• Communications for Statistical Applications and Methods
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• 제10권2호
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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.

• Communications for Statistical Applications and Methods
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• 제19권6호
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• pp.809-817
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• 2012

We propose a semiparametric inference for a generalized varying coefficient partially linear model(VCPLM) for negative binomial data. The VCPLM is useful to model real data in that varying coefficients are a special type of interaction between explanatory variables and partially linear models fit both parametric and nonparametric terms. The negative binomial distribution often arise in modelling count data which usually are overdispersed. The varying coefficient function estimators and regression parameters in generalized VCPLM are obtained by formulating a penalized likelihood through smoothing splines for negative binomial data when the shape parameter is known. The performance of the proposed method is then evaluated by simulations.

• 한국농공학회지
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• 제30권4호
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• pp.109-116
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• 1988

This study was aimed at determining a regime of flow through rubble mound dike consisted of all sized quarrystons, and deriving a relationship between hydraulic gradient (I) and mean flow velocity (V) through the dike. The analysis was carried out with the data observed after final gap closing of the Haenam Sea dike from May, 6 to May, 14, 1987. The resu]ts are summarized as follows: 1. The regime of flow would be defined as the turbulent flow. 2. As to the relationships, two kinds of formula that are exponential and binomial were obtained. Exponential formula: I=2.099V 1.2888 Binomial formula: I=0.6113V+5.5235V$^2$ 3. Correlation coefficient of the former was 0.824 and that of the latter was 0.821, and the deviations between observed data and estimated were 0.0070 and 0.0064 respectively. 4. Comparing the correlation coefficient, both the equations have the same correlation coefficients, but in case of the deviation the binomial equation was better than the exponential equation. Therefore, the binomial equation is proposed for analyzing the flow through rubble mound dike.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제29권2호
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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.

• Communications for Statistical Applications and Methods
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• 제17권6호
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• pp.829-843
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• 2010

To test for the serial dependence in time series of counts data, Jung and Tremayne (2003) evaluated the size and power of several tests under the class of INARMA models based on binomial thinning operations for Poisson marginal distributions. The overdispersion phenomenon(i.e., a variance greater than the expectation) is common in the real world. Overdispersed count data can be modeled by using alternative thinning operations such as random coefficient thinning, iterated thinning, and quasi-binomial thinning. Such thinning operations can lead to time series models of counts with negative binomial or generalized Poisson marginal distributions. This paper examines whether the test statistics used by Jung and Tremayne (2003) on serial dependence in time series of counts data are affected by overdispersion.

• 대한수학회보
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• 제54권1호
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• pp.225-242
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• 2017

A family of $Ap{\acute{e}}ry$-like series involving reciprocals of central binomial coefficients is studied and it is shown that they represent transcendental numbers. The structure of such series is further examined in terms of finite combinations of logarithms and arctangents with arguments and coefficients belonging to a suitable algebraic extension of rationals. Monotonicity of certain quotients of weighted binomial sums which arise in the study of competitive cheap talk models is established with the help of a continuous extension of the discrete model at hand. The monotonic behavior of such quotients turns out to have important applications in game theory.

• 한국학교수학회논문집
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• 제19권3호
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• pp.291-308
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• 2016

본 연구는 수학영재의 심화학습을 위한 주제로 사용해 볼 수 있는 이항계수의 정의와 성질을 탐구하고, 이로부터 수학적 귀납법, 이항정리, 조합의 정의, 도로망 상황 모델 등을 이용한 이항계수가 포함된 등식의 문제해결방법을 연구하였다. 그리고 이러한 내용들이 수학영재 학생들에게는 충분히 탐구의 대상이 될 수 있어 수학영재 교육의 심화학습 주제로 적절하게 다루어질 수 있다는 것과, 수학의 깊은 의미를 경험할 수 있는 학습주제로 사용될 수 있다는 것을 학생들에게 지도한 예시로 소개한다.

• 한국안전학회지
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• 제29권6호
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• pp.151-157
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• 2014

This study deals with the rear-end collision at roundabouts. The purpose of this study is to develop the accident models of rear-end collision in Korea. In pursuing the above, this study gives particular attention to developing the appropriate models using Poisson, negative binomial model, ZAM, multiple linear and nonlinear regression models, and statistical analysis tools. The main results are as follows. First, the Vuong statistics and overdispersion parameters indicate that ZIP is the most appropriate model among count data models. Second, RMSE, MPB, MAD and correlation coefficient tests show that the multiple nonlinear model is the most suitable to the rear-end collision data. Finally, such the independent variables as traffic volume, ratio of heavy vehicle, number of circulatory roadway lane, number of crosswalk and stop line are adopted in the optimal model.

• 호남수학학술지
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• 제46권3호
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• pp.365-377
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• 2024

In this paper, we give some sums involving the generalized harmonic numbers H^r_n (σ) and the (q, r)-binomial coefficient $\left({L \atop k}\right)_{q,r}$ by using Euler's transform. For example, for (c, r) ∈ ℤ⁺ × ℝ⁺, $${\sum_{n=0}^{\infty}}{\sum_{k=0}^{n}}\,(-1)^k\,\left({n+r \atop n-k}\right)\frac{c^{n+1}H^{r-1}_k({\sigma})}{(n+1)(1+c)^{n+1}}=-(c+{\frac{1}{{\sigma}}})\,{\ln}\,(1+c{\sigma})+c,$$ and $${\sum_{k=0}^{n}}\left({n \atop k}\right)\left({L \atop k}\right)_{2,r}={\sum_{j=0}^{n}}{\sum_{k=0}^{j}}(-1)^k\left({j-k+2L+r \atop j-k}\right)\left({r \atop n-j}\right)\left({L \atop k}\right)_2,$$ where σ is appropriate parameter, H^r_n (σ) is the generalized hyperharmonic number of order r and $\left({L \atop k}\right)_q$ is the q-binomial coefficient.