• Communications for Statistical Applications and Methods
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• v.14 no.3
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• pp.583-592
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• 2007

Given the moments of a symmetric random variable, its density and distribution functions can be accurately approximated by making use of the algorithm proposed in this paper. This algorithm is specially designed for approximating symmetric distributions and comprises of four phases. This approach is essentially based on the transformation of variable technique and moment-based density approximants expressed in terms of the product of an appropriate initial approximant and a polynomial adjustment. Probabilistic quantities such as percentage points and percentiles can also be accurately determined from approximation of the corresponding distribution functions. This algorithm is not only conceptually simple but also easy to implement. As illustrated by the first two numerical examples, the density functions so obtained are in good agreement with the exact values. Moreover, the proposed approximation algorithm can provide the more accurate quantities than direct approximation as shown in the last example.

• Journal of The Korean Astronomical Society
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• v.22 no.2
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• pp.127-140
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• 1989

Problem of the diffuse radiation (DFR) transfer is solved exactly for pure hydrogen nebulae of uniform density, and accuracies of the on-the-spot (OTS) approximation are critically examined. For different values of density and spectral types of the central star, we have calculated the degree of ionization and the kinetic temperature of electrons as functions of distance from the central star, and compared them with the corresponding results of the OTS approximation. At most locations inside an HII region. the DFR ionizes considerable amount of hydrogen; therefore, the OTS approximation under-estimates the size of ionized regions. The exact treatment of the DFR transfer results in an about 10 to 20 percent increase in the classical $Str{\ddot{o}}mgren$ radius. The OTS approximation overestimates the local heating rate by raising the density of neutral hydogens. Consequently, it predicts higher values for the local electron temperature. The OTS approximation also exaggerates the dependence of electron temperature on density. When the hydrogen density is changed from $10/cm^3$ to $10^3/cm^3$ with an 06.5V star, the OTS approximation shows an about 3,000 K difference in the electron temperature, while the exact treatment of the DFR-transfer reduces the difference to about 1,000 K. The OTS approximation fails to demonstrate the brightening of the electron temperature close to the ionization boundary.

• East Asian mathematical journal
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• v.39 no.5
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• pp.537-547
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• 2023

In this paper, we introduce the optimal approximation by a Gaussian function for a probability density function. We show that the approximation can be obtained by solving a non-linear system of parameters of Gaussian function. Then, to understand the non-normality of the empirical distributions observed in financial markets, we consider the nearly Gaussian function that consists of an optimally approximated Gaussian function and a small periodically oscillating density function. We show that, depending on the parameters of the oscillation, the nearly Gaussian functions can have fairly thick heavy tails.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.147-158
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• 2004

Sufficient conditions are given under which a generalized class of kernel-type estimators allows asymptotic approximation on the modulus of continuity. This generalized class includes sample distribution function, kernel-type estimator of density function, and an estimator that may apply to the censored case. In addition, an application is given to asymptotic normality of recursive density estimators of density function at an unknown point.

• The Korean Journal of Applied Statistics
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• v.24 no.6
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• pp.1161-1168
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• 2011

The approximation for the distribution functions of nonparametric test statistics is a significant step in statistical inference. A rank sum test for dispersions proposed by Ansari and Bradley (1960), which is widely used to distinguish the variation between two populations, has been considered as one of the most popular nonparametric statistics. In this paper, the statistical tables for the distribution of the nonparametric Ansari-Bradley statistic is produced by use of polynomially adjusted normal approximation as a semi parametric density approximation technique. Polynomial adjustment can significantly improve approximation precision from normal approximation. The normal-polynomial density approximation for Ansari-Bradley statistic under finite sample sizes is utilized to provide the statistical table for various combination of its sample sizes. In order to find the optimal degree of polynomial adjustment of the proposed technique, the sum of squared probability mass function(PMF) difference between the exact distribution and its approximant is measured. It was observed that the approximation utilizing only two more moments of Ansari-Bradley statistic (in addition to the first two moments for normal approximation provide) more accurate approximations for various combinations of parameters. For instance, four degree polynomially adjusted normal approximant is about 117 times more accurate than normal approximation with respect to the sum of the squared PMF difference.

• Journal of Industrial Technology
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• v.10
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• pp.9-13
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• 1990

The approximation of the radial distribution functions(RDF) of mixture plays an important role in deriving the mixing rules for the corresponding states principle(CSP). The mean density approximation(MDA), one of the most successful approximations, fails to predict the radial distribution functions when the size ratio in terms of the Lennard-Jones size parameters is greater than 1.5. To get a better prediction of important structural integrals over the radial distribution functions that arise in the asymmetrical attraction contribution of the perturbaton theory, the new type of mean density approximation(NTMDA) is proposed. With this NTMDA, quite reliable results for those integrals for systems with comparatively large ratio of the size parameters are obtained.

• Journal of the Korean Data and Information Science Society
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• v.7 no.2
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• pp.189-201
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• 1996

In this paper, we study the saddlepoint approximations for the ratio of independent random variables. In Section 2, we derive the saddlepoint approximation to the density. And in Section 3, we derive two approximation formulae for the tail probability, one by following Daniels'(1987) method and the other by following Lugannani and Rice's (1980). In Section 4, we represent some numerical examples which show that the errors are small even for small sample size.

• Journal of the Korean Data and Information Science Society
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• v.10 no.2
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• pp.415-428
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• 1999

In this paper, we study the saddlepoint approximations for the ratio of two independent sequences of random variables. In Section 2, we review the saddlepoint approximation to the probability density function. In section 3, we derive an saddlepoint approximation formular for the tail probability by following Daniels'(1987) method. In Section 4, we represent a numerical example which shows that the errors are small even for small sample size.

• Journal of the Korean Data and Information Science Society
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• v.9 no.2
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• pp.255-262
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• 1998

In this paper, we study the saddlepoint approximations for the ratio of independent random variables. In Section 2, we derive the saddlepoint approximation to the probability density function. In Section 3, we represent a numerical example which shows that the errors are small even for small sample size.

• Communications for Statistical Applications and Methods
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• v.19 no.3
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• pp.451-457
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• 2012

Baumgartner et al. (1998) proposed a novel statistical test for the null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for more than two samples. Whereas the expressions of the exact density and distribution functions of the generalized Baumgartner statistic are not yet found, the characteristic function of its limiting distribution has been obtained. Due to the development of computational power, the Fourier series approximation can be readily utilized to accurately and efficiently approximate its density function based on its Laplace transform. Numerical examples show that the Fourier series method provides an accurate approximation for statistical quantities of the generalized Baumgartner statistic.