• Journal of Environmental Health Sciences
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• v.45 no.2
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• pp.192-202
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• 2019

Objectives: According to the central limit theorem, the samples in population might be considered to follow normal distribution if a large number of samples are available. Once we assume that toxicity dataset follow normal distribution, we can treat and process data statistically to calculate genus or species mean value with standard deviation. However, little is known and only limited studies are conducted to investigate whether toxicity dataset follows normal distribution or not. Therefore, the purpose of study is to evaluate the generally accepted normality hypothesis of aquatic toxicity dataset Methods: We selected the 8 chemicals, which consist of 4 organic and 4 inorganic chemical compounds considering data availability for the development of species sensitivity distribution. Toxicity data were collected at the US EPA ECOTOX Knowledgebase by simple search with target chemicals. Toxicity data were re-arranged to a proper format based on the endpoint and test duration, where we conducted normality test according to the Shapiro-Wilk test. Also we investigated the degree of normality by simple log transformation of toxicity data Results: Despite of the central limit theorem, only one large dataset (n>25) follow normal distribution out of 25 large dataset. By log transforming, more 7 large dataset show normality. As a result of normality test on small dataset (n<25), log transformation of toxicity value generally increases normality. Both organic and inorganic chemicals show normality growth for 26 species and 30 species, respectively. Those 56 species shows normality growth by log transformation in the taxonomic groups such as amphibian (1), crustacean (21), fish (22), insect (5), rotifer (2), and worm (5). In contrast, mollusca shows normality decrease at 1 species out of 23 that originally show normality. Conclusions: The normality of large toxicity dataset was not always satisfactory to the central limit theorem. Normality of those data could be improved through log transformation. Therefore, care should be taken when using toxicity data to induce, for example, mean value for risk assessment.

• Journal of the Korean Data and Information Science Society
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• v.17 no.1
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• pp.221-231
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• 2006

A chi-squared test of multivariate normality is suggested which is oriented for detecting deviations from elliptical symmetry. We derive the limiting distribution of the test statistic via a central limit theorem on empirical processes. A simulation study is conducted to study the accuracy of the limiting distribution in finite samples. Finally, we compare the power of our method with those of other popular tests of multivariate normality under a non-normal distribution.

• Journal of the Korean Statistical Society
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• v.28 no.4
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• pp.479-488
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• 1999

Moore and Stubblebine(1981) suggested a chi-square test for multivariate normality based on cell counts calculated from the sample Mahalanobis distances. They derived the limiting distribution of the test statistic only when equiprobable cells are employed. Using conditional limit theorems, we derive the limiting distribution of the statistic as well as the asymptotic normality of the cell counts. These distributions are valid even when equiprobable cells are not employed. We finally apply this method to a real data set.

• Proceedings of the Safety Management and Science Conference
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• 2011.11a
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• pp.551-556
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• 2011

This research presents an implementation strategy of Process Capability Index (PCI) according to the types of process characteristics. The types of process feature are classified as four perspectives of variation range, time period, error position, and process stage. The paper examines short-term or long-term PCI, within or between variation, position of precision or accuracy, and inclusion of measurement or calibration stage. Moreover, the study proposes normality test of unilateral PCI.

• Journal of the Korean Data and Information Science Society
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• v.27 no.5
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• pp.1399-1412
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• 2016

We compare many normality tests consisting of different sources of information extracted from the given data: Anderson-Darling test, Kolmogorov-Smirnov test, Cramervon Mises test, Shapiro-Wilk test, Shaprio-Francia test, Lilliefors, Jarque-Bera test, D'Agostino' D, Doornik-Hansen test, Energy test and Martinzez-Iglewicz test. For the purpose of comparison, those tests are applied to the various types of data generated from skewed distribution, unsymmetric distribution, and distribution with different length of support. We then summarize comparison results in terms of two things: type I error control and power. The selection of the best test depends on the shape of the distribution of the data, implying that there is no test which is the most powerful for all distributions.

• Journal of the Korean Data and Information Science Society
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• v.18 no.3
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• pp.809-815
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• 2007

Since Bollerslev(1986), the GARCH model has been popular in analysing the volatility of the financial time series. In real data analysis, practitioners conventionally put the normal assumption on the innovation random variables of the GARCH model, which is often violated. In this paper, we analyse the domestic financial data based on the GARCH(1,1) model and among existing normality tests, perform the Jarque-Bera test based on the residuals. It is shown that the innovation based on the GARCH(1,1) model dose not follow the normality assumption.

• The Korean Journal of Applied Statistics
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• v.20 no.1
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• pp.79-89
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• 2007

A test for normality introduced by Arizono and Ohta(1989) is based on fullback-Leibler discrimination information. The test statistic is derived from the discrimination information estimated using sample entropy of Vasicek(1976) and the maximum likelihood estimator of the variance. However, these estimators are biased and so it is reasonable to make use of unbiased estimators to accurately estimate the discrimination information. In this paper, Arizono-Ohta test for normality is improved. The derived test statistic is based on the bias-corrected entropy estimator and the uniformly minimum variance unbiased estimator of the variance. The properties of the improved KL test are investigated and Monte Carlo simulation is performed for power comparison.

• Communications for Statistical Applications and Methods
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• v.14 no.3
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• pp.609-621
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• 2007

Diagnostic test results, which are approximately normal with a few number of outliers, but non-normal probability distribution, are frequently observed in practice. In the evaluation of two diagnostic tests, Greenhouse and Mantel (1950) proposed a parametric test under the assumption of normality but this test is inappropriate for the above non-normal case. In this paper, we propose a computationally simple nonparametric test that is based on quantile estimators of mean and standard deviation, instead of the moment-based mean and standard deviation as in some parametric tests. Parametric and nonparametric tests are compared with simulations under the assumption of, respectively, normality and non-normality, and under various combinations of the probability distributions for the normal and diseased groups.

• Communications for Statistical Applications and Methods
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• v.28 no.5
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• pp.463-475
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• 2021

Desgagné and de Micheaux (2018) proposed an alternative univariate normality test to the Jarque-Bera test. The proposed statistic is based on the sample second power skewness and kurtosis while the Jarque-Bera statistic uses sample Pearson's skewness and kurtosis that are the third and fourth standardized sample moments, respectively. In this paper, we generalize their statistic to a multivariate version based on orthogonalization or an empirical standardization of data. The proposed multivariate statistic follows chi-squared distribution approximately. A simulation study shows that the proposed statistic has good control of type I error even for a very small sample size when critical values from the approximate distribution are used. It has comparable power to the multivariate version of the Jarque-Bera test with exactly the same idea of the orthogonalization. It also shows much better power for some mixed normal alternatives.

• Communications for Statistical Applications and Methods
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• v.25 no.1
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• pp.71-78
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• 2018

In this study, we consider the likelihood ratio test for the covariance matrix of the multivariate normal data. For this, we propose a method for obtaining null distributions of the likelihood ratio statistics by the Monte-Carlo approach when it is difficult to derive the exact null distributions theoretically. Then we compare the performance and precision of distributions obtained by the asymptotic normality and the Monte-Carlo method for the likelihood ratio test through a simulation study. Finally we discuss some interesting features related to the likelihood ratio test for the covariance matrix and the Monte-Carlo method for obtaining null distributions for the likelihood ratio statistics.