MCEM을 이용한 정규혼합분포 추정

이승찬,이재준,전수영 한국자료분석학회 2017 Journal of the Korean Data Analysis Society Vol.19 No.1

정규혼합분포는 정규분포의 혼합으로 이루어져있으며 위험관리 분야에서 많이 이용되고 있는 극단값분포에 대한 대안으로 활용할 수 있다. 본 연구는 KOSPI 200 수익률의 정규혼합분포를 추정하기 위해 MCEM(Monte Carlo EM) 알고리즘을 적용하고자 한다. MCEM은 각 분포의 성분을 결측치로 보고, 결측치를 메트로폴리스-헤스팅스 알고리즘을 통해 생성하고 EM 알고리즘을 통해 각 분포의 모수를 결정한다. MCEM 알고리즘의 효율성은 여러 성분을 이용한 모의실험과 실증 자료를 통해 알 수 있었다. 본 연구의 실증 자료 표본은 2008년 외환위기 1년 후인 2009년 10월 15일부터 2017년 1월 7일 동안의 일별 수익률 자료이다. MCEM을 통해 분포의 추정을 실시한 결과 대다수의 분포는 2개의 성분을 갖고 있는 정규혼합분포로 나타났다. 실 자료 중 1000개의 표본을 1일씩 연속적으로 이동하여 1017개 표본기간들의 정규혼합분포를 추정하였고, VaR의 추정을 실시한 결과 하나의 정규분포보다 정규혼합분포를 통해 VaR를 더 잘 추정함을 알 수 있었다. 따라서 본 연구 결과는 자료의 특성에 따른 정규혼합분포를 통하여 VaR를 구하는 것이 다중최빈값을 갖는 경우에서 더 활용성이 높으며 충분히 리스크 측정도구로서 사용될 수 있음을 보였다. The normal mixture distribution is a mixture of normal distributions and can be used as an alternative to extreme value distributions which are widely used in risk management. This study proposes a Monte Carlo EM (MCEM) algorithm to estimate the normal mixed distribution of KOSPI 200 returns. MCEM considers the components of each distribution as missing values, generating through a Metropolis-Hastings algorithm, and determines the parameters of each distribution through EM. The effectiveness of MCEM was demonstrated by simulation and empirical data. The empirical data of this study are daily return data from October 15, 2009 to January 7, 2017, one year after the 2008 financial crisis. The result through MCEM indicates that the data follow a normal mixed distribution with two components. We estimated a normal mixed distribution of 1017 sample periods using 1000 samples of real data which are continuously moved by one day. In addition, the result of VaR estimation indicates that a normal mixed distribution is better than one normal distribution. Therefore, obtaining VaR through the normal mixed distribution is more usable in the case of multiple modes and can be used sufficiently as a risk measurement tool.

A Study of Mann-Whitney Test and Median Test in Several Distributions

박찬근,신성민,허태영 한국자료분석학회 2008 Journal of the Korean Data Analysis Society Vol.10 No.2

The Mann-Whitney test and the Median test are two tests that can be used to test for a difference in location parameters. This paper compared powers of the two tests under a variety of population distributions through a simulation study. Both tests require that two underlying populations have the same variance, but this assumption was relaxed in some of the comparisons. In every cases, equal sizes of 10 and 20 were used. We compared the those two nonparametric tests when underlying distributions were mixed normal, lognormal, beta, gamma and chi-square distributions. Each test was performed 5,000 times and the SAS/MACRO used.

Variable Sampling Inspection with Screening When Lot Quality Follows Mixed Normal Distribution

Suzuki, Yuichiro,Takemoto, Yasuhiko,Arizono, Ikuo Korean Institute of Industrial Engineers 2009 Industrial Engineeering & Management Systems Vol.8 No.3

The variable sampling inspection scheme with screening for the purpose of assuring the upper limit of maximum expected surplus loss after inspection has been proposed. In this inspection scheme, it has been assumed that a product lot consists of products manufactured through a single production line and lot quality characteristics follow a normal distribution. In the previous literature with respect to inspection schemes, it has been commonly assumed that lot quality characteristics obey a single normal distribution under the condition that all products are manufactured in the same condition. On the other hand, the production line is designed in order that the workload of respective processes becomes uniform from the viewpoint of line balancing. One of the solutions for the bottleneck process is to arrange the workshops in parallel. The lot quality characteristics from such a production line with the process consisting of some parallel workshops might not follow strictly the single normal distribution. Therefore, we expand an applicable scope of the above mentioned variable sampling inspection scheme with screening in this article. Concretely, we consider the variable sampling inspection with screening for the purpose of assuring the upper limit of average outgoing surplus quality loss in the production lots when the lot quality follows the mixed normal distribution.

Variable Sampling Inspection with Screening When Lot Quality Follows Mixed Normal Distribution

Yuichiro Suzuki,Yasuhiko Takemoto,Ikuo Arizono 대한산업공학회 2009 Industrial Engineeering & Management Systems Vol.8 No.3

The variable sampling inspection scheme with screening for the purpose of assuring the upper limit of maximum expected surplus loss after inspection has been proposed. In this inspection scheme, it has been assumed that a product lot consists of products manufactured through a single production line and lot quality characteristics follow a normal distribution. In the previous literature with respect to inspection schemes, it has been commonly assumed that lot quality characteristics obey a single normal distribution under the condition that all products are manufactured in the same condition. On the other hand, the production line is designed in order that the workload of respective processes becomes uniform from the viewpoint of line balancing. One of the solutions for the bottleneck process is to arrange the workshops in parallel. The lot quality characteristics from such a production line with the process consisting of some parallel workshops might not follow strictly the single normal distribution. Therefore, we expand an applicable scope of the above mentioned variable sampling inspection scheme with screening in this article. Concretely, we consider the variable sampling inspection with screening for the purpose of assuring the upper limit of average outgoing surplus quality loss in the production lots when the lot quality follows the mixed normal distribution.

A mixed repetitive sampling plan based on process capability index

Aslam, M.,Azam, M.,Jun, C.H. Elsevier Science Ltd 2013 Applied mathematical modelling Vol.37 No.24

A repetitive mixed sampling plan based on the process capability index is proposed in this manuscript. The proposed plan is applicable to the inspection of the product whose lifetime follows the normal distribution. The plan parameters of the proposed mixed plan are determined by satisfying the given producer's risk and consumer's risk at the same time for the specified acceptable quality level and limiting quality level. Symmetric and asymmetric cases in quality levels beyond specification limits are discussed and tables are presented for both cases. The comparison of the proposed mixed plan is made with the attributes repetitive plan and variables repetitive plan. Real example is given to explain the proposed plan.

Use of mixed bivariate distributions for deriving inter-station correlation coefficients of rain rate

Ha, Eunho,Yoo, Chulsang Wiley 2007 Hydrological processes Vol.21 No.22

<P>Even though rain rate is notorious for its spatial and temporal intermittency, its effect on the second-order statistics of rain rate, especially the inter-station correlation coefficients, has not been intensively evaluated before. This study has derived and compared the inter-station correlation coefficient of rain rate for three cases of data: (1) only the positive measurements at both locations; (2) the positive measurements at either one or both locations; (3) all the measurements including zero measurement at both locations. For these three cases, the inter-station correlation coefficients are analytically derived by applying the mixed bivariate log-normal distribution. As an application example, the model parameters are estimated using the rain rate data collected at the Geum River basin, Korea, and the resulting inter-station correlation coefficients are evaluated and compared with those estimated by applying the Gaussian distribution. We could find that highly biased inter-station correlation coefficients are unavoidable when simply estimating them under the assumption of Gaussian distribution, or even when using the log-transformed rain rate data. Copyright © 2007 John Wiley & Sons, Ltd.</P>

Skewed factor models using selection mechanisms

Kim, H.M.,Maadooliat, M.,Arellano-Valle, R.B.,Genton, M.G. Academic Press 2016 Journal of multivariate analysis Vol.145 No.-

<P>Traditional factor models explicitly or implicitly assume that the factors follow a multivariate normal distribution; that is, only moments up to order two are involved. However, it may happen in real data problems that the first two moments cannot explain the factors. Based on this motivation, here we devise three new skewed factor models, the skew-normal, the skew-t, and the generalized skew-normal factor models depending on a selection mechanism on the factors. The ECME algorithms are adopted to estimate related parameters for statistical inference. Monte Carlo simulations validate our new models and we demonstrate the need for skewed factor models using the classic open/closed book exam scores dataset. (C) 2015 Elsevier Inc. All rights reserved.</P>

Joint Modeling with Bivariate Random Effects

Ha, Il-Do 경산대학교 기초과학연구소 2001 基礎科學 Vol.5 No.1

최근에, 맡은 장기적 임상시험은 각 개체에 대한 연관된 반응인 반복측정과 생존시간의 자료를 수집한다. 한 예는 환자의 신장이식에 관한 소실시간을 갖는 추적 임상 시험이다. 본 논문에서는 이러한 연관된 자료를 분석하기 위한 결합 변량효과모형을 제안하여, 새로운 다단계우도 추론법을 개발한다. Recently, many long-term clinical trials collect both a vector of repeated measures and a survival time on each subject, which the two responses are dependent. One example s longitudinal trial on patient's kidney graft, with graft failure time. We develop and inferential method of joint random effects model for these correlated data, using hierarchical-likelihood (h-likelihood). Keywords : Bivariate normal distribution; Frailty model; H-likelihood; Mixed lingear model; Random effects.

A linear mixed model for analyzing longitudinal skew-normal responses with random dropout

M. Ganjali,T. Baghfalaki,M. Khazaei 한국통계학회 2013 Journal of the Korean Statistical Society Vol.42 No.2

In this paper, a linear mixed effects model is used to fit skewed longitudinal data in the presence of dropout. Two distributional assumptions are considered to produce background for heavy tailed models. One is the linear mixed model with skew-normal random effects and normal errors and the other one is the linear mixed model with skewnormal errors and normal random effects. An ECM algorithm is developed to obtain the parameter estimates. Also an empirical Bayes approach is used for estimating random effects. A simulation study is implemented to investigate the performance of the presented algorithm. Results of an application are also reported where standard errors of estimates are calculated using the Bootstrap approach.

혼합정규분포를 가정한 조건부 상호정보의 준모수적 추정량을 이용한 고차원 자료에서의 변수선택

안치경(Chikyung Ahn),김동욱(Donguk Kim) 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.6

변수간의 비선형적인 연관성을 감지할 수 있는 상호정보 (mutual information)는 변수선택의 좋은 기준이 되지만 고차원 자료에서는 적용하기 쉽지 않아 많은 연구가 진행되어 왔다. Cai 등 (2009)은 일반적인 상호정보가 아닌 최대 2차원까지만 고려하여 추정하는 조건부 상호정보를 이용하여 추정의 어려움을 해결하였으며, 고차원자료에 SVM을 적용하기 위한 변수선택에서 기존의 필터링 방법과 SVM-RFE로 선택된 변수들보다 더 분류 성능이 뛰어난 변수들을 선택하는 것을 보였다. Ahn과 Kim (2014)은 조건부 상호정보의 추정에 대한 계산효율을 높이기 위해 설명변수간에는 모수적으로 분포가정을 하는 준모수적 조건부 상호정보 추정량을 제안하였다. 하지만 설명변수간에 정규분포라는 가정이 심하게 위배되면 분류성능이 매우 저하될 수 있는 단점이 있다. 본 연구에서는 설명변수의 분포를 혼합정규분포로 가정하여 조건부 상호정보를 가중치를 활용하여 준모수적인 방법으로 추정하는 방법을 제시하였다. 반응변수와 설명변수 간에는 모수적 분포를 가정하지 않으므로 비모수적 연관성을 측정하는 상호정보의 특징을 보존하며 설명변수간에는 모수적 분포가정을 하여 추정의 효율을 향상시킬 수 있다. 모의실험결과 혼합정규분포를 가정한 조건부 상호정보의 준모수적 추정법이 유의변수 선택능력에서 매우 우수하였다. We propose a method of estimating conditional mutual information by semiparametric method using mixed normal distribution assumption between explanatory variables. In order to maintain the advantage of mutual information that keeps the nonparametric relationship between the explanatory variable and the response variable, the mutual information between the explanatory variable and the response variable is estimated in a nonparametric manner. Furthermore, to improve the efficiency of mutual information estimation, the mutual information between the explanatory variables is to be estimated parametrically. Since the estimated density function is used as a weight for conditional mutual information estimation, the outliers with relatively small density estimate have little effect on the semi-parametric estimator of conditional mutual information. Experimental results show that the semi-parametric estimation method of conditional mutual information assuming mixed normal distribution have shown excellent performance in terms of significant variable selection.