• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Communications for Statistical Applications and Methods
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• v.10 no.2
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• pp.585-594
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• 2003

It has been generally recognized that conventional binomial or Poisson model provides poor fits to the actual correlated binary data due to the extra-binomial variation. A number of generalized statistical models have been proposed to account for this additional variation. Among them, beta-binomial, correlated-binomial, and modified-binomial models are binomial-related models which are frequently used in modeling the sum of n correlated binary data. In many situations, it is reasonable to assume that n correlated binary data are exchangeable, which is a special case of correlated binary data. The sum of n exchangeable correlated binary data is modeled relatively well when the above three binomial-related models are applied. But the estimation results of correlation coefficient turn to be quite different. Hence, it is important to identify which model provides better estimates of model parameters(success probability, correlation coefficient). For this purpose, a small-scale simulation study is performed to compare the behavior of above three models.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.61-68
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• 2022

Let M_p := 2^p - 1 be a Mersenne prime. In this article, we find integers a, b, c, d, e and n satisfying $\sum_{t=0}^{n}\;\({an+b\\ct+d}\)\;=\;M_{p^e}$ given a Mersenne prime number M_p. In order to find a special case that satisfies the above results, we reprove an well-known relation of a certain sum of binomial coefficients and a divisor function.

• International Journal of Contents
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• v.3 no.1
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• pp.1-8
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• 2007

The distributions of the minimum correlated F-variable arises in many applied statistical problems including simultaneous analysis of variance (SANOVA), equality of variance, selection and ranking populations, and reliability analysis. In this paper, negative binomial sampling technique is employed to derive the distributions of the minimum of chi-square variables and hence the distributions of the minimum correlated F-variables. The work presented in this paper is divided in two parts. The first part is devoted to develop some combinatorial identities arised from the negative binomial sampling. These identities are constructed and justified to serve important purpose, when we deal with these distributions or their characteristics. Other important results including cumulants and moments of these distributions are also given in somewhat simple forms. Second, the distributions of minimum, chisquare variable and hence the distribution of the minimum correlated F-variables are then derived within the negative binomial sampling framework. Although, multinomial theory applied to order statistics and standard transformation techniques can be used to derive these distributions, the negative binomial sampling approach provides more information regarding the nature of the relationship between the sampling vehicle and the probability distributions of these functions of chi-square variables. We also provide an algorithm to compute the percentage points of the distributions. The computation methods we adopted are exact and no interpolations are involved.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.493-502
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• 2017

We propose an accurate approximation method via discrete Krawtchouk orthogonal polynomials to the distribution of a sum of independent but non-identically distributed binomial random variables. This approximation is a weighted binomial distribution with no need for continuity correction unlike commonly used density approximation methods such as saddlepoint, Gram-Charlier A type(GC), and Gaussian approximation methods. The accuracy obtained from the proposed approximation is compared with saddlepoint approximations applied by Eisinga et al. [4], which are the most accurate method among higher order asymptotic approximation methods. The numerical results show that the proposed approximation in general provide more accurate estimates over the entire range for the target probability mass function including the right-tail probabilities. In addition, the method is mathematically tractable and computationally easy to program.

• Journal of the Korean Statistical Society
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• v.21 no.2
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• pp.167-177
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• 1992

In this paper, we investigate the limiting distribution of $M_n = max (X_1, X-2, \cdots, X_n)$ in the infinite moving average process ${X_t = \sum c_i Z_{t-i}}$ generated from i.i.d. negative binomial variables $Z_i$'s. While no limit result is possible, nonetheless asymptotic bounds are derived. We also present the tail behavior of $X_t$, i.e., weighted sum of i.i.d. random variables. This continues a study made by Rootzen (1986) for discrete innovation sequences.

• The Pure and Applied Mathematics
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• v.29 no.2
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• pp.171-177
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• 2022

The aim of this note is to establish two known sums involving central binomial coefficients via a hypergeometric series approach. As an application, we discover two new closed-form evaluations of generalized hypergeometric function.

• International Journal of Contents
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• v.2 no.2
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• pp.17-24
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• 2006

The work presented in this paper is divided into two parts. The first part presents finite urn problems which generate truncated negative binomial random variables. Some combinatorial identities that arose from the negative binomial sampling and truncated negative binomial sampling are established. These identities are constructed and serve important roles when we deal with these distributions and their characteristics. Other important results including cumulants and moments of the distributions are given in somewhat simple forms. Second, the distributions of the maximum of two chi-square variables and the distributions of the maximum correlated F-variables are then derived within the negative binomial sampling scheme. Although multinomial theory applied to order statistics and standard transformation techniques can be used to derive these distributions, the negative binomial sampling approach provides more information and deeper insight regarding the nature of the relationship between the sampling vehicle and the probability distributions of these functions of chi-square variables. We also provide an algorithm to compute the percentage points of these distributions. We supplement our findings with exact simple computational methods where no interpolations are involved.

• The Korean Journal of Applied Statistics
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• v.29 no.5
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• pp.823-833
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• 2016

The lifetime is a key characteristic of product quality. It is best to obtain the lifetime data of all samples, but they are often censored due to time or expense limitations. In this paper, we propose a binomial cumulative sum (CUSUM) chart to monitor the mean of type I right-censored Weibull lifetime data, for a xed value of the Weibull shape parameter. We compare the performance of the proposed binomial CUSUM chart with CUSUM charts studied previously using the steady-state average run length (ARL). The results show that the performance of the binomial CUSUM chart is better when the censoring rate is high and/or the sample size is small.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.119-128
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• 2010

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_n}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1], [4], [2], [3], [5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_n}$, in cases when the $N_n$, $n\geq1$ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.