A View on the Validity of Central Limit Theorem: An Empirical Study Using Random Samples from Uniform Distribution

Lee, Chanmi,Kim, Seungah,Jeong, Jaesik The Korean Statistical Society 2014 Communications for statistical applications and me Vol.21 No.6

We derive the exact distribution of summation for random samples from uniform distribution and then compare the exact distribution with the approximated normal distribution obtained by the central limit theorem. To check the similarity between two distributions, we consider five existing normality tests based on the difference between the target normal distribution and empirical distribution: Anderson-Darling test, Kolmogorov-Smirnov test, Cramer-von Mises test, Shapiro-Wilk test and Shaprio-Francia test. For the purpose of comparison, those normality tests are applied to the simulated data. It can sometimes be difficult to derive an exact distribution. Thus, we try two different transformations to find out which transform is easier to get the exact distribution in terms of calculation complexity. We compare two transformations and comment on the advantages and disadvantages for each transformation.

Convergence of statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions and their generalizations

V.Yu. Korolev,A.I. Zeifman 한국통계학회 2017 Journal of the Korean Statistical Society Vol.46 No.2

We present some mixture representations for the Linnik, Mittag-Leffler and Weibull distributions in terms of normal, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the scheme of geometric summation is far not the only asymptotic setting (even for sums of independent random variables) in which the Mittag-Leffler and Linnik laws appear as limit distributions. The two-sided Mittag-Leffler and the one-sided Linnik distribution are introduced and also proved to be limit laws for some statistics constructed from samples with random sizes.

On-line Generation of Three-Dimensional Core Power Distribution Using Incore Detector Signals to Monitor Safety Limits

Jang, Jin-Wook,Lee, Ki-Bog,Na, Man-Gyun,Lee, Yoon-Joon Korean Nuclear Society 2004 Nuclear Engineering and Technology Vol.36 No.6

It is essential in commercial reactors that the safety limits imposed on the fuel pellets and fuel clad barriers, such as the linear power density (LPD) and the departure from nucleate boiling ratio (DNBR), are not violated during reactor operations. In order to accurately monitor the safety limits of current reactor states, a detailed three-dimensional (3D) core power distribution should be estimated from the in-core detector signals. In this paper, we propose a calculation methodology for detailed 3D core power distribution, using in-core detector signals and core monitoring constants such as the 3D Coupling Coefficients (3DCC), node power fraction, and pin-to-node factors. Also, the calculation method for several core safety parameters is introduced. The core monitoring constants for the real core state are promptly provided by the core design code and on-line MASTER (Multi-purpose Analyzer for Static and Transient Effects of Reactors), coupled with the core monitoring program. through the plant computer, core state variables, which include reactor thermal power, control rod bank position, boron concentration, inlet moderator temperature, and flow rate, are supplied as input data for MASTER. MASTER performs the core calculation based on the neutron balance equation and generates several core monitoring constants corresponding to the real core state in addition to the expected core power distribution. The accuracy of the developed method is verified through a comparison with the current CECOR method. Because in all the verification calculation cases the proposed method shows a more conservative value than the best estimated value and a less conservative one than the current CECOR and COLSS methods, it is also confirmed that this method secures a greater operating margin through the simulation of the YGN-3 Cycle-1 core from the viewpoint of the power peaking factor for the LPD and the pseudo hot pin axial power distribution for the DNBR calculation.

ON THE WEAK LIMIT THEOREMS FOR GEOMETRIC SUMMATIONS OF INDEPENDENT RANDOM VARIABLES TOGETHER WITH CONVERGENCE RATES TO ASYMMETRIC LAPLACE DISTRIBUTIONS

Hung, Tran Loc Korean Mathematical Society 2021 대한수학회보 Vol.58 No.6

The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.

수생태 독성자료의 정규성 분포 특성 확인을 통해 통계분석 시 분포 특성 적용에 대한 타당성 확인 연구

옥승엽,문효방,나진성 한국환경보건학회 2019 한국환경보건학회지 Vol.45 No.2

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 datasetMethods: 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 dataResults: 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.

A Modification to HL-RF Method for Computation of Structural Reliability Index in Problems with Skew-distributed Variables

Mohsen A. Shayanfar,Mohammad A. Barkhordari,Mohammad A. Roudak 대한토목학회 2018 KSCE JOURNAL OF CIVIL ENGINEERING Vol.22 No.8

The Hasofer-Lind and Rackwitz-Fiessler (HL-RF) method in reliability analysis is a popular iterative method for obtaining thereliability index. However, in the cases of limit state functions with skew-distributed variables, HL-RF method may giveinappropriate answers. This paper represents a modification to HL-RF method in order to improve its performance in such problems. Based on this modification, non-normal distributions are replaced with equivalent skew-normal distributions instead of equivalentnormal distributions. By this modification, asymmetric non-normal distributions are not replaced with symmetric distributionsanymore. It is demonstrated that this consideration of skewness of non-normal distributions improves the behavior of HL-RF methodand makes the proposed method more reliable. This improvement is shown through illustrative examples.

ON LIMIT BEHAVIOURS FOR FELLER'S UNFAIR-FAIR-GAME AND ITS RELATED MODEL

An, Jun Korean Mathematical Society 2022 대한수학회지 Vol.59 No.6

Feller introduced an unfair-fair-game in his famous book [3]. In this game, at each trial, player will win 2^k yuan with probability p_k = 1/2^kk(k + 1), k ∈ ℕ, and zero yuan with probability p₀ = 1 - Σ^∞_k=1 p_k. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed "fair", Feller [2] proved that when the total trial number n is large enough, player will loss n yuan with its probability approximate 1. So it's an "unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take 2^k = m to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.

Spatial-temporal distribution of carabid beetles in wetlands

Do, Yu-No,Jo, Hyun-Bin,Kang, Ji-Hoon,Joo, Gea-Jae The Ecological Society of Korea 2012 Journal of Ecology and Environment Vol.35 No.1

In this study, we investigated carabid beetles residing in the wetlands to understand their ecological adaptation and strategy selection associated with restricted resources and habitat limitation. The species richness, abundance, seasonal activity, and spatial distribution of the carabid beetles between the Mujechi Wetlands (wetland sites) and Mt. Jeongjok (mountain sites) have been compared. A total of 1,733 individual beetles from 30 species were collected and classified at the studied sites. The wetland sites were identified as having lower species richness and abundance for carabid beetles when compared with the adjacent mountain sites, whereas these beetles were observed to be dominant in the wetland sites than in the adjacent mountain sites. Calosoma inquisitor cyanescens, Carabus sternbergi sternbergi, and Carabus jankowskii jankowskii species were dominant in both the wetland and mountain sites. These species showed significantly different seasonal activity patterns in the wetland sites relative to the mountain sites. Although the three listed carabid species were observed to be widely distributed throughout the wetland sites, they still showed preference for drier sites, which clearly shows a distinction in their habitats. The results of the spatial-temporal distribution of carabid beetles in the wetland sites reflect their special strategies regarding space and time partitioning for maintaining their population. The distribution patterns of carabid beetles in the wetland sites also showed the desiccation gradient and environmental changes prevalent in wetlands. Ecological surveys, which use carabid beetles in the wetlands, can then be performed when restoring wetlands and for establishing management practices for improving the habitat quality.

Price Limit Expansion and Volatility: A Theoretical Perspective

유진,이정환,Xin Su 한국증권학회 2021 Asia-Pacific Journal of Financial Studies Vol.50 No.3

We theoretically examine whether and how price limit expansion changes return volatility. This study incorporates competing hypotheses regarding investor reactions to limit-hit events into a model that considers trader irrationality; we then conduct several simulations. We find that, when price limits are widened, stock return volatility tends to increase but may also remain unchanged or decrease. We consider the implications of the study’s main findings, which shed light on the mixed empirical results found in the price limit literature so far.

Asymptotically Distribution-Free Procedure in a Two-Way Layout

Park, Young-Hun The Korean Statistical Society 1995 Journal of the Korean Statistical Society Vol.24 No.2

Main purpose of this article is to consider the asymptotic distribution of the rank transformed F statistic for interaction in a two-way layout. Some theorems and sufficient conditions are derived to have the rank transformed F statistic converged in distribution to a chi-squared random variable with (I-1)(J-1) degrees of freedom divided by (I-1)(J-1). These results will be useful for the other theoretical studies of the rank transform procedure in experimental designs.