• Journal of the Korean Data and Information Science Society
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• v.25 no.4
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• pp.903-914
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• 2014

The inverse Weibull distribution has been proposed as a model in the analysis of life testing data. Also, inverse Weibull distribution has been recently derived as a suitable model to describe degradation phenomena of mechanical components such as the dynamic components (pistons, crankshaft, etc.) of diesel engines. In this paper, we derive the approximate maximum likelihood estimators of the scale parameter and the shape parameter in the inverse Weibull distribution under multiply type-II censoring. We also develop four modified empirical distribution function (EDF) type tests for the inverse Weibull or extreme value distribution based on multiply type-II censored samples. We also propose modified normalized sample Lorenz curve plot and new test statistic.

• Structural Engineering and Mechanics
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• v.74 no.1
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• pp.55-67
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• 2020

The common practice to predict the characteristic structural load effects (LEs) in long reference periods is to employ the extreme value theory (EVT) for building limit distributions. However, most applications ignore that LEs are driven by multiple loading events and thus do not have the identical distribution, a prerequisite for EVT. In this study, we propose the composite extreme value modeling approach using clustering to (a) cluster initial blended samples into finite identical distributed subsamples using the finite mixture model, expectation-maximization algorithm, and the Akaike information criterion; (b) combine limit distributions of subsamples into a composite prediction equation using the generalized Pareto distribution based on a joint threshold. The proposed approach was validated both through numerical examples with known solutions and engineering applications of bridge traffic LEs on a long-span bridge. The results indicate that a joint threshold largely benefits the composite extreme value modeling, many appropriate tail approaching models can be used, and the equation form is simply the sum of the weighted models. In numerical examples, the proposed approach using clustering generated accurate extrema prediction of any reference period compared with the known solutions, whereas the common practice of employing EVT without clustering on the mixture data showed large deviations. Real-world bridge traffic LEs are driven by multi-events and present multipeak distributions, and the proposed approach is more capable of capturing the tendency of tailed LEs than the conventional approach. The proposed approach is expected to have wide applications to general problems such as samples that are driven by multiple events and that do not have the identical distribution.

• The Korean Journal of Applied Statistics
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• v.23 no.2
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• pp.219-234
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• 2010

In index investing according to KOSPI, we estimate Value at Risk(VaR) from the extreme losses of the daily returns which are obtained from KOSPI. To this end, we apply Block Maxima(BM) model which is one of the useful models in the extreme value theory. We also estimate the extremal index to consider the dependency in the occurrence of extreme losses. From the back-testing based on the failure rate method, we can see that the model is adaptable for the VaR estimation. We also compare this model with the GARCH model which is commonly used for the VaR estimation. Back-testing says that there is no meaningful difference between the two models if we assume that the conditional returns follow the t-distribution. However, the estimated VaR based on GARCH model is sensitive to the extreme losses occurred near the epoch of estimation, while that on BM model is not. Thus, estimating the VaR based on GARCH model is preferred for the short-term prediction. However, for the long-term prediction, BM model is better.

• The Korean Journal of Financial Management
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• v.22 no.1
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• pp.119-146
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• 2005

This study applies extreme value theory to get extreme value-VAR for Korean Stock market and showed the usefulness of the approach. Block maxima model and POT model were used as extreme value models and tested which model was more appropriate through back testing. It was shown that the block maxima model was unstable as the variation of the estimate was very large depending on the confidence level and the magnitude of the estimates depended largely on the block size. This shows that block maxima model was not appropriate for Korean Stock market. On the other hand POT model was relatively stable even though extreme value VAR depended on the selection of the critical value. Back test also showed VAR showed a better result than delta VAR above 97.5% confidence level. POT model performs better the higher the confidence level, which suggests that POT model is useful as a risk management tool especially for VAR estimates with a confidence level higher than 99%. This study picks up the right tail and left tail of the return distribution and estimates the EVT-VAR for each, which reflects the asymmetry of the return distribution of the Korean Stock market.

• The Korean Journal of Applied Statistics
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• v.24 no.5
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• pp.833-845
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• 2011

This paper conducts a statistical analysis of extreme values for both daily log-returns and daily negative log-returns, which are computed using a collection of KOSPI data from January 3, 1998 to August 31, 2011. The Poisson-GPD model is used as a statistical analysis model for extreme values and the maximum likelihood method is applied for the estimation of parameters and extreme quantiles. To the Poisson-GPD model is also added the Bayesian method that assumes the usual noninformative prior distribution for the parameters, where the Markov chain Monte Carlo method is applied for the estimation of parameters and extreme quantiles. According to this analysis, both the maximum likelihood method and the Bayesian method form the same conclusion that the distribution of the log-returns has a shorter right tail than the normal distribution, but that the distribution of the negative log-returns has a heavier right tail than the normal distribution. An advantage of using the Bayesian method in extreme value analysis is that there is nothing to worry about the classical asymptotic properties of the maximum likelihood estimators even when the regularity conditions are not satisfied, and that in prediction it is effective to reflect the uncertainties from both the parameters and a future observation.

• Journal of Korea Water Resources Association
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• v.50 no.3
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• pp.211-221
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• 2017

In statistical hydrology, various extreme distributions such as the generalized extreme value (GEV), generalized logistic (GLO) and Gumbel (GUM) models have been widely used to analyze the extreme events. In the case of rainfall events in South Korea, the GEV and Gumbel distributions are known to be appropriate among various extreme distribution models. However, the proper probability distribution model may be different depending on the type of extreme events, rainfall duration, region, and statistical characteristics of extreme events. In this regard, it is necessary to apply a wide range of statistical properties that can be represented by the distribution model because it has two shape parameters. In this study, the statistical applicability of rainfall data is analyzed using the Burr XII distribution and the dimensionless L-moment ratio for 620 stations in South Korea. For this purpose, L-skewness and L-kurtosis of the Burr XII distribution are derived and L-moment ratio diagram is drawn and then the applicability of 620 stations was analyzed. As a result, it is found that the Burr XII distribution for the stations of the Han River basin in which L-skewness is relatively larger than L-kurtosis is appropriate, It is possibility of replacing the distribution of commonly used Gumbel or GEV distributions. Therefore, the Burr XII model can be replaced as an appropriate probability model in this basin.

• The Korean Journal of Applied Statistics
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• v.28 no.2
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• pp.137-149
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• 2015

Flood planning needs to recognize trends for extreme precipitation events. Especially, the r-year return level is a common measure for extreme events. In this paper, we present a nonstationary temporal model for precipitation return levels using a hierarchical Bayesian modeling. For intensity, we model annual maximum daily precipitation measured in Korea with a generalized extreme value (GEV). The temporal dependence among the return levels is incorporated to the model for GEV model parameters and a linear model with autoregressive error terms. We apply the proposed model to precipitation data collected from various stations in Korea from 1973 to 2011.

• Korean Management Science Review
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• v.24 no.2
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• pp.115-126
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• 2007

Weekly minima of daily log returns of Korean composite stock price index 200 and its five industry-based business divisions over the period from January 1990 to December 2005 are fitted using two block-based extreme distributions: Generalized Extreme Value(GEV) and Generalized Logistic(GLO). Parameters are estimated using the probability weighted moments. Applicability of two distributions is investigated using the Monte Carlo simulation based empirical p-values of Anderson Darling test. Our empirical results indicate that both the GLO and GEV models seem to be comparably applicable to the weekly minima. These findings are against the evidences in Gettinby et al.[7], who claimed that the GEV model was not valid in many cases, and supported the significant superiority of the GLO model.

• The Korean Journal of Applied Statistics
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• v.20 no.2
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• pp.291-311
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• 2007

With a series of unexpected huge losses in the financial markets around the world recently, especially in the insurance market with extreme loss cases such as catastrophes, there is an increasing demand for risk management for extreme loss exposures due to high unpredictability of those risks. For extreme risk management, to make a maximum use of the information concerning the tail part of a loss distribution, EVT(Extreme Value Theory) modelling nay be the best to analyze extreme values. The Extreme Value Theory is widely used in practice and, especially in financal markets, EVT modelling is getting popular to analyBe the effects of extreme risks. This study is to review the significance of the Extreme Value Theory in risk management and, focusing on analyzing insurer's risk capital, extreme risk is measured using the real fire loss data and insurer's specific amount of risk capital is figured out to buffer the extreme risk.

• Journal of Korea Water Resources Association
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• v.43 no.4
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• pp.337-351
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• 2010

There is a growing dissatisfaction with use of conventional statistical methods for the prediction of extreme events. Conventional methodology for modeling extreme event consists of adopting an asymptotic model to describe stochastic variation. However asymptotically motivated models remain the centerpiece of our modeling strategy, since without such an asymptotic basis, models have no rational for extrapolation beyond the level of observed data. Also, this asymptotic models ignored or overestimate the uncertainty and finally decrease the reliability of uncertainty. Therefore this article provide the research example of the extreme rainfall event and the methodology to reduce the uncertainty. In this study, the Bayesian MCMC (Bayesian Markov Chain Monte Carlo) and the MLE (Maximum Likelihood Estimation) methods using a quadratic approximation are applied to perform the at-site rainfall frequency analysis. Especially, the GEV distribution and Gumbel distribution which frequently used distribution in the fields of rainfall frequency distribution are used and compared. Also, the results of two distribution are analyzed and compared in the aspect of uncertainty.