• Water Engineering Research
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• v.3 no.3
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• pp.195-202
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• 2002

The mean is generally used as a point estimator in water-quality data. Unfortunately, the nonnormal and skewed distributions of data hinder the direct application of the mean, which is inappropriate statistics in this case. The use of robust statistics such as L, M, and R-estimators are recommended and become more efficient. The median (L-estimator), the biweight (M-estimator), and the Hodges-Lehmann method (R-estimator) are briefly introduced and applied in this paper. From the actual data analyses, it is known that the median does not guarantee robustness for a small number of data sets, and robust measures of location or the arithmetic mean without outliers are highly recommended if the distribution has tails or outliers. Care must be taken to measure the location because water quality level within a water body can change depending on the selected point estimator.

• Journal of the Korean Data and Information Science Society
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• v.24 no.4
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• pp.825-833
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• 2013

An estimation of confidence intervals is essential to calculate uncertainty for greenhouse gases inventory. It is generally assumed that the population has a normal distribution for the confidence interval of parameters. However, in case data distribution is asymmetric, like nonnormal distribution or positively skewness distribution, the traditional estimation method of confidence intervals is not adequate. This study compares two estimation methods of confidence interval; parametric and non-parametric method for exponential distribution as an asymmetric distribution. In simulation study, coverage probability, confidence interval length, and relative bias for the evaluation of the computed confidence intervals. As a result, the chi-square method and the standardized t-bootstrap method are better methods in parametric methods and non-parametric methods respectively.

• Communications for Statistical Applications and Methods
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• v.19 no.5
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• pp.685-696
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• 2012

These days the interest of quality leads to the necessity of control charts for monitoring the process in various fields of practical applications. The $\overline{X}$ chart is one of the most widely used tools for quality control that also performs well under the normality of quality characteristics. However, quality characteristics tend to have nonnormal properties in real applications. Numerous recent studies have tried to find and explore the performance of $\overline{X}$ chart due to non-normality; however previous studies numerically examined the effects of non-normality and did not provide any theoretical justification. Moreover, numerical studies are restricted to specific type of distributions such as Burr or gamma distribution that are known to be flexible but can hardly replace other general distributions. In this paper, we approximate the false alarm rate(FAR) of the $\overline{X}$ chart using the Edgeworth expansion up to 1/n-order with the fourth cumulant. This allows us to examine the theoretical effects of nonnormality, as measured by the skewness and kurtosis, on $\overline{X}$ chart. In addition, we investigate the effect of skewness and kurtosis on $\overline{X}$ chart in numerical studies. We use a skewed-normal distribution with a skew parameter to comprehensively investigate the effect of skewness.

• The Korean Journal of Applied Statistics
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• v.29 no.2
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• pp.301-308
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• 2016

The process capability index is used to determine whether a production process is capable of producing items within a specified tolerance. Some process capability indices $C_p$, $C_{pk}$ and $C_{pm}$ have been of particular interest as useful management tools for tracking process performance. Most evaluations on process capability indices focus on statistical estimation and test of hypothesis. It is necessary to investigate their asymptotic correlationship among basic estimators ${\hat{C}}_p$, ${\hat{C}}_{pk}$ and ${\hat{C}}_{pm}$ of process capability indices $C_p$, $C_{pk}$ and $C_{pm}$. In this paper, we study their asymptotic correlationship for three process capability indices ${\hat{C}}_p$, ${\hat{C}}_{pk}$ and ${\hat{C}}_{pm}$ under bivariate normal distribution BN(${\mu}_x,{\mu}_y,{\sigma}^2_x,{\sigma}^2_y,{\rho}$). With some nonnormal processes, the asymptotic correlation coefficient of any two respective process capability index estimators could be established.

• 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.

• Geotechnical Engineering
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• v.2 no.1
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• pp.27-44
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• 1986

The measured soil data are analyzed to the descriptive statistics and classified into the four models of uncorrelated-normal (UNNO), uncorrelated-nonnormal (VNNN), correlatedonnormal(CONN), and correlated-nonnormal(CONN) . This paper presents the comparisons of reliability index and check points using the advanced first-order second-moment method with respect to the four models as well as BASIC Program. A sin91e-mode Performance function is consisted of the basic design variables of bearing capacity and settlements on shallow foundations and input the above analyzed soil informations. The main conclusions obtained in this study are summarized as follows: 1. In the bearing capacity mode, cohesion and bearing-capacity factors by C-U test are accepted for normal and lognormal distribution, respectively, and negatively low correlated to each other. Since the reliability index of the CONN model is the lowest one of the four model, which could be recommended a reliability.based design, whereas the other model might overestimate the geotechnical conditions. 2. In the case of settlements mode, the virgin compression ratio and preccnsolidation pressure are fitted for normal and lognormal distribution, respectively. Constraining settlements to the lower ones computed by deterministic method, The CONN model is the lowest reliability of the four models.

• Communications for Statistical Applications and Methods
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• v.15 no.3
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• pp.451-464
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• 2008

Six sigma is the rating that signifies "best in clas", with only 3.4 defects per million units or operations. Higher sigma quality level is generally perceived by customers as improved performance by assigning a correspondingly higher satisfaction score. The process capability indices and the sigma level $Z_{st}$ have been widely used in six sigma industries to assess process performance. Most evaluations on process capability indices focus on point estimates, which may result in unreliable assessments of process performance. In this paper, we consider statistical inference for process capability indices $C_p$, $C_{pk}$ and $C_{pm}$. Also, we study better testing procedure on assessing sigma level $Z_{st}$ and capability index $C_{pm}$, for practitioners to use in determining whether a given process is capable. The proposed method is easy to use and the decision making is more reliable. Whether a process is clearly normal or nonnormal, our bootstrap testing procedure could be applied effectively without the complexity of calculation. A numerical result based on our proposed method is illustrated.

• Communications for Statistical Applications and Methods
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• v.17 no.3
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• pp.459-471
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• 2010

Higher quality level is generally perceived by customers as improved performance by assigning a correspondingly higher satisfaction score. Usually, the quality level is measured by process capability indices. The index is used to determine whether a production process is capable of producing items within a specified tolerance. The third generation index $C_{pmk}$ is more powerful than two useful indices $C_p$ and $C_{pk}$. which have been widely used in six sigma industries to assess process performance. Most evaluations on process capability indices focus on point estimates, which may result in unreliable assessments of process performance. In this paper, we consider better testing procedure on assessing process capability index $C_{pmk}$ for practitioners to use in determining whether a given process is capable. It is easy to use the proposed method for assessing process capability index $C_{pmk}$. Whether a process is clearly normal or nonnormal, our bootstrap testing procedure could be applied effectively without the complexity of calculation. A numerical result based on our proposed method is illustrated.