• Journal of the Korean Statistical Society
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• v.34 no.1
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• pp.35-38
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• 2005

A single, tractable, special case of the problem of continuous mixtures of beta distributions over their parameters is considered. This is the uniform mixture of generalized arc-sine distributions which, curiously, turns out to be linked by transformation to the Cauchy distribution.

• Communications for Statistical Applications and Methods
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• v.16 no.2
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• pp.389-396
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• 2009

Approximations for the null distribution of a test statistic arising in multivariate analysis to test homogeneity of variances and a mixture of two beta distributions by making use of a product of beta baseline density function and a polynomial adjustment, so called beta-polynomial density approximant, are discussed. Explicit representations of density and distribution approximants of interest in each case can easily be obtained. Beta-polynomial density approximants produce good approximation over the entire range of the test statistic and also accommodate even the bimodal distribution using an artificial example of a mixture of two beta distributions.

• Journal of Forest and Environmental Science
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• v.36 no.1
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• pp.7-16
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• 2020

Tree size distribution modelling is an integral part of forest management. Most distribution yield systems rely on some flexible probability models. In this study, a simple finite mixture of two components two-parameter Weibull distribution was compared with complex four-parameter distributions in terms of their fitness to predict tree size distribution of teak (Tectona grandis Linn f) plantations. Also, a system of equation was developed using Seemingly Unrelated Regression wherein the size distributions of the stand were predicted. Generalized beta, Johnson's SB, Logit-Logistic and generalized Weibull distributions were the four-parameter distributions considered. The Kolmogorov-Smirnov test and negative log-likelihood value were used to assess the distributions. The results show that the simple finite mixture outperformed the four-parameter distributions especially in stands that are bimodal and heavily skewed. Twelve models were developed in the system of equation-one for predicting mean diameter, seven for predicting percentiles and four for predicting the parameters of the finite mixture distribution. Predictions from the system of equation are reasonable and compare well with observed distributions of the stand. This simplified mixture would allow for wider application in distribution modelling and can also be integrated as component model in stand density management diagram.

• Communications for Statistical Applications and Methods
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• v.31 no.3
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• pp.263-277
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• 2024

This paper proposes new parameterizations of the tilted beta binomial distribution, obtained from the combination of the binomial distribution and the tilted beta distribution, where the beta component of the mixture is parameterized as a function of their mean and variance. These new parameterized distributions include as particular cases the beta rectangular binomial and the beta binomial distributions. After that, we propose new linear regression models to deal with overdispersed binomial datasets. These new models are defined from the proposed new parameterization of the tilted beta binomial distribution, and assume regression structures for the mean and variance parameters. These new linear regression models are fitted by applying Bayesian methods and using the OpenBUGS software. The proposed regression models are fitted to a school absenteeism dataset and to the seeds germination rate according to the type seed and root.

• Journal of Radiation Protection and Research
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• v.26 no.4
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• pp.375-384
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• 2001

It is possible to count and perform quench correction on two ${\beta}$ -label samples so long as the maximum ${\beta}$-energies are sufficiently different. However, when 4he conventional technique is applied to the radioassay of a mixture of more than three nuclides, the reliability of the activities determined is considerably reduced, resulting from the large overlapping of liquid scintillation pulse height distributions of each nuclide. A technique that allows the activities of multiple ${\beta}$-labeled samples to be radioassayed was proposed by using the least square method. The technique was applied to mixture samples of $^3H,\;^{14}C,\;^{36}Cl$, and $^{90}Sr$. The analytical values were in good agreement with the reference values within 7% relative error.

• Journal of Korea Foundry Society
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• v.17 no.3
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• pp.267-275
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• 1997

Fabrication of high strength Mg-Li-Al alloys by squeeze casting was established by the stabilization of melt and mold temperatures, applied pressure and the refining method. The entrapment of inclusions during pouring was prevented using 30 ppi alumina foam filter. The as-cast microstructure consists of a mixture of ${\alpha}$ and ${\beta}$ phases including AILi and $MgLi_2$, Al particles, which are distributed in the ${\beta}$ matrix. The grain sizes of gravity and squeeze casting alloys were 288 ${\mu}m$ and 207 ${\mu}m$ respectively. The addition of Al in Mg-Li alloys promoted the formation of second phase particles, which were adjusted to optimize the properties of Mg-Li-Al alloys. The Mg-10wt%Li-5wt%Al alloy after heat treatment at $350^{\circ}C$ for 1 hour showed the maximum hardness value. This is due to the facts that the amounts of ${\alpha}$ and ${\beta}$ phases and their distributions are dependent upon the solution treatment temperature, and that the amounts of AILi and $MgLi_2Al$ particles are dependent upon the Al content.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.2
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• pp.81-95
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• 2008

In this paper, the negative Polya-Eggenberger distribution has been introduced by compounding negative binomial distribution with beta distribution of I-kind which generates a number of univariate contagious or compound (or mixture of) distributions as its particular cases. The distribution is unimode, over dispersed and all of its positive and negative integer moments exist. The difference equation of the proposed model shows that it is a member of the Ord's family of distribution. Some of its interesting properties have been explored besides different methods of estimation been discussed. Finally, the parameters of the proposed model have been estimated by using a computer programme in R-software. Application of the proposed model to some data, available in the literature, has been given and its goodness of fit demonstrated.

• Journal of the Korean Statistical Society
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• v.21 no.2
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• pp.187-200
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• 1992

Samples are often found to be too heterogeneous to be explained by a one-parameter family of models in the sense that the implicit mean-variance relationship in such a family is violated by the data. This phenomenon is often called over-dispersion. The most frequently used method in dealing with over-dispersion is to mix a one-parameter family creating a two parameter marginal mixture family for the data. In this paper, we investigate performance of estimators such as maximum likelihood estimator, method of moment estimator, and maximum quasi-likelihood estimator in negative binomial and beta-binomial distribution. Simulations are done for various mean parameter and dispersion parameter in both distributions, and we conclude that the moment estimators are very superior in the sense of bias and asymptotic relative efficiency.

• Smart Structures and Systems
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• v.19 no.3
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• pp.309-322
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• 2017

In this paper, the buckling, and free vibration analysis of tapered functionally graded carbon nanotube reinforced composite (FG-CNTRC) micro Reddy beam under longitudinal magnetic field using finite element method (FEM) is investigated. It is noted that the material properties of matrix is considered as Poly methyl methacrylate (PMMA). Using Hamilton's principle, the governing equations of motion are derived by applying a modified strain gradient theory and the rule of mixture approach for micro-composite beam. Micro-composite beam are subjected to longitudinal magnetic field. Then, using the FEM, the critical buckling load, and natural frequency of micro-composite Reddy beam is solved. Also, the influences of various parameters including ${\alpha}$ and ${\beta}$ (the constant coefficients to control the thickness), three material length scale parameters, aspect ratio, different boundary conditions, and various distributions of CNT such as uniform distribution (UD), unsymmetrical functionally graded distribution of CNT (USFG) and symmetrically linear distribution of CNT (SFG) on the critical buckling load and non-dimensional natural frequency are obtained. It can be seen that the non-dimensional natural frequency and critical buckling load decreases with increasing of ${\beta}$ for UD, USFG and SFG micro-composite beam and vice versa for ${\alpha}$. Also, it is shown that at the specified value of ${\alpha}$ and ${\beta}$, the dimensionless natural frequency and critical buckling load for SGT beam is more than for the other state. Moreover, it can be observed from the results that employing magnetic field in longitudinal direction of the micro-composite beam increases the natural frequency and critical buckling load. On the other hands, by increasing the imposed magnetic field significantly increases the stability of the system that can behave as an actuator.

• Archives of Pharmacal Research
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• v.26 no.9
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• pp.763-767
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• 2003

The differences in pharmacokinetic behavior and tissue distribution of verapamil and its enantiomers were investigated in rats. In high-performance liquid chromatographic method, an achiral ODS column (150 mm $\times$ 4.6 mm i.d.) with the mobile phase consisting of methanol-water (73:30, v/v) was used for the determination of the concentration for racemic verapamil, and a Chiralcel OJ column (250 mm$\times$4.6 mm i.d.) with the mixture of n-haxane-ethanol-triethylamine (85:15:0.2, v/v/v) as mobile phase was used to determine the concentrations of verapamil enantiomers. A fluorescence detector in the analytical system was set at excitation and emission wavelengths of 275 nm and 315 nm. The differences between enantiomers were apparent in the pharmacokinetics in rats. The area under the concentration-time curve (AUC) of S-(-) verapamil was higher than that of R-(+) verapamil. The half-distribution time ($T_{1/2(\alpha)}$) of S-(-) verapamil which distributing to tissue from blood was shorter than that of R-(+) verapamil, but the elimination half-time ($T_{1/2(\beta)}$) was longer in rat following oral administration of racemic verapamil. At 1.3 h after oral administration of racemic verapamil, however, there were no significant differences between enantiomers for the distributions in major tissues such as heart, cerebrum, cerebellum, liver, spleen and kidney.