• Journal of the Korean Data and Information Science Society
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• v.10 no.1
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• pp.11-18
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• 1999

To estimate the mean and the variance of a lognormal distribution, Finney (1941) derived the uniformly minimun variance unbiased estimators(UMVUE) in the form of infinite series. However, the conditions ${\sigma}^{2}\;>\;n\;and\;{\sigma}^{2}\;<\;\frac{n}{4}$ for computing $E(\hat{\theta}_{AM})\;and\;E(\hat{\eta}^{2}_{AM})$ are necessary. In this paper, we give an alternative derivation of the UMVUE's.

• Communications for Statistical Applications and Methods
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• v.28 no.4
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• pp.351-368
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• 2021

For a probabilistic model with positively skewed data, a lognormal distribution is one of the key distributions that play a critical role. Several lognormal models can be found in various areas, such as medical science, engineering, and finance. In this paper, we propose a new estimator for a lognormal mean and depict the performance of the proposed estimator in terms of the relative mean squared error (RMSE) compared with Shen's estimator (Shen et al., 2006), which is considered the best estimator among the existing methods. The proposed estimator includes a tuning parameter. By finding the optimal value of the tuning parameter, we can improve the average performance of the proposed estimator over the typical range of σ². The bias reduction of the proposed estimator tends to exceed the increased variance, and it results in a smaller RMSE than Shen's estimator. A numerical study reveals that the proposed estimator has performance comparable with Shen's estimator when σ² is small and exhibits a meaningful decrease in the RMSE under moderate and large σ² values.

• Communications for Statistical Applications and Methods
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• v.18 no.5
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• pp.625-636
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• 2011

This paper proposes two parametric entropy estimators, the minimum variance unbiased estimator and the maximum likelihood estimator, for the lognormal distribution for a comparison of the properties of the two estimators. The variances of both estimators are derived. The influence of the bias of the maximum likelihood estimator on estimation is analytically revealed. The distributions of the proposed estimators obtained by the delta approximation method are also presented. Performance comparisons are made with the two estimators. The following observations are made from the results. The MSE efficacy of the minimum variance unbiased estimator appears consistently high and increases rapidly as the sample size and variance, n and ${\sigma}^2$, become simultaneously small. To conclude, the minimum variance unbiased estimator outperforms the maximum likelihood estimator.

• Journal of the Korean Data and Information Science Society
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• v.16 no.4
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• pp.1119-1128
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• 2005

The Bayes factors with improper noninformative priors are defined only up to arbitrary constants. So it is known that Bayes factors are not well defined due to this arbitrariness in Bayesian hypothesis testing and model selections. The intrinsic Bayes factor and the fractional Bayes factor have been used to overcome this problem. In this paper, we suggest a Bayesian hypothesis testing based on the intrinsic Bayes factor and the fractional Bayes factor for the comparison of two lognormal variances. Using the proposed two Bayes factors, we demonstrate our results with some examples.

• Journal of Applied Reliability
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• v.18 no.1
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• pp.80-86
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• 2018

Purpose: This article provides a mathematical model for the accelerated degradation test when the performance degradation characteristic follows the lognormal distribution. Method: For developing test plans, the total number of test units and the test time are determined based on the minimization of the asymptotic variance of the q-th quantile of the lifetime distribution at the use condition. Results: The mathematical model for the accelerated degradation test is provided. Conclusion: Accelerated degradation test method is widely used to evaluate the product lifetime within a resonable amount of cost and time. In this article. a mathematical model for the accelerated degradation test method is newly developed for this purposes.

• Journal of the Korea Safety Management & Science
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• v.9 no.2
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• pp.123-134
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• 2007

This paper focuses on the in-house reliability assurance plan for the bulk materials of each company. The reliability assurance needs in essence a long time and high cost for testing the materials. In order to reduce the time and cost, accelerated life test is adopted. The bulk sampling technique was used for acceptance. Design parameters might be total sample size(segments and increments}, stress level and so on. We focus on deciding the sample size by minimizing the asymptotic variance of test statistics as well as satisfying the consumer's risk. In bulk sampling, we also induce the sample size by adapting the normal life time distribution model when the variable of the lognormal life time distribution is transformed and adapted to the model. In addition, the sample size for both the segments and increments can be induced by minimizing the asymptotic variance of test statistics of the segments and increments with consumer's risk met. We can assure the reliability of the mean life and B100p life time of the bulk materials by using the calculated minimum sample size.

• Communications for Statistical Applications and Methods
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• v.14 no.3
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• pp.633-640
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• 2007

We develop noninformative priors for the ratio of the lognormal means in equal variances case. The Jeffreys' prior and reference priors are derived. We find a first order matching prior and a second order matching prior. It turns out that Jeffreys' prior and all of the reference priors are first order matching priors and in particular, one-at-a-time reference prior is a second order matching prior. One-at-a-time reference prior meets very well the target coverage probabilities. We consider the bioequivalence problem. We calculate the posterior probabilities of the hypotheses and Bayes factors under Jeffreys' prior, reference prior and matching prior using a real-life example.

• The Korean Journal of Applied Statistics
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• v.23 no.2
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• pp.325-333
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• 2010

Protection against disclosure of survey respondents' identifiable and/or sensitive information is a prerequisite for statistical agencies that release microdata files from their sample surveys. Coarsening is one of popular methods for protecting the confidentiality of the data. Grouped data can be released in the form of microdata or tabular data. Instead of releasing the data in a tabular form only, having microdata available to the public with interval codes with their representative values greatly enhances the utility of the data. It allows the researchers to compute covariance between the variables and build statistical models or to run a variety of statistical tests on the data. It may be conjectured that the variance of the interval data is lower that of the ungrouped data in the sense that the coarsened data do not have the within interval variance. This conjecture will be investigated using the uniform and triangular distributions. Traditionally, midpoint is used to represent all the values in an interval. This approach implicitly assumes that the data is uniformly distributed within each interval. However, this assumption may not hold, especially in the last interval of the economic data. In this paper, we will use three distributional assumptions - uniform, Pareto and lognormal distribution - in the last interval and use either midpoint or median for other intervals for wage and food costs of the Statistics Korea's 2006 Household Income and Expenditure Survey(HIES) data and compare these approaches in terms of the first two moments.

• Journal of the Korean Data and Information Science Society
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• v.7 no.1
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• pp.47-57
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• 1996

This paper on planning constant accelerated life test is assumed that parameters for a lognormal life distribution are depended on changes of stresses. The proposed test plans are optimum in that they minimize the asymptotic variance of maximum likelihood estimator of a specified quantile at the design stress. The optimal amount of low stress level ${\xi}_{L}$ and optimal sample proportion ${\pi}$ to be allocated at low stress level are obtained when the ratio of scales at high stress level and design stress level is unknown.

• Journal of Korean Institute of Industrial Engineers
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• v.23 no.1
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• pp.177-196
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• 1997

In this paper, statistically optimal accelerated life test(ALT) plans considering statistical efficiency only and new compromise ALT plans to sacrifice some statistical efficiency in return for improved overall properties including estimobility probability and robustness for the model assumptions are developed under the assumptions of constant stress, intermittent inspection, Type I censoring and lognormal failure distribution which has been one of the popular choices of failure distributions in the extensive engineering applications of ALT. Computational experiments are conducted to compare with four ALT plans including two proposed ones under continuous and intermittent inspections over a range of parameter values in terms of asymptotic variance, sensitivities for guessed input values, and proportion of estimable samples, etc. The small and moderate sample properties for the proposed ALT plans designed under asymptotic criterion are also investigated by Monte Carlo simulation.