical Applications and Methods
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• 제6권2호
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• pp.585-594
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• 1999

Let {$X_{nj}$, 1$\leq$j$\leq$n,j$\geq$1} be a triangular array of random variables which are neither independent nor identically distributed. The almost sure convergences of randomly weighted partial sums of the form $$\sum_n^{j=1}$$ $W_{nj}$$X_{nj} are studied where {Wnj 1$\leq$j$\leq$n, j$\geq$1} is a triangular array of random weights. Application regarding the Efron bootstrap is also introduced.

• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.327-336
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• 2012

Let {$X_{ni}$, $i{\geq}1$, $n{\geq}1$} be an array of rowwise and pairwise negatively quadrant dependent random variables with mean zero, {$a_{ni}$, $i{\geq}1$, $n{\geq}1$} an array of weights and {$b_n$, $n{\geq}1$} an increasing sequence of positive integers. In this paper we consider some results concerning complete convergence of ${\sum}_{i=1}^{bn}a_{ni}X_{ni}$.

• Communications for Statistical Applications and Methods
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• 제16권5호
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• pp.841-849
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• 2009

Let {$X_n$, n ${\ge}$ 1} be a negatively associated sequence of identically distributed random variables with mean zeros and positive finite variances. Set $S_n$ = ${\Sigma}^n_{i=1}\;X_i$. Suppose that 0 < ${\sigma}^2=EX^2_1+2{\Sigma}^{\infty}_{i=2}\;Cov(X_1,\;X_i)$ < ${\infty}$. We prove that, if $EX^2_1(log^+{\mid}X_1{\mid})^{\delta}$ < ${\infty}$ for any 0< ${\delta}{\le}1$, then $\lim_{{\epsilon}\downarrow0}{\epsilon}^{2{\delta}}\sum_{{n=2}}^{\infty}\frac{(logn)^{\delta-1}}{n^2}ES^2_nI({\mid}S_n{\mid}\geq{\epsilon}{\sigma}\sqrt{nlogn}=\frac{E{\mid}N{\mid}^{2\delta+2}}{\delta}$, where N is the standard normal random variable. We also prove that if $S_n$ is replaced by $M_n=max_{1{\le}k{\le}n}{\mid}S_k{\mid}$ then the precise rate still holds. Some results in Fu and Zhang (2007) are improved to the complete moment case.

• 한국국방경영분석학회지
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• 제10권1호
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• pp.23-39
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• 1984

Consider a failure model for a stochastic system. A shock is any perturbation to the system which causes a random amount of damage to the system. Any of the shocks can cause the system to fail at shock times. The amount of damage at each shock is a function of the sum of the magnitudes of damage caused from all previous shocks. The times between shocks form a sequence of independent and identically distributed random variables. The system must be replaced upon failure at some cost but it also can be replaced before failure at a lower cost. The long term expected cost per unit time criterion is used. Structural relationships of the optimal replacement policy under the appropriate regularity conditions will be developed. And these relationships will provide theoretical background for the algorithm development.

• 한국경영과학회지
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• 제19권1호
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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.

• Communications for Statistical Applications and Methods
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• 제27권5호
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• pp.523-533
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• 2020

This paper suggests three available methods for finding nonnegative estimates of variance components of the random effects in mixed models. The three proposed methods based on the concepts of projections are called projection method I, II, and III. Each method derives sums of squares uniquely based on its own method of projections. All the sums of squares in quadratic forms are calculated as the squared lengths of projections of an observation vector; therefore, there is discussion on the decomposition of the observation vector into the sum of orthogonal projections for establishing a projection model. The projection model in matrix form is constructed by ascertaining the orthogonal projections defined on vector subspaces. Nonnegative estimates are then obtained by the projection model where all the coefficient matrices of the effects in the model are orthogonal to each other. Each method provides its own system of linear equations in a different way for the estimation of variance components; however, the estimates are given as the same regardless of the methods, whichever is used. Hartley's synthesis is used as a method for finding the coefficients of variance components.