• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.87-97
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• 1992

Sequential procedure with ${\beta}$-protection for the mean vector ${\mu}(\theta)$ of a p(> 1)-variate multivariate distribution $P_{\theta}$, ${\theta}{\in}{\Theta}$, with covariance matrix ${\sum}(\theta)$ is considered when the only nuisance parameters is ${\sum}(\theta)$. We obtain a confidence set for ${\mu}(\theta)$ with coverage probability condition and ${\beta}$-protection at ${\mu}-{\delta}(\mu)$ for some imprecision function ${\delta}:\mathbb{R}^p{\rightarrow}\mathbb{R}^p$.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.485-506
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• 2023

This paper investigates an extended form Hurwitz-Lerch zeta function, as well as related integral images, ordinary and fractional derivatives, and series expansions, using the term extended beta function. We establish a connection between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. Furthermore, we present a probability distribution application of the extended Hurwitz-Lerch zeta function ζ^𝛿,𝜇_𝜈,λ. Several results, both known and new, are shown to follow as special cases of our findings.

• The Korean Journal of Applied Statistics
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• v.21 no.5
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• pp.835-843
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• 2008

In this paper, we investigate a Bayesian inference for software reliability models based on mean value functions which take the form of the mixture of beta distribution functions. The posterior simulation via the Markov chain Monte Carlo approach is used to produce estimates of posterior properties. Its applicability is illustrated with two real data sets. We compute the predictive distribution and the marginal likelihood of various models to compare the performance of them. The model comparison results show that the model based on the beta-mixture performs better than other models.

• Journal of the Korean Statistical Society
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• v.27 no.3
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• pp.349-358
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• 1998

Consider the AR(1) model $X_{t}$=$\beta$ $X_{t-1}$+$\varepsilon$$_{t}$ where $\beta$ < 1 is an unknown parameter to be estimated and {$\varepsilon$$_{t}$} denotes the independent and identically distributed error terms with unknown common distribution function F. In this paper, a strong representation for the least absolute deviation (LAD) estimate of $\beta$ in AR(1) models is obtained under some mild conditions on F. on F.F.

• Journal of the Korean Society of Combustion
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• v.6 no.2
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• pp.23-28
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• 2001

In the present work, the effect of PDF selection and intermittency on the result of the numerical simulation are examined by the simulation of a turbulent methane-air jet diffusion flame. As to the PDFs, beta-function and clipped Gaussian are considered. Results for the pure mixing jet are compared with experimental results. Then, the turbulent flame is calculated for the same conditions and the results obtained for the several models are compared. It is found that the clipped Gaussian distribution coupled with consideration of intermittency recovers the experimental data very well. As to the reacting flow results, the main overall properties of the turbulent jet diffusion flame such as maximum flame temperature are less affected by the choice of the PDF. Flame height and NO emissions, on the contrary, appear to be significantly influenced.

• Proceedings of the Korean Society of Machine Tool Engineers Conference
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• 1999.10a
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• pp.49-56
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• 1999

This paper present a study on the application of sound pressure and vibration signals to detect the presence of defects in a rolling element bearing using a statistical analysis method. The well established statistical parameters such as the crest factor and the distribution of moments including kurtosis and skewless are utilized in this study. In addition, other statistical parameters derived from the beta distribution function are also used. A comparison study on the performance of the different types of parameter used is also performed. The statistical analysis is used because of its simplicity and quick computation. Under ideal conditions, the statistical method can be used to identify the different types of defect present in the bearing. In addition, the results also reveal that there is no significant advantages in using the beta function parameters when compared to using kurtosis and the crest factor for detecting and identifying defects in rolling element bearings from both sound and vibration signals.

• Journal of Advanced Marine Engineering and Technology
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• v.24 no.4
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• pp.405-413
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• 2000

This paper present study on the application of sound pressure and vibration signals to detect the presence of defects in a rolling element bearing using a statistical analysis method. The well established statistical parameters such as the crest factor and the distribution of moments including kurtosis and skew are utilized in this study. In addition, other statistical parameters derived from the beta distribution function are also used. A comparison study on the performance of the different types of parameter used is also performed. The statistical analysis is used because of its simplicity and quick computation. Under ideal conditions, the statistical method can be used to identify the different types of defect present in the bearing. In addition, the results also reveal that there is no significant advantages in using the beta function parameters when compared to using kurtosis and the crest factor for detecting and identifying defects in rolling element bearings from both sound and vibration signals.

• Journal of Korea Water Resources Association
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• v.45 no.6
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• pp.531-544
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• 2012

This study suggested the parameter estimation method for given rainfall events to be properly expressed by the beta distribution. For this purpose, this study compared the characteristics of probability density function with the parameter proposed considering the cases with and without addition to the rainfall peak, and the cases of using the real hyetograph and the rearranged hyetograph about the rainfall peak. As an example, this study analyzed the independent rainfall events at Seoul in 2010 and the annual maximum independent rainfall events from 1961 to 2010. The results derived are as follows. First, this study confirmed the necessity of additional consideration on rainfall peak to mimic the real hyetograph of rainfall events by the beta distribution. Second, this study confirmed the case of using rearranged hyetograph about the rainfall peak derived a better beta distribution to well mimic the characteristics of real rainfall than the case using the real hyetograph.

• Magazine of the Korean Society of Agricultural Engineers
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• v.21 no.4
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• pp.108-117
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• 1979

In order to obtain the basic data for design of water structures which can be contributed to the planning of water use. Best fitted distribution function and the equations for the probable minimum flow were derived to the annual minimum flow of five subwatersheds along Geum River basin. The result were analyzed and summarized as follows. 1. Type III extremal distribution was considered as a best fit one among some other distributions such as exponential and two parameter lognormal distribution by $x^2$-goodness of fit test. 2. The minimum flow are analyzed by Type III extremal distribution which contains a shape parameter $\lambda$, a location parameter ${\beta}$ and a minimum drought $\gamma$. If a minimum drought $\gamma=0$, equations for the probable minimum flow, $D_T$, were derived as $D_T={\beta}e^{\lambda}1^{y'}$, with two parameters and as $D_T=\gamma+(\^{\beta}-\gamma)e^{{\lambda}y'}$ with three parameters in case of a minimum drought ${\gamma}>0$ respectively. 3. Probable minimum flow following the return periods for each stations were also obtained by above mentioned equations. Frequency curves for each station are drawn in the text. 4. Mathematical equation with three parameters is more suitable one than that of two parameters if much difference exist between the maximum and the minimum value among observed data.

• The Pure and Applied Mathematics
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• v.5 no.2
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• pp.123-132
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• 1998

Let F and G denote the distribution functions of the failure times and the censoring variables in a random censorship model. Susarla and Van Ryzin(1978) verified consistency of $F_{\alpha}$, he NPBE of F with respect to the Dirichlet process prior D($\alpha$), in which they assumed F and G are continuous. Assuming that A, the cumulative hazard function, is distributed according to a beta process with parameters c, $\alpha$, Hjort(1990) obtained the Bayes estimator $A_{c,\alpha}$ of A under a squared error loss function. By the theory of product-integral developed by Gill and Johansen(1990), the Bayes estimator $F_{c,\alpha}$ is recovered from $A_{c,\alpha}$. Continuity assumption on F and G is removed in our proof of the consistency of $A_{c,\alpha}$ and $F_{c,\alpha}$. Our result extends Susarla and Van Ryzin(1978) since a particular transform of a beta process is a Dirichlet process and the class of beta processes forms a much larger class than the class of Dirichlet processes.