• Communications for Statistical Applications and Methods
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• v.31 no.1
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• pp.1-35
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• 2024

The assumption of homoscedasticity is one of the most crucial assumptions for many parametric tests used in the biological sciences. The aim of this paper is to compare the empirical probability of type I error and the power of ten parametric and two non-parametric tests for homoscedasticity with simulations under different types of distributions, number of groups, number of samples per group, variance ratio and significance levels, as well as through empirical data from an agricultural experiment. According to the findings of the simulation study, when there is no violation of the assumption of normality and the groups have equal variances and equal number of samples, the Bhandary-Dai, Cochran's C, Hartley's Fmax, Levene (trimmed mean) and Bartlett tests are considered robust. The Levene (absolute and square deviations) tests show a high probability of type I error in a small number of samples, which increases as the number of groups rises. When data groups display a nonnormal distribution, researchers should utilize the Levene (trimmed mean), O'Brien and Brown-Forsythe tests. On the other hand, if the assumption of normality is not violated but diagnostic plots indicate unequal variances between groups, researchers are advised to use the Bartlett, Z-variance, Bhandary-Dai and Levene (trimmed mean) tests. Assessing the tests being considered, the test that stands out as the most well-rounded choice is the Levene's test (trimmed mean), which provides satisfactory type I error control and relatively high power. According to the findings of the study and for the scenarios considered, the two non-parametric tests are not recommended. In conclusion, it is suggested to initially check for normality and consider the number of samples per group before choosing the most appropriate test for homoscedasticity.

• Journal of the Korean Statistical Society
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• v.7 no.1
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• pp.11-16
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• 1978

Bayesian procedures are in vogue to revise the parameter estimates of the forecasting model in the light of actual time series data. In this paper, we study the Bayes forecast for demand and the risk when (a) 'noise' and (b) mean demand rate in a constant process model have moderately non-normal probability distributions.

• Communications for Statistical Applications and Methods
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• v.29 no.3
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• pp.373-391
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• 2022

The distribution of observations in most econometric studies with spatial heterogeneity is skewed. Usually, a single transformation of the data is used to approximate normality and to model the transformed data with a normal assumption. This assumption is however not always appropriate due to the fact that panel data often exhibit non-normal characteristics. In this work, the normality assumption is relaxed in spatial mixed models, allowing for spatial heterogeneity. An inference procedure based on Bayesian mixed modeling is carried out with a multivariate skew-elliptical distribution, which includes the skew-t, skew-normal, student-t, and normal distributions as special cases. The methodology is illustrated through a simulation study and according to the empirical literature, we fit our models to non-life insurance consumption observed between 1998 and 2002 across a spatial panel of 103 Italian provinces in order to determine its determinants. Analyzing the posterior distribution of some parameters and comparing various model comparison criteria indicate the proposed model to be superior to conventional ones.

• Journal of Intelligence and Information Systems
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• v.28 no.2
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• pp.19-31
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• 2022

In Machine learning, we usually divide the entire data into training data and test data, train the model using training data, and use test data to determine the accuracy and generalization performance of the model. In the case of models with low generalization performance, the prediction accuracy of newly data is significantly reduced, and the model is said to be overfit. This study is about a method of generating training data based on central limit theorem and combining it with existed training data to increase normality and using this data to train models and increase generalization performance. To this, data were generated using sample mean and standard deviation for each feature of the data by utilizing the characteristic of central limit theorem, and new training data was constructed by combining them with existed training data. To determine the degree of increase in normality, the Kolmogorov-Smirnov normality test was conducted, and it was confirmed that the new training data showed increased normality compared to the existed data. Generalization performance was measured through differences in prediction accuracy for training data and test data. As a result of measuring the degree of increase in generalization performance by applying this to K-Nearest Neighbors (KNN), Logistic Regression, and Linear Discriminant Analysis (LDA), it was confirmed that generalization performance was improved for KNN, a non-parametric technique, and LDA, which assumes normality between model building.

• Communications for Statistical Applications and Methods
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• v.18 no.4
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• pp.477-483
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• 2011

This article is concerned with a class of threshold-asymmetric GARCH models both for stationary case and for non-stationary case. We investigate large sample properties of estimators from QML(quasi-maximum likelihood) and QL(quasilikelihood) methods. Asymptotic distributions are derived and it is interesting to note for non-stationary case that both QML and QL give asymptotic normal distributions.

• Proceedings of the Korean Society for Quality Management Conference
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• 1998.11a
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• pp.159-173
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• 1998

Quality characteristics on the properties of process capability indices (PCIs) are often required to be normally distributed. But, if a characteristic is not normally distributed, serious errors can result from normal-based techniques. In this case, we may well consider the use of new PCIs specially designed to be robust for non-normality. In this paper, a newly proposed measure of process capability is introduced and compared with existing PCIs using the simulated non-normal data.

• 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 Korean Society of Industrial and Systems Engineering
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• v.42 no.2
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• pp.18-27
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• 2019

The process control methods based on the statistical analysis apply the analysis method or mathematical model under the assumption that the process characteristic is normally distributed. However, the distribution of data collected by the automatic measurement system in real time is often not followed by normal distribution. As the statistical analysis tools, the process capability index (PCI) has been used a lot as a measure of process capability analysis in the production site. However, PCI has been usually used without checking the normality test for the process data. Even though the normality assumption is violated, if the analysis method under the assumption of the normal distribution is performed, this will be an incorrect result and take a wrong action. When the normality assumption is violated, we can transform the non-normal data into the normal data by using an appropriate normal transformation method. There are various methods of the normal transformation. In this paper, we consider the Box-Cox transformation among them. Hence, the purpose of the study is to expand the analysis method for the multivariate process capability index using Box-Cox transformation. This study proposes the multivariate process capability index to be able to use according to both methodologies whether data is normally distributed or not. Through the computational examples, we compare and discuss the multivariate process capability index between before and after Box-Cox transformation when the process data is not normally distributed.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.379-387
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• 2001

In this paper, we study the value distribution of meromorphic functions and prove the following theorem: Let f(z) be a transcendental meromorphic function. If f and f'have the same zeros, then f'(z) takes any non-zero value b infinitely many times.

• Communications for Statistical Applications and Methods
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• v.8 no.3
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• pp.797-803
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• 2001

This paper obtained Bayes prediction density for the spatial linear model with non-informative prior. It showed the results that predictive inferences is completely unaffected by departures from the normality assumption in the direction of the elliptical family and the structure of prediction density is unchanged by more than one additional future observations.